A Walk Through Weak Hyperstructures: Hv-Structures 9813278862, 9789813278868

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A Walk Through

Weak Hyperstructures Hv-Structures

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b2530   International Strategic Relations and China’s National Security: World at the Crossroads

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A Walk Through

Weak Hyperstructures Hv-Structures Bijan Davvaz Yazd University, Iran

Thomas Vougiouklis

Democritus University of Thrace, Greece

World Scientific NEW JERSEY

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SINGAPORE

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15/11/18 12:09 PM

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Names: Davvaz, Bijan, author. | Vougiouklis, Thomas, author. Title: A walk through weak hyperstructures : Hv-structures / by Bijan Davvaz (Yazd University, Iran), Thomas Vougiouklis (Democritus University of Thrace, Greece). Description: New Jersey : World Scientific, 2019. | Includes bibliographical references and index. Identifiers: LCCN 2018048126 | ISBN 9789813278868 (hardcover : alk. paper) Subjects: LCSH: Hypergroups. | Group theory. | Ordered algebraic structures. Classification: LCC QA174.2 .D3845 2019 | DDC 511.3/3--dc23 LC record available at https://lccn.loc.gov/2018048126

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2019 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

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Preface

Hyperstructures were born as a generalization of an operation by the hyperoperation, from the single-valued operation to the multi-valued one. It was then that the problem of generalizations was transferred into the generalizations of axioms. In 1934 Frederick Marty, who introduced the hyperoperation and gave the definition of the hypergroup, used the ‘double’ axiom of reproductivity instead of the two axioms: the existence of the unit element and the existence of the inverses. This is a revolutionary generalization since the majority of the hyperstructures do not have unit elements. In mathematics, any generalization of a structure should contain the generalized one as a sub-case. In hyperstructures the problem becomes complicated as in the result we replace the elements by sets, in fact, we replace a set by a power set. Therefore, we need new tools to achieve the connection of the hyperstructures with the classical structures. This new tool is the fundamental relation of each new hyperstructure. It is a fact that any fundamental relation is based on the ‘result’. For example, in hypergroups the fundamental relation β ∗ is the transitive closure of the relation β, where two elements are in β-relation if they belong to a hyperproduct of two elements. In the fundamental relation β ∗ , introduced by M. Koskas in 1970, a classical group corresponds to any hypergroup. In other words, any hypergroup hides a group. The largest generalization in order to have this correspondence, the existence of the β ∗ fundamental relation, is the one by using the so called weak axioms. In the weak axioms, defined in all known classical structures as introduced by Vougiouklis in 1990, the ‘equality’ in any relation is replaced by the ‘non-empty intersection’ and this leads to the largest class of hyperstructures called Hv -structures. The main way to prove theorems in this topic is the reduction to absurdity. Since the weak generalization is the most general, the number of Hv -structure is v

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dramatically big. Therefore, many problems in life and in other sciences can be expressed by models using Hv -structures. In order to specify the appropriate Hv -structure in models, one can use more restrictions or axioms to reduce the number of possible cases. In generalizations new concepts appear. Moreover, new axioms, new properties and new classes of hyperstructures, are discovered. Consequently, new classifications are needed and very interesting mathematical problems are revealed. The present book consists of seven chapters. Chapter 1 contains a fairly detailed discussion of the basic ideas underlying the fundamentals of algebraic structures such as semigroups, groups, rings, modules and vector spaces. Chapter 2 gives a brief introduction to algebraic hyperstructures to be used in the next chapters. Many readers, already familiar with these theories, may wish to skip them or to regard them as a survey. In Chapter 3, the concept of Hv -semigroups, Hv -groups and some examples are presented. Fundamental relations on Hv -groups are discussed. Reversible Hv -groups, a sequence of finite Hv -groups, fuzzy Hv -groups and Hv -semigroups as noise problem are studied. In Chapter 4, we present the notion of Hv -rings. Hv -rings are the largest class of algebraic hyperstructures that satisfy ring-like axioms. We consider the fundamental relation γ ∗ defined on Hv -rings and give some properties of this important relation. The fundamental relation on an Hv ring is the smallest equivalence relation such that the quotient would be the (fundamental) ring. Then, we present several kinds of Hv -rings. In particular, we investigate multiplicative Hv -rings, Hv -fields, P -hyperoperations, ∂-hyperoperations, Hv -ring of fractions, Hv -near rings and fuzzy Hv -ideals. Chapter 5 begins with the definition of Hv -module. Then the concepts of Hv -module of fractions, direct system and direct limit of Hv -modules are provided. It is proved that direct limit always exists in the category of Hv -modules. We study M [−] and −[M ] functors and investigate the exactness and some related concepts. Next, we prove Five Short Lemma and Shanuel’s lemma in Hv -modules. At the end of this chapter, the concepts of fuzzy and intuitionistic fuzzy Hv -submodules are presented. In Chapter 6, we cover Hv -vector space, hyperalgebra, e-hyperstructures and Hv -matrix representations. Moreover, we study Lie-Santilli theory. In the quiver of hyperstructures Santilli, in early 90’es, tried to find algebraic structures in order to express his pioneer Lie-Santilli’s Theory. Santilli’s theory on ‘isotopies’ and ‘genotopies’, born in 1960’s, desperately needs ‘units e’ on left or right, which are nowhere singular, symmetric, real-valued, positivedefined for n-dimensional matrices based on the so called isofields. These elements can be found in hyperstructure theory, especially in Hv -structure

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theory introduced. This connection appeared first in 1996 and actually several Hv -fields, the e-hyperfields, can be used as isofields or genofields, in such way that they should cover additional properties and satisfy more restrictions. Several large classes of hyperstructures as the P -hyperfields, can be used in Lie-Santilli’s theory when multivalued problems appeared, either in finite or in infinite case. Chapter 7, which is novel in a book of this kind, illustrates the use of weak hyperstructures. We present examples of weak hyperstructures associated with chain reactions and dismutation reactions. For the first time Davvaz and Dehghan-Nezhad provided examples of hyperstructures associated with chain reactions. Also, we investigate the examples of hyperstructures and weak hyperstructures associated with redox reactions and electrochemical cells. Another motivation for the study of hyperstructures comes from biology, more specifically from Mendel, the father of genetics, who took the first steps in defining “contrasting characters, genotypes in F1 and F2 ... and setting different laws”. The genotypes of F2 are dependent on the type of its parents genotype and it follows certain rules. Also, inheritance issue based on genetic information is examined carefully via a new hyperalgebraic approach. Several examples are provided from different biology points of view, and we show that the theory of hyperstructures exactly fits the inheritance issue. Moreover, we provide a physical example of hyperstructures associated with the elementary particle physics, the leptons. We consider this important group of the elementary particles and show that this set along with the interactions between its members can be described by the algebraic hyperstructures.

Bijan Davvaz Department of Mathematics, Yazd University, Yazd, Iran Thomas Vougiouklis School of Science of Education, Democritus University of Thrace, Alexandroupolis, Greece

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Contents

Preface

v

1.

1

Fundamentals of algebraic structures 1.1 1.2 1.3 1.4

2.

Semigroups and Rings . . . . . Modules . . . . Vector space . .

groups . . . . . . . . . . . . . . .

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Algebraic hyperstructures 2.1 2.2 2.3 2.4

Semihypergroup Hypergroups . . Hyperrings . . . Hypermodules . .

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3. Hv -groups 3.1 3.2 3.3 3.4 3.5 3.6 3.7

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Hv -groups and some examples . . . Enumeration of Hv -groups . . . . . . Fundamental relation on Hv -groups Reversible Hv -groups . . . . . . . . . A sequence of finite Hv -groups . . . Fuzzy Hv -groups . . . . . . . . . . . Hv -semigroups and noise problem . .

4. Hv -rings 4.1 4.2

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Hv -rings and some examples . . . . . . . . . . . . . . . . 115 Fundamental relations on Hv -rings . . . . . . . . . . . . . 123 ix

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4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12

Uniting elements . . . . . . . . . . . . . . Multiplicative Hv -rings . . . . . . . . . . . Hv -fields . . . . . . . . . . . . . . . . . . . Hv -rings endowed with P -hyperoperations ∂-hyperoperations and Hv -rings . . . . . . (H, R)-Hv -rings . . . . . . . . . . . . . . . The Hv -ring of fractions . . . . . . . . . . Hv -group rings . . . . . . . . . . . . . . . Hv -near rings . . . . . . . . . . . . . . . . Fuzzy Hv -ideals . . . . . . . . . . . . . . .

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5. Hv -modules 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 6.

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Hv -modules and fundamental relations . . . . . . . Hv -module of fractions . . . . . . . . . . . . . . . . Direct system and direct limit of Hv -modules . . . M[-] and -[M] Functors . . . . . . . . . . . . . . . . Five short lemma and snake lemma in Hv -modules Shanuel’s lemma in Hv -modules . . . . . . . . . . . Product and direct sum in Hv -modules . . . . . . . Fuzzy and intuitionistic fuzzy Hv -submodules . . .

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Hyperalgebra and Lie-Santilli theory 6.1 6.2 6.3 6.4

Hv -vector space . . . . . . . . . e-hyperstructures . . . . . . . . The Lie-Santilli’s admissibility Hv -matrix representations . . .

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Outline of applications and modeling 7.1

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Chemical examples . . . . . . . . . 7.1.1 Chain reactions . . . . . . 7.1.2 Dismutation reactions . . . 7.1.3 Redox reactions . . . . . . 7.1.4 Galvanic cell . . . . . . . . 7.1.5 Electrolytic cell . . . . . . 7.1.6 Galvanic/Electrolytic cells Biological examples . . . . . . . . . 7.2.1 Inheritance examples . . .

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Contents

7.2.2

7.3

Examples of different types of non-Mendelian inheritance . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Hyperstructures in second generation genotype . . 7.2.4 The hypothetical cross of n different traits, case of simple dominance . . . . . . . . . . . . . . . . . . 7.2.5 The hypothetical cross of n different traits, case of incomplete dominance . . . . . . . . . . . . . . . . 7.2.6 The hypothetical cross of m + n different traits, case of simple and incomplete dominance combined together . . . . . . . . . . . . . . . . . . . . . . . Physical examples . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Leptons . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 The algebraic hyperstructure of Leptons . . . . .

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Bibliography

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Index

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Chapter 1

Fundamentals of algebraic structures

1.1

Semigroups and groups

Let S be a non-empty set and ζ : S × S → S a binary operation that maps each ordered pair (x, y) of S to an element ζ(x, y) of S. The pair (S, ζ) (or just S, if there is no fear of confusion) is called a groupoid. Definition 1.1. A semigroup is a pair (S, ·) in which S is a non-empty set and · is a binary associative operation on S, i.e., the equation (x · y) · z = x · (y · z) holds for all x, y, z ∈ S. For an element x ∈ S we let xn be the product of x with itself n times. So, x1 = x, x2 = x · x and xn+1 = x · xn for n ≥ 1. A semigroup S is finite if it has only a finitely many elements. A semigroup S is commutative, if it satisfies x·y =y·x for all x, y ∈ S. If there exists e in S such that for all x ∈ S, e·x=x·e=x we say that S is a semigroup with identity or (more usual) a monoid. The element e of S is called identity. Proposition 1.1. A semigroup can have at most one identity. Proof.

If e and e0 are both identities, then e = e · e0 = e0 .



The description of the binary operation in a semigroup (S, ·) can be carried out in various ways. The most natural is simply to list all results of 1

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the operation for arbitrary pairs of elements. This method of describing the operation can be presented as a multiplication table, also called a Cayley table. Example 1.1. (1) Let S = {a, b, c} be a set of three elements and define the following table. · a b c

a a b c

b b a b

c c c c

Then, S is a finite semigroup. (2) Let N = {0, 1, . . .} be the set of all non-negative integers and N∗ = {1, 2, . . .} the set of all positive integers. Then, (N, ·) is a semigroup for the usual multiplication of integers. Also, (N, +) is a semigroup, when + is the ordinary addition of integers. Define (N, ?) by n ? m = max{n, m}. Then, (N, ?) is a semigroup, since n ? (m ? k) = max{n, max{m, k}} = max{n, m, k} = max{max{n, m}, k} = (n ? m) ? k. (3) The set Mn (R) of n × n square matrices over real numbers, with matrix multiplication, is a semigroup. (4) The direct product S × T of two semigroups (S, ·) and (T, ◦) is defined by (s1 , t1 ) ? (s2 , t2 ) = (s1 · s2 , t1 ◦ t2 ) (s1 , s2 ∈ S, t1 , t2 ∈ T ). It is easy to show that the so defined product is associative and hence the direct product is, indeed, a semigroup. The direct product is a convenient way of combining two semigroup operations. The new semigroup S × T inherits properties of both S and T . (5) The bicyclic semigroup B = N × N with binary operation (a, b) ∗ (c, d) = (a − b + max{b, c}, d − c + max{b, c}). This is a monoid with identity (0, 0). Proposition 1.2. Let (S, ·) be a finite semigroup. Then, there exists a ∈ S such that a2 = a.

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Proof. Suppose that x is an arbitrary element of S. Since S is finite, it follows that x, x2 , x3 , . . . are distinct, where xn = x · ... · x (n times). So, there exist integers m, n with n < m such that xm = xn . Hence, xn+k = xn , where k = m − n. Now, we have x2n+k = xn · xn+k = x2n . By mathematical induction, we obtain xrn+k = xrn , for all r ∈ N. Also, we have xrn+2k = xrn+k · xk = xrn · xk = xrn+k = xrn , xrn+3k = xrn+2k · xk = xrn · xk = xrn+k = xrn , and so on. Again, by mathematical induction, we obtain xrn+lk = xrn , for all l ∈ N. In particular, we obtain xkn+nk = xkn or x2nk = xkn . Now, we set xnk = a.  Let us consider the set consisting of all the integers {0, ±1, ±2, . . .}. The sum m + n of any two integers m and n is also an integer, and the following two rules of addition hold for any arbitrary integers m, n and p: (1) m + n = n + m, (2) (m + n) + p = m + (n + p). Furthermore, for any two given integers m and n, the equation m+x=n has a unique solution x = n − m, which is also an integer. Similar situations often occur in many different fields of mathematics, and they do not necessarily concern only the integers. Consider, for instance, the set of all nonsingular 2 × 2 matrices (that is, all 2 × 2 matrices A such that the determinant of A is not zero). Let A and B be any 2 × 2 matrices,     a1 a2 b1 b2 A= and B = . a3 a4 b3 b4 Then, the product of A and B is also a 2×2 matrix. This product is defined as   a b + a2 b3 a1 b2 + a2 b4 AB = 1 1 . a3 b1 + a4 b3 a3 b2 + a4 b4

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With respect to the binary operation of the multiplication of matrices, we note that, in general, AB is not necessarily equal to BA, but the associative law (AB)C = A(BC) is valid for any three 2 × 2 matrices A, B and C. If the 2 × 2 matrix A is nonsingular, then the equation AX = B and Y A = B have unique solutions, X = A−1 B and Y = BA−1 , respectively, where X and Y are both 2 × 2 matrices. A given set of elements together with an operation satisfying the associative law is said to be a group or to form a group if any linear equation has a unique solution which is in the set. Thus, the totality of nonsingular 2 × 2 matrices together with multiplication is said to be group, as is the set of all the integers with addition. We will now state the formal definition of a group. Definition 1.2. Let G be a non-empty set together with a binary operation that assigns to each ordered pair (a, b) of elements of G an element a · b in G. We say G is a group under this operation if the following two properties are satisfied: (1) For any three elements a, b and c of G, the associative law holds: (a · b) · c = a · (b · c). (2) For two arbitrary elements a and b, there exist x and y of G which satisfy the equations a · x = b and y · a = b. The following properties of a group are important. Theorem 1.1. (20 ) There is a unique element e in G such that for all g ∈ G, g · e = e · g = g. (200 ) For any element a ∈ G, there is a unique element a0 ∈ G such that a · a0 = a0 · a = e, where e is the element of G defined in (20 ). (3) The solutions x and y of the equations a · x = b and y · a = b are unique and we have x = a0 · b and y = b · a0 , where a0 is the element associated with the element a in (200 ). Proof. Since the set G is not empty, we take an element a of G. By the property (2), there are solutions x = e and y = e0 of the equations a · x = a and y·a = a. Also, if g is an arbitrary element of G, there are elements u and v of G such that a·u = g = v·a; so we have g·e = (v·a)·e = v·(a·e) = v·a = g. The second equality, (v · a) · e = v · (a · e), follows from the associative law. Similarly, we obtain e0 · g = g. Since the element g is arbitrary, we may

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take g = e to obtain e0 · e = e. On the other hand, the element e satisfies g · e = g for any g ∈ G, so e0 · e = e0 . Therefore, we have e0 = e0 · e = e. This proves that an arbitrary solution of the equation a · x = a is equal to a solution of y · a = a. Thus, the uniqueness of the element e is proved, and (20 ) holds. The proof of (200 ) is similar. By (2), there are elements a0 and a00 of G such that a · a0 = e = a00 · a. Using (1) and (20 ), we have a00 = a00 · e = a00 · (a · a0 ) = (a00 · a) · a0 = e · a0 = a0 . Hence, the proof of the uniqueness of a0 is similar to that of the uniqueness of e in (20 ). The proof of (3). If a · x = b, then the left multiplication of a0 of (200 ) gives us a0 · b = a0 · (a · x) = (a0 · a) · x = e · x = x. Thus, a solution of a · x = b is x = a0 · b, and it is unique; similarly, the solution of y · a = b is uniquely determined to be y = b · a0 .  Corollary 1.1. A non-empty set G with an operation is a group if the conditions (1), (20 ) and (200 ) are satisfied. The element e defined in (20 ) is called the identity of G, the element a0 defined in (200 ) is called the inverse of a. The inverse of an element a is customary denoted by a−1 . Theorem 1.2. We have (a−1 )−1 = a and (a · b)−1 = b−1 · a−1 , for all a, b ∈ G. Proof. The first equality follows from (200 )(the uniqueness of the inverse). The second one is proved by the equality (a·b)·(b−1 ·a−1 ) = ((a·b)·b−1 )·a−1 = (a·(b·b−1 ))·a−1 = (a·e)·a−1 = a·a−1 = e and the uniqueness of the inverse. Notice the change in the order of the factors from a · b to b−1 · a−1 .  The product of the elements a1 , a2 , ..., an (n ≥ 3) of G is defined inductively by a1 · ... · an = (a1 · ... · an−1 ) · an . The general associative law holds in any group. That is, if x1 , x2 , ..., xn are n arbitrary elements of a group, then the product of x1 , ..., xn is uniquely determined irrespective of the ways the product is taken, provided that the order of factors is unchanged. For example, (x · y) · ((z · (u · v)) · w) = x · (((y · (z · u)) · v) · w). If a1 = a2 = ... = an , then we use the power notation (for a = a1 ), a1 · a2 · ... · an = an . If n = −m is a negative integer, then we define an = (a−1 )m ; also, we define a0 = e. The formulas am · an = am+n and (am )n = amn hold for any element a of G and any pair of integers m and n.

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We say that two elements a and b of a group G are commutative or commute if a · b = b · a. A group is said to be abelian or commutative, if any two elements commute. The number of elements in a group G is called the order of G and is denoted by |G|. If |G| is finite, then G is said to be a finite group; otherwise G is an infinite group. Before going on to work out some properties of groups, we pause to examine some examples. Motivated by these examples we shall define various special types of groups which are important. Example 1.2. (1) The set of integers Z, the set of rational numbers Q and the set of real numbers R are all groups under ordinary addition. (2) The set Zn = {0, 1, ..., n − 1} for n ≥ 1 is a group under addition modulo n. For any i in Zn , the inverse of i is n − i. This group usually referred to as the group of integers modulo n. (3) For a positive integer n, consider the set Cn = {a0 , a1 , ..., an−1 }. On Cn define a binary operation as follows:  l+m a if l + m < n l m aa = a(l+m)−n if l + m ≥ n. For every positive integer n, Cn is an abelian group. The group Cn is called the cyclic group of order n. (4) In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. (5) The quaternion group is a non-abelian group of order 8. It is often denoted by Q or Q8 and written in multiplicative form, with the following 8 elements Q = {1, −1, i, −i, j, −j, k, −k}. Here 1 is the identity element, (−1)2 = 1 and (−1) · a = a · (−1) = −a for all a in Q. The remaining multiplication rules can be obtained from the following relation: i2 = j 2 = k 2 = i · j · k = −1.

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The concept of subgroups is one of the most basic ideas in group theory. Definition 1.3. A non-empty subset H of a group G is said to be a subgroup of G if the following conditions are satisfied: (1) a, b ∈ H implies a · b ∈ H; (2) a ∈ H implies a−1 ∈ H. If H is a subgroup of G, then H is a group in its own right. Lemma 1.1. Let G be a group and H be a non-empty subset of G. Then, H is a subgroup of G if H is closed under division, i.e., if a · b−1 is in H, whenever a, b are in H. Proof.

It is straightforward.



Lemma 1.2. Let H be a non-empty finite subset of a group G. Then, H is a subgroup of G if H is closed under the operation of G. Proof.

It is straightforward.



Example 1.3. (1) Let G be the group of all real numbers under addition, and let H be the set of all integers. Then, H is a subgroup of G. (2) Let G be the group of all nonzero real numbers under multiplication, and let H be the set of positive rational numbers. Then, H is a subgroup of G. (3) Let G be the group of all non-zero complex numbers under multiplication, and let H = {a+bi | a2 +b2 = 1}. Then, H is a subgroup of G. (4) Let G be an abelian group. Then, H = {x ∈ G | x2 = e} is a subgroup of G. (5) The center Z(G) of a group G is the subset of elements in G that commute with every element of G. In symbols, Z(G) = {a ∈ G | a · x = x · a, for all x ∈ G}. The center of a group G is a subgroup of G. (6) If G is a group and a ∈ G, then the cyclic subgroup generated by a, denoted by hai, is the set of all the powers of a. Among the subgroups of G, the subgroups G and {e} are said to be trivial. A subgroup H is said to be a proper subgroup of G if H 6= G. If M is a proper subgroup of G and if M ⊆ H ⊆ G for a subgroup H of G

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implies that G = H or H = M , then M is said to be a maximal subgroup of G. Proposition 1.3. Let H and K be two subgroups of a group G. The intersection H ∩ K of H and K is a subgroup of G. In general if {Hi }i∈I is T a family of subgroups of G, then Hi is a subgroup of G. i∈I

Proof.

It is straightforward.



Definition 1.4. If X is a subset of a group G, then the smallest subgroup of G containing X, denoted by hXi, is called the subgroup generated by X. If X consists of a single element a, then hXi = hai, the cyclic subgroup generated by a. Theorem 1.3. If X is a non-empty subset of a group G, then the subgroup hXi is the set of all finite products of the form u1 · u2 · ... · un , where for each i, either ui ∈ X or u−1 ∈ X. i Proof. Let H be the set of all finite products of the form u1 · u2 · ... · un , where ui or u−1 ∈ X and n any positive integer. Consider x = a1 · a2 · ... · an i and y = b1 · b2 · ... · bm in H. Then, x · y = a1 · a2 · ... · an · b1 · b2 · ... · bm is a product of finite number of elements ai , bj such that either the factor or its −1 −1 inverse is in X, consequently x · y ∈ H. Further, x−1 = a−1 n · ... · a2 · a1 . −1 −1 −1 −1 −1 −1 Since ai or ai is in X, and ai = (ai ) , we see that either ai or (ai ) is in X, and so x−1 ∈ H. This proves that H is a subgroup of G. Clearly, X ⊆ H. Consider any subgroup K of G containing X. Then, for each u ∈ X, u ∈ K and hence u−1 ∈ K. Thus, if x = u1 · u2 · ... · un , where ui ∈ X or u−1 ∈ X, is any element of H, then x ∈ K, since ui ∈ K for all i i. Hence, H ⊆ K. This proves that H is the subgroup of G generated by X.  Definition 1.5. Let G be a group and H be a subgroup of G. For a, b ∈ G we say a is congruent to b mod H, written as a ≡ b mod H if a · b−1 ∈ H. Lemma 1.3. The relation a ≡ b mod H is an equivalence relation. Proof.

It is straightforward.



Definition 1.6. If H is a subgroup of G and a ∈ G, then H ·a = {h·a | h ∈ H}. Then, H ·a is called a right coset of H in G. A left coset a·H is defined similarly. The number of distinct right cosets of H is called the index of H in G and denoted by [G : H].

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Lemma 1.4. For all a ∈ G, we have H · a = {x ∈ G | a ≡ x mod H}. Proof.

It is straightforward.



The following corollary contains the basic properties of right cosets and it is useful in many applications. Corollary 1.2. Let H be a subgroup of G. (1) Every element a of G contained in exactly one coset of H. This coset is H · a. (2) Two distinct cosets of H have no common element. (3) The group G is partitioned into a disjoint union of cosets of H. (4) There is a one to one correspondence between any two right cosets of H in G. (5) There is a one to one correspondence between the set of left cosets of H in G and the set of right cosets of H in G. Theorem 1.4. (Lagrange’s theorem) If G is a finite group and H is a subgroup of G, then |H| is a divisor of |G|. The above theorem of Lagrange is one of the basic results in finite group theory. Definition 1.7. If G is a group and a ∈ G, then the order of a is the least positive integer n such that an = e. If no such integer exists we say that a is of infinite order. There are many useful corollaries of Lagrange theorem. Corollary 1.3. A finite cyclic group of prime order contains no non-trivial subgroup. Corollary 1.4. The order of an element of a finite group G divides the order |G|. Definition 1.8. Let A and B be two subsets of a group G. The set A · B = {a · b | a ∈ A, b ∈ B} consisting of the products of elements a ∈ A and b ∈ B is said to be the product of A and B. The associative law of multiplication gives us (A · B) · C = A · (B · C) for any three subsets A, B and C. The product of two subgroups is not necessarily a subgroup. We have the following theorem.

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Theorem 1.5. Let A and B be subgroups of a group G. Then, the following two conditions are equivalent. (1) The product A · B is a subgroup of G; (2) We have A · B = B · A. Proof. Suppose that A · B is a subgroup of G. Then, for any a ∈ A, b ∈ B, we have a−1 · b−1 ∈ A · B and so b · a = (a−1 · b−1 )−1 ∈ A · B. Thus, B · A ⊆ A · B. Now, if x is any element of A · B, then x−1 = a · b ∈ A · B and so x = (x−1 )−1 = (a · b)−1 = b−1 · a−1 ∈ B · A, so A · B ⊆ B · A. Thus, A · B = B · A. On the other hand, suppose that A · B = B · A, i.e., if a ∈ A and b ∈ B, then a · b = b1 · a1 for some a1 ∈ A, b1 ∈ B. In order to prove that A · B is a subgroup we must verify that it is closed and every element in A · B has its inverse in A · B. Suppose that x = a · b ∈ A · B and y = a0 · b0 ∈ A · B. Then, x · y = a · b · a0 · b0 , but since b · a0 ∈ B · A = A · B, b · a0 = a2 · b2 with a2 ∈ A, b2 ∈ B. Hence, x · y = a · (a2 · b2 ) · b0 = (a · a2 ) · (b2 · b0 ) ∈ A · B. Clearly, x−1 = b−1 · a−1 ∈ B · A = A · B. Thus, A · B is a subgroup of G.  There is one kind of subgroup that is especially interesting. If G is a group and H is a subgroup of G, it is not always true that a · H = H · a for all a ∈ G. There are certain situations where this does hold, however, and these cases turn out to be of critical importance in the theory of groups. It was Galois, who first recognized that such subgroups were worthy of special attention. Definition 1.9. A subgroup N of a group G is called a normal subgroup of G if aN = N a for all a ∈ G. A group G is said to be simple if G 6= {e} and G contains no non-trivial normal subgroup. The only simple abelian groups are Zp with p prime. There are several equivalent formulations of the definition of normality. Lemma 1.5. Let G be a group and N be a subgroup of G. Then, (1) N is normal in G if and only if a−1 na ∈ N for all a ∈ G and n ∈ N. (2) N is normal in G if and only if the product of two right cosets of N in G is again a right coset of N in G. Proof.

It is straightforward.



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Example 1.4. (1) The center Z(G) of a group is always normal. Indeed, any subgroup of Z(G) is normal in G. (2) If H has only two left cosets in G, then  H is normal in G. ab (3) Let G be the set of all real matrices where ad 6= 0, under 0d n1 b o matrix multiplication. Then, N = | b ∈ R is a normal 01 subgroup of G. (4) Let Q be the set of all rational numbers and G = {(a, b) | a, b ∈ Q, a 6= 0}. Define ? on G as follows: (a, b) ? (c, d) = (ac, ad + b). Then, (G, ?) is a non-abelian group. If we consider N = {(1, b) | b ∈ Q}, then N is a normal subgroup of G. Theorem 1.6. Let N be a normal subgroup of a group G, and let G denote the set of all cosets of N . For any two elements X and Y of G, we define their product X · Y as the subset of G obtained by taking the product of the two subsets X and Y of G. Then, X · Y is a coset of N . With respect to this multiplication on G, the set G forms a group. Proof. Let X and Y be two elements of G. Then, there are elements x and y of G such that X = N · x and Y = N · y. By assumption, N is normal so that N · x = x · N for any x ∈ G. Whence X · Y = (N · x) · (N · y) = N · (x · N ) · y = N · (N · x) · y = N · x · y. This proves that X · Y is a coset of N . If Z ∈ G, then by the associative law for the product of subsets, we have (X · Y ) · Z = X · (Y · Z). Thus, the multiplication defined on G satisfies the associative law. By definition, we obtain (N · x) · (N · y) = N · x · y; so the coset which contains the identity e of G, namely N , is the identity of G, and the inverse of N · x is the coset N · x−1 . Therefore, G forms a group.  The group G which was defined in the above theorem is called the quotient group of G by N and is written G = G/N . The mapping x → N x from G into G is called the canonical map. The order of the quotient group G/N is equal to the index of the normal subgroup N , i.e., |G/N | = [G : N ]. Example 1.5. (1) Let G = Z18 and N =< 6 >. Then, G/N = {0 + N, 1 + N, 2 + N, 3 + N, 4 + N, 5 + N }.

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(2) Let G be a group such that (ab)p = ap bp for all a, b ∈ G, m where p is a prime number. Let N = {x ∈ G | xp = e for some m depending on x}. Then, N is a normal subgroup of G. If G = G/N and if x ∈ G is such that xp = e, then x = e. We close this section with the following correspondence theorem. Theorem 1.7. Let N be a normal subgroup of a group G, and let G = G/N . For any subgroup V of G, there corresponds a subgroup V of G such that N ⊆ V and V = V /N. The subgroup V consists of those elements of G which are contained in some elements of V and is uniquely determined by V . Thus, between the set G of subgroups of G and the set G of subgroups of G which contain N , there exists a one to one correspondence, V ↔ V . Proof.

It is straightforward.



Let G be a finite group with n elements a1 , a2 , ..., an . A multiplication table for G is the n × n matrix with i, j entry ai ∗ aj : G a1 a2 ... an

a1 a1 ∗ a1 a2 ∗ a1 ... an ∗ a1

a2 a1 ∗ a2 a2 ∗ a2 ... an ∗ a2

... an ... a1 ∗ an ... a2 ∗ an ... ... ... an ∗ an

Informally, we say that we “know” a finite group G if we can write a multiplication table for it. In this section, we consider one of the most fundamental notions of group theory-“homomorphism”. The homomorphism term comes from the Greek words “homo”, which means like and “morphe”, which means form. In our presentation about groups we see that one way to discover information about a group is to examine its interaction with other groups using homomorphisms. A group homomorphism preserves the group operation. Let us consider two almost trivial examples of groups. Let G be the group whose elements are the numbers 1 and −1 with operation multipliˆ be the additive group Z2 . Compare multiplication tables cation and let G of these two groups: ˆ 0 1 G 1 −1 G 1 1 −1 −1 −1 1

0 1

0 1 1 0

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ˆ are distinct groups; on the other hand, it is It is quite clear that G and G equally clear that there is no significant difference between them. Let us make this idea precise. ˆ (not Definition 1.10. A function f defined on a group G to a group G necessarily distinct from G) is said to be a (group) homomorphism from G ˆ if f (xy) = f (x)f (y) for all x, y ∈ G. If f is a surjective homomorinto G ˆ then G ˆ is said to be homomorphic to G. If f is a phism, i.e., f (G) = G, surjective and one to one homomorphism, then f is called an isomorphism ˆ If there is an isomorphism from G onto G, ˆ we say that G from G onto G. ˆ are isomorphic and write G ∼ ˆ and G = G. ˆ The subset Let f be a homomorphism from G into G. ˆ H = {x ∈ G | f (x) is the identity of G} is called the kernel of f and is denoted by Kerf . Example 1.6. (1) Every canonical mapping is a homomorphism. (2) Let G be the group of all positive real numbers under the mulˆ be the group of all tiplication of the real numbers and let G ˆ be defined by real numbers under addition. Let f : G → G f (x) = log10 x for all x ∈ G. Since log10 (xy) = log10 x + log10 y, we have f (xy) = f (x) + f (y), so f is a homomorphism. Also, it happens to be onto and one to one. (3) Let GL(n) be the multiplicative group of all nonsingular n × n matrices over the real numbers. Let R∗ be the multiplicative group of all nonzero real numbers. We define f : G → R∗ by f (A) = detA for all A ∈ GL(n). Since for any two n × n matrices A, B, det(AB) = detA · detB, we obtain f (AB) = f (A)f (B). Hence, f is a homomorphism of GL(n) into R∗ . Also, f is onto. (4) Let D2n be the dihedral group defined as the set of all formal symbols ai bj , i = 0, 1, j = 0, 1, ..., n − 1, where a2 = e, bn = e and ab = b−1 a. Then, the subgroup N = {e, b, b2 , ..., bn−1 } is normal in G and D2n /N ∼ = H, where H = {1, −1} is the group under the multiplication of the real numbers. Proposition 1.4. Let f be a homomorphism from a group G into a group ˆ The following propositions hold. G. ˆ (1) f (e) = e0 , the identity element of G.

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(2) f (x−1 ) = f (x)−1 for all x ∈ G. (3) The kernel of f is a normal subgroup of G. (4) Let H be a subgroup of G. The image f (H) = {f (x) | x ∈ H} is a ˆ For a subgroup H ˆ of G, ˆ the inverse image subgroup of G. ˆ = {x ∈ G | f (x) ∈ H} ˆ f −1 (H) is a subgroup of G. (5) For two elements x and y of G, f (x) = f (y) if and only if x and y lie in the same coset of the kernel f . In particular, if f is surjective, then f is an isomorphism if and only if the kernel of f is {e}. (6) If H is a normal subgroup of G, then f (H) is a normal subgroup of f (G). Proof.

It is straightforward.



We are in a position to establish an important connection between homomorphisms and quotient groups. Many authors prefer to call the next theorem the Fundamental theorem of group homomorphism. Theorem 1.8. Let f be a homomorphism from a group G onto a group ˆ Then, there exists an isomorphism g from G/kerf onto G ˆ such that G. f = gϕ, where ϕ is the canonical homomorphism from G onto G/kerf . In this case, we say the following diagram is commutative. ϕ

G f

Proof.



ˆ G

|

/ G/kerf g

ˆ by Suppose that K = kerf . Define a function g from G/K to G g(K · x) = f (x).

The function g is well defined and does not depends on the choice of a representative from K · x. Since f is a homomorphism, we have g(K · x · K · y) = g(K · x · y) = f (x · y) = f (x) · f (y) = g(K · x) · g(K · y). ˆ If the coset Kx lies in Hence, g is a homomorphism from G/K onto G. the kernel of g, then e = g(K · x) = f (x), so that x ∈ K. Thus, g is an isomorphism. It is clear that f = gϕ holds; so the diagram is commutative. 

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Corollary 1.5. Let N be a normal subgroup of a group G, and let ϕ be the canonical homomorphism from G onto G/N . Let f be a homomorphism ˆ Then, there exists a homomorphism g from G/N from G into a group G. ˆ such that f = gϕ if and only if N ⊆ kerf . In this case, we have into G f (G) ∼ = (G/N )/(kerf /N ). Proof. If there is a homomorphism g satisfying f = gϕ, we have f (N ) = e; so N ⊆ kerf . Conversely, suppose that N ⊆ kerf . Set K = kerf and ˆ by G = G/N . We define a function g from G into G g(N · x) = f (x). The function g is uniquely determined independent of the choice of a repˆ Clearly, resentative x from N · x, and g is a homomorphism from G into G. f = gϕ by definition, and we have kerg = K. Now, we obtain G/K = f (G) ∼ = G/K = (G/N )/(K/N ). = g(G) ∼



Corollary 1.6. Let f be a homomorphism from a group G onto a group ˆ Let N ˆ be a normal subgroup of G ˆ and set N = f −1 (N ˆ ). Then, N is a G. ˆ ˆ normal subgroup of G and G/N ∼ G/ N . = Theorem 1.9. Let H be a normal subgroup of a group G, and let K be any subgroup of G. Then, the following hold. (1) HK is a subgroup of G; (2) H ∩ K is a normal subgroup of K; (3) H · K/H ∼ = K/H ∩ K. Proof. Since H is a normal subgroup of G, H ·K = K·H and so according to Theorem 1.5, H · K is a subgroup of G. Now, consider the canonical mapping from G onto the factor group G/H. Let f be the restriction of the canonical mapping on K. Then, f is a homomorphism from K into G/H. By definition, f is a homomorphism from K onto H · K/H. Thus, we have H · K/H ∼ = K/kerf. But, the kernel of the canonical homomorphism is K, whence we get kerf = H ∩ K. This completes the proof.  Example 1.7. (1) Any finite cyclic group of order n is isomorphic to Zn . Any infinite cyclic group is isomorphic to Z.

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(2) Let G be the group of all real valued functions on the unit interval [0, 1], where for any f, g ∈ G, we define addition f + g by (f + g)(x) = f (x) + g(x) for all x ∈ [0, 1]. If N = {f ∈ G | f ( 41 ) = 0}, then G/N ∼ = R under +. (3) The quotient group R/Z is isomorphic to the group S 1 of complex numbers of absolute value 1 (with multiplication). An isomorphism is given by f (x + Z) = e2πxi for all x ∈ R. (4) The quotient groups and subgroups of a cyclic groups are cyclic. Definition 1.11. By an automorphism of a group G we shall mean an isomorphism of G onto itself. If g is an element of G, then the function ig : x 7→ g −1 xg is an automorphism of G, which is called the inner automorphism by g. Let Aut(G) denote the set of all automorphisms of G. For the product of elements of Aut(G) we can use the composition of mappings. Lemma 1.6. If G is a group, then Aut(G) is also a group. Aut(G) is called the group of automorphism of G. The set of all inner automorphisms is a subgroup of Aut(G), which is written Inn(G), and is called the group of inner automorphism of G. If G is abelian, then Inn(G) = {e}. Theorem 1.10. The group of inner automorphisms of G is isomorphic to the quotient group G/Z(G), where Z(G) is the center of G. Furthermore, Inn(G) is a normal subgroup of Aut(G). Proof. The function g 7→ ig is a homomorphism from G onto Inn(G). Thus, G/K ∼ = Inn(G), where K is the kernel of the above homomorphism. An element g of G lies in the kernel K if and only if ig = e, i.e., g −1 ·x·g = x, for all x ∈ G. Therefore, we have K = Z(G). The last assertion follows immediately from the formula σ −1 ig σ = iσ(g) , which holds for any g ∈ G and σ ∈ Aut(G). Example 1.8. (1) Aut(Z) ∼ = Z2 ; (2) Aut(G) = 1 if and only if |G| ≤ 2.



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Rings

In this section, the definition of a ring and numerous examples are given. A ring is a two-operational system and these operations are usually called addition and multiplication. Definition 1.12. A nonempty set R is said to be a ring if in R there are defined two binary operations, denoted by + and · respectively, such that for all a, b, c in R: (1) (2) (3) (4) (5) (6)

a + b = b + a, (a + b) + c = a + (b + c), there is an element 0 in R such that a + 0 = a, there exists an element −a in R such that a + (−a) = 0, (a · b) · c = a · (b · c), · is distributive with respect to +, i.e., x · (y + z) = x · y + x · z and (x + y) · z = x · z + y · z.

Axioms (1) through (4) merely state that R is an abelian group under the operation +. The additive identity of a ring is called the zero element. If a ∈ R and n ∈ Z, then na has its usual meaning for additive groups. If in addition: (7) a · b = b · a, for all a, b in R, then R is said to be a commutative ring. If R contains an element 1 such that (8) 1 · a = a · 1 = a for all a in R, then R is said to be a ring with unit element. If R is a system with unit satisfying all the axioms of a ring expect possibly a + b = b + a for all a, b ∈ R, then one can show that R is a ring. For any two elements a, b of a ring R, we shall denote a + (−b) by a − b and for convenience sake we shall usually write ab instead of a · b. Before going on to work out some properties of rings, we pause to examine some examples. Motivated by these examples we shall define various special types of rings which are of importance. Example 1.9. (Some examples of commutative rings). (1) Each of the number sets Z, Q, R and C forms a ring with respect to ordinary addition and multiplication.

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(2) For every m ∈ Z, {ma | a ∈ Z} forms a ring with respect to ordinary addition and multiplication. (3) The set Zn is a ring with respect to addition and multiplication modulo n. (4) We say that a ring R is a Boolean ring (after the English Mathematician George Boole) if x2 = x for all x ∈ R. A Boolean ring is commutative. Let X be a set and A, B be subsets of X. The symmetric difference between two subsets A and B, denoted by A 4 B, is the set of all x such that either x ∈ A or x ∈ B but not both. The set P(X) of all subsets of a set X is a ring. The addition is the symmetric difference 4 and the multiplication is the set operation intersection ∩. Its zero element is the empty set, and its unit element is the set X. This is an example of a Boolean ring. (5) Let Z[i] denote the set of all complex numbers of the form a + bi where a and b are integers. Under the usual addition and multiplication of complex numbers, Z[i] forms a ring called the ring of Gaussian integers. (6) The set of all continuous real-valued functions defined on the interval [a, b] forms a ring, the operations are addition and multiplication of functions. (7) A polynomial is a formal expression of the form p(x) = a0 + a1 x + ... + an−1 xn−1 + an xn , where a0 , ..., an ∈ R and x is a variable. Polynomials can be added and multiplied as usual. With these operations the set R[x] of all polynomials is a ring. In fact, given any commutative ring R, one can construct the ring R[x] of polynomials over R in a similar way. We define now the ring of polynomials in the n-variables x1 , ..., xn over R, R[x1 , ..., xn ], as follows: let R1 = [x1 ], R2 = R1 [x2 ],..., Rn = Rn−1 [xn ]. Rn is called the ring of polynomials in x1 , ..., xn over R. (8) Let R be a commutative ring with unit element and denoted by R[[x]] the set of all formal power series over the ring R. Then R[[x]] is a ring with addition and multiplication defined by ∞ ∞ ∞ P P P ai xi + bi xi = (ai + bi )xi , i=0 ∞ P i=0

i

ai x ·

i=0 ∞ P i=0

i

bi x =

i=0 ∞ P i=0

ci xi ,

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n P

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ai bn−i . The ring R[[x]] is called the ring of power

i=0

series. (9) Let R be a commutative ring with unit. A nonempty subset S of R is called a multiplicative subset if 0 6∈ S and s1 , s2 ∈ S implies s1 s2 ∈ S. Let R×S be the set of all ordered pairs (r, s) where r ∈ R and s ∈ S. In R × S we define now a relation as follows: (r1 , s1 ) ∼ (r2 , s2 ) if and only if there exists s ∈ S such that s(r1 s2 −s1 r2 ) = 0. The relation ∼ is an equivalence relation on R × S. Let [r, s] be the equivalence class of (r, s) in R × S, and let S −1 R be the set of all such equivalence classes [r, s] where r ∈ R and s ∈ S. The quotient set S −1 R is a commutative ring with unit under addition and multiplication defined by [r1 , s1 ] + [r2 , s2 ] = [r1 s2 + r2 s1 , s1 s2 ], [r1 , s1 ] · [r2 , s2 ] = [r1 r2 , s1 s2 ], for all r1 , r2 ∈ R and s1 , s2 ∈ S. S −1 R is usually called the ring of fractions of R. In the special case in which R is the ring of integers, the S −1 R so constructed is, of course, the ring of rational numbers. Example 1.10. (Some examples of noncommutative rings). (1) One of the smallest noncommutative rings is the Klein 4-ring (R, +, ·), where (R, +) is the Klein 4-group {0, a, b, c} with 0 the neutral element and the binary operation · given by the following table: · 0 a b c

0 0 0 0 0

a 0 a b c

b 0 0 0 0

c 0 a b c

(2) The set Mn (R) of all n×n matrices with entries from R forms a ring with respect to the usual addition and multiplication of matrices. In fact, given an arbitrary ring R, one can consider the ring Mn (R) of n × n matrices with entries from R. (3) If G is an abelian group, then End(G), the set of endomorphisms of G, forms a ring, the operations in this ring are the addition and composition of endomorphisms.

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(4) Let Ω consist of all complex valued functions f of real variable x such that Z ∞ |f (x)|dx < ∞. −∞

Ω is an additive abelian group with respect to the ordinary addition. We consider the binary operation ∗ called convolution, h = f ∗ g, where (f ∗ g)(x) is defined by the equation Z ∞ (f ∗ g)(x) = f (x − t)g(t)dt. −∞

It can be shown that if f and g are in Ω, then h is also in Ω (it follows from Fubini’s theorem in analysis). The remaining axioms are easy to verify and we conclude that Ω is a ring with respect to the ordinary addition + and convolution ∗. This ring lacks a unit element. (5) Let G be a group and R a ring. Firstly, we define the set R[G] to be one of the following: • The set of all formal R-linear combinations of elements of G. • The set of all functions f : G → R with f (g) = 0 for all but finitely many g in G. No matter which definition is used, we can write the elements of P R[G] in the form ag g, with all but finitely many of the ag being g∈G

0, and the addition on R[G] is the addition of formal linear combinations or addition of functions, respectively. The multiplication of elements of R[G] is defined by setting P P P ( ag g)( bh h) = (ag bh )gh. g∈G

h∈G

g,h∈G

If R has a unit element, this is the unique bilinear multiplication for which (1g)(1h) = (1gh). In this case, G is commonly identified with the set of elements 1g of R[G]. The identity element of G then serves as the 1 in R[G]. It is not difficult to verify that R[G] is a ring. This ring is called the group ring of G over R. Note that: If R and G are both commutative (i.e., R is commutative and G is an abelian group), then R[G] is commutative.

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(6) This last example is often called the ring of real quaternions. This ring was firstly described by the Irish mathematician Hamilton. Initially it was extensively used in the study of mechanics; today its primary interest is that of an important example, although still it plays key roles in geometry and number theory. Let Q be the set of all symbols a0 + a1 i + a2 j + a3 k, where all the numbers a0 , a1 , a2 and a3 are real numbers. Define the equality between two elements of Q as follows: a0 + a1 i + a2 j + a3 k = b0 + b1 i + b2 j + b3 k if and only if a0 = b0 , a1 = b1 , a2 = b2 and a3 = b3 . We define the addition and multiplication on Q by (a0 + a1 i + a2 j + a3 k) + (b0 + b1 i + b2 j + b3 k) = (a0 + b0 ) + (a1 + b1 )i + (a2 + b2 )j + (a3 + b3 )k, (a0 + a1 i + a2 j + a3 k) · (b0 + b1 i + b2 j + b3 k) = (a0 b0 − a1 b1 − a2 b2 − a3 b3 ) + (a0 b1 + a1 b0 + a2 b3 − a3 b2 )i +(a0 b2 + a2 b0 + a3 b1 − a1 b3 )j + (a0 b3 + a3 b0 + a1 b2 − a2 b1 )k. It is easy to see that Q is a noncommutative ring in which 0 = 0 + 0i + 0j + 0k and 1 = 1 + 0i + 0j + 0k are the zero and unit elements respectively. Note that the set {1, −1, i, −i, j, −j, k, −k} forms a non-abelian group of order 8 under this product. (7) (Differential operator rings). Consider the homogeneous linear differential equation an (x)Dn y + ... + a1 (x)Dy+a0 (x)y=0, where the solution y(x) is a polynomial with complex coefficients, and also the terms ai (x) belong to C[x]. The equation can be written in compact form as L(y)=0, where L is the differential operator an (x)Dn +...+ d . Thus the differential operator can be a1 (x)D+a0 (x), with D = dx thought as a polynomial in the two indeterminates x and D, but in this case the indeterminates do not commute, since D(xy(x)) = y(x) + xD(y(x)), yielding the identity Dx = 1 + xD. The repeated use of this identity makes possible to write the composition of two differential operators in the standard form a0 (x) + a1 (x)D + ... + an (x)Dn , and we denote the resulting ring by C[x][D]. We wish to be able to compute in rings in the same manner in which we compute with real numbers, keeping in mind always that there are different. It may happen that ab 6= ba, or a does not divide b. To this end we mention some preliminary results, which assert that certain something we should like to be true in rings are indeed true. Lemma 1.7. Let R be a ring. Then

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(1) Since a ring is an abelian group under +, there are certain things we know from the group theory, for instance, −(−a) = a and −(a+ b) = −a − b, for all a, b in R and so on, (2) 0a = a0 = 0, for all a in R, (3) (−a)b = a(−b) = −(ab), for all a, b in R, (4) (−a)(−b) = ab, for all a, b in R, (5) (na)b = a(nb) = n(ab), for all n ∈ Z and a, b in R, P  P  P n m n P m (6) ai bj = ai bj , for all ai , bj in R. i=1

j=1

i=1 j=1

Moreover, if R has a unit element 1, then (7) (−1)a = −a, for all a ∈ R, (8) (−1)(−1) = 1. Proof.

It is straightforward.



In dealing with an arbitrary ring R there may exist non-zero elements a and b in R, such that their product is zero. Such elements are called zero-divisors. Definition 1.13. A non-zero element a is called a zero-divisor if there exists a non-zero element b ∈ R such that either ab = 0 or ba = 0. Example 1.11. As examples of such rings, we have (1) In the ring Z6 we have 2 · 3 = 0 and so 2 and 3 are zero-divisors. More generally, if n is not prime then Zn contains zero-divisors. (2) Consider the ring R of all order pairs of real numbers (a, b). If (a, b) and (c, d) are two elements in R, we define the addition and multiplication in R by the equalities: (a, b) + (c, d) = (a + c, b + d) and (a, b) · (c, d) = (ac, bd). Then R is a ring. The zero element is (0, 0) and the ring has zero-divisors. Definition 1.14. A commutative ring is an integral domain if it has no zero-divisors. The ring of integers, is an example of an integral domain. It is easy to verify that a ring R has no zero-divisors if and only if the right and left cancellation laws hold in R. Definition 1.15. If the non-zero elements of a ring R form a multiplicative group, i.e., R has unit element and every element except the zero element has an inverse, then we shall call the ring a skew field or a division ring.

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Definition 1.16. A field is a commutative division ring. The inverse of an element a under multiplication will be denoted by a−1 . Example 1.12. (1) If p is prime, then Zp is a field. (2) Q, R and C are examples of fields whereas Z is not. (3) In Example 1.9(9), let R be an integral domain and S = R\{0}. Then S −1 R = F is a field. F is usually called the field of fractions. (4) Consider the set {a + bx | a, b ∈ Z2 } with x a “indeterminate”. We use the arithmetic addition modulo 2 and multiplication using the “rule” x2 = x + 1. Then we obtain a field with 4 elements: {0, 1, x, 1 + x}. (5) Consider the set {a + bx + cx2 | a, b, c ∈ Z2 }, where we now use the rule x3 = 1 + x. This gives a field with 8 elements: {0, 1, x, 1 + x, x2 , 1 + x2 , x + x2 , 1 + x + x2 }. (6) Consider the set {a + bx |a, b ∈ Z3 } with arithmetic modulo 3 and the “rule” x2 = −1 (so it is similar as the multiplication in C). Then we obtain a field with 9 elements: {0, 1, 2, x, 1 + x, 2 + x, 2x, 1 + 2x, 2 + 2x}. More generally, using “tricks” like the above ones, we can construct a finite field with pk elements for any prime p and positive integer k. This is denoted by GF (pk ) and it is called the Galois Field named after the French mathematician Evariste Galois. The ring of quaternions is a division ring which is not a field. Many other examples of non-commutative rings exist, for instance see the following example. o n a b | a, b ∈ C , where a, b Example 1.13. Consider the set M = −b a are conjugates of a, b. M is a ring with unit under matrix addition and  x + iy u + iv multiplication. If A = is a nonzero matrix in M , then −u + iv x − iy   x − iy u + iv −  x2 + y 2 + u2 + v 2 x2 + y 2 + u2 + v 2  . A−1 =    u − iv x + iy x2 + y 2 + u2 + v 2 x2 + y 2 + u2 + v 2

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Hence M is a division ring, but is not commutative, since       01 i 0 i 0 01 6= . −1 0 0 −i 0 −i −1 0 Clearly, every field is an integral domain, but, in general, an integral domain is not a field. For example, the ring of integers is an integral domain, but not all nonzero elements have inverse under multiplication. However, for finite domains of integrity, we have the following theorem. Theorem 1.11. Any finite ring without zero-divisors is a division ring. Proof. Let R = {x1 , x2 , ..., xn } be a finite ring without zero-divisors and suppose that a(6= 0) ∈ R. Then ax1 , ax2 , ..., axn are all n distinct elements lying in R, as cancellation laws hold in R. Since a ∈ R, there exists xi ∈ R such that a = axi . Then we have a(xi a − a) = a2 − a2 = 0, and so xi a = a. Now, for every b ∈ R we have ab = (axi )b = a(xi b), hence b = xi b and further ba = b(xi a) = (bxi )a which implies that b = bxi . Hence xi is the unit element for R and we denote it by 1. Now, 1 ∈ R, so there exists c ∈ R such that 1 = ac. Also a(ca − 1) = (ac)a − a = a − a = 0, and so ca = 1. Consequently, R is a division ring.  Corollary 1.7. A finite integral domain is a field. By a famous theorem of Wedderburn, “every finite division ring is a field”. Therefore, we can say that “any finite ring without zero-divisors is a field. Definition 1.17. Let R be a ring. Then D is said to be of finite characteristic if there exists a positive integer n such that na = 0 for all a ∈ R. If no such of n exists, R is said to be of characteristic 0. If R is of finite characteristic, then we define the characteristic of R as the smallest positive integer n such that na = 0 for all a ∈ R. The characteristic of Zn is equal to n, whereas Z, Q, R and C are of characteristic 0. Proposition 1.5. (1) Any finite field is of finite characteristic. However, an integral domain may be infinite and with a finite characteristic. (2) The characteristic of an integral domain with unit element is either zero or a prime number.

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(3) If D is an integral domain and if na = 0 for some a 6= 0 in D and some integer n 6= 0, then D is of finite characteristic. Note that, it is not true for an arbitrary ring; it is enough to consider the ring Z2 × Z. (4) Let R be a ring with unit element. Then the characteristic of R is equal to n if and only if n is the least positive integer such that n · 1 = 1. Proof.

It is straightforward.



In the study of groups, subgroups play a crucial role. Subrings, the analogous notion in ring theory, play a much less important role than their counterparts in group theory. Nevertheless, subrings are important. Definition 1.18. Let R be a ring and S be a nonempty subset of R, which is closed under the addition and multiplication in R. If S is itself a ring under these operations then S is called a subring of R; more formally, S is a subring of R if the following conditions hold. a, b ∈ S ⇒ a − b ∈ S and a · b ∈ S. Example 1.14. (1) For each positive integer n, the set nZ = {0, ±n, ±2n, ±3n, ...} is a subring of Z. (2) Z is a subring of the ring of real numbers and also a subring of the ring of polynomials Z[X]. (3) The ring of Gaussian integers is a subring of the complex numbers. a0 (4) The set A of all 2 × 2 matrices of the type , where a, b and c bc are integers, is a subring of the ring M2 (Z). (5) The polynomial ring R[x] is a subring of R[[x]]. (6) If R is any ring, then the center of R is the set Z(R)={x ∈ R | xy=yx, ∀y ∈ R}. Clearly, the center of R is a subring of R. Let R be a ring and S be a proper subring of it. Then there exists the following five cases: • R and S have a common unit element. • R has a unit element but S does not. • R and S both have their own nonzero unities but these are distinct.

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• R has no unit element but S has a unit element. • Neither R nor S have unit element. Example 1.15. (1) The ring Q and its subring Z have the common unit element 1. (2) The subring S of even integers of the ring Z has no unit element. Actually, the only subring with unit of Z is Z. (3) Let S be the subring of all pairs (a, 0) of the ring Z × Z for which the operations + and · are defined component by component. Then S and Z × Z have the unities (1, 0) and (1, 1), respectively. (4) Let S be the subring of all pairs (a, 0) of the ring R={(a, 2b) | a, b ∈ Z} (operations are defined component by component). Now S has the unit element (1, 0) but R has no unit element. (5) Neither the ring {(2a, 2b) | a, b ∈ Z} (operations are defined component by component) nor its subring consisting of the pairs (2a, 0) have unit element. In group theory, normal subgroups play a special role, they permit us to construct quotient groups. Now, we introduce the analogous concept for rings. Definition 1.19. A non-empty subset I of a ring R is said to be an ideal of R if (1) I is a subgroup of R under addition, (2) for every a ∈ I and r ∈ R, both ar and ra are in I. Clearly, each ideal is a subring. For any ring R, {0} and R are ideals of R. The ideal {0} is called the trivial ideal. An ideal I of R such that I 6= 0 and I 6= R is called a proper ideal. Observe that if R has a unit element and I is an ideal of R, then I = R if and only if 1 ∈ I. Consequently, a nonzero ideal I of R is proper if and only if I contains no invertible elements of R. It is easy to see that the intersection of any family of ideals of R is also an ideal. Example 1.16. (1) For any positive integer n, the set nZ is an ideal of Z. In fact, every ideal of Z has this form, for suitable n. (2) Let I be the set of all polynomials over R with zero constant term. Then I is an ideal of R[x].

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(3) Let R be the ring of all real-valued functions of a real variable. The subset S of all differentiable functions is a subring of R but not an ideal of R. (4) Let f ∈ Q[x]. Then the set {f g | g ∈ Q[x]} is an ideal of Q[x]. In fact, every ideal, though not every subring, of Q[x] has this form. (5) Let   n a b c o R =  d e f  | a, b, c, d, e, f, g ∈ Z 0 0 g then R is a ring under matrix addition and multiplication. The set   n 0 0 x o I =  0 0 y  | x, y ∈ Z 0 0 0 is an ideal of R. (6) Let R be a ring and let Mn (R) be the ring of matrices over R. If I is an ideal of R then the set Mn (I) of all matrices with entries in I is an ideal of Mn (R). Conversely, every ideal of Mn (R) is of this type. (7) Let m be a positive integer such that m is not a square in Z. If √ R = {a + mb | a, b ∈ Z}, then R is a ring under the operations of sum and product of real numbers. If p is an odd prime number, √ √ consider the set Ip = {a + mb | p|a and p|b}, where a + mb ∈ R. Then Ip is an ideal of R. (8) For ideals I1 , I2 of a ring R define I1 +I2 to be the set {a+b | a ∈ I1 , nP o n b ∈ I2 } and I1 I2 to be the set ai bi | n ∈ Z+ , ai ∈ I1 , bi ∈ I2 . i=1

Then I1 + I2 and I1 I2 are ideals of R. (9) Let R be an arbitrary ring and let a1 , a2 , ...am ∈ R. Then the set of all elements of the form  ni m m m m  P P P P P zi ai + si ai + a i ti + ui,k ai vi,k , i=1

i=1

i=1

i=1

i=1

where m, zi , ni ∈ Z, si , ti , ui,k , vi,k ∈ R, is an ideal. In fact it is the smallest ideal of R which contains a1 , a2 , ...am . Hence it is called the ideal generated by a1 , a2 , ...am . If R is commutative and has a unit element, the above set reduces to the set {a1 r1 + a2 r2 + ... + am rm | ri ∈ R}. We denote this ideal briefly by ha1 , a2 , ...am i. If m = 1 the ideal ha1 i is called the principal ideal generated by a1 . In particular, h1i = R.

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(10) The subset E of Z[x] composed by all polynomials with even constant term is an ideal of Z[x]. In fact E = hx, 2i and it is not principal. (11) Let X be a nonempty set and P(X) denotes the ring of power set of X. Then a nonempty subset I of P(X) is an ideal of P(X) if and only if P(A ∪ B) ⊆ I for all A, B ∈ I. (12) Let R be a commutative ring and let A be an arbitrary subset of R. Then the annihilator of A, Ann(A) = {r ∈ R | ra = 0 for all a ∈ A} is an ideal. Lemma 1.8. Let R be a commutative ring with unit element whose only ideals are the trivial ideal and R. Then R is a field. Proof. In order to prove this lemma, for any nonzero element a ∈ R we must find an element b ∈ R such that ab = 1. The set Ra = {xa | x ∈ R} is an ideal of R. By our assumptions on R, Ra = {0} or Ra = R. Since 0 6= a = 1 · a ∈ Ra, Ra 6= {0}, and so Ra = R. Since 1 ∈ R, there exists b ∈ R such that 1 = ba.  Definition 1.20. (Quotient ring) Let R be a ring and let I be an ideal of R. In order to define the quotient ring, we consider firstly an equivalence relation on R. We say that the elements a, b ∈ R are equivalent, and we write a ∼ b, if and only if a − b ∈ I. If a is an element of R, we denote the corresponding equivalence class by [a]. The quotient ring of modulo I is the set R/I = {[a] | a ∈ R}, with a ring structure defined as follows. If [a], [b] are equivalence classes in R/I, then [a] + [b] = [a + b] and [a] · [b] = [ab]. Since I is closed under addition and multiplication, it follows that the ring structure in R/I is well defined. Clearly, a + I = [a]. Example 1.17. Let us present some quotient rings. (1) Z/6Z = {6Z, 1 + 6Z, 2 + 6Z, 3 + 6Z, 4 + 6Z, 5 + 6Z}. (2) We consider the ring of polynomials R[x] with real coefficients and hx2 + 1i generated by x2 + 1. Then R[x]/hx2 + 1i = {ax + b + hx2 + 1i | a, b ∈ R}. (3) If R = Z[x, y] and I = hx2 , y 2 + 1i, then every element of R/I has the form a + bx + cy + dxy + I, where a, b, c, d ∈ Z.

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Now, as a group homomorphism preserves the group operation, a ring homomorphism preserves the ring operations. Definition 1.21. A mapping ϕ from the ring R into the ring R0 is said to be a ring homomorphism if (1) ϕ(a + b) = ϕ(a) + ϕ(b), (2) ϕ(ab) = ϕ(a)ϕ(b), for all a, b ∈ R. If ϕ is a ring homomorphism from R to R0 , then ϕ(0) = 0 and ϕ(−a) = −ϕ(a) for every a ∈ R. A ring homomorphism ϕ : R → R0 is called an epimorphism if ϕ is onto. It is called a monomorphism if it is one to one, and an isomorphism if it is both one to one and onto. A homomorphism ϕ of a ring R into itself is called an endomorphism. An endomorphism is called an automorphism if it is an isomorphism. The rings R and R0 are said to be isomorphic if there exists an isomorphism between them, in this case, we write R ∼ = R0 . Before going on we examine these concepts for certain examples. Example 1.18. (1) For any positive integer n, the mapping k → k mod n is a ring homomorphism from Z onto Zn . (2) Let I be an ideal of a ring R. We define ϕ : R → R/I by ϕ(a) = a + I for all a ∈ R. Then ϕ is an epimorphism. This map is called a natural √ √ homomorphism. (3) Let Z( 2) be the set√of real numbers of the form m + n 2 where m, n are integers; Z( 2) forms a ring under the usual √ addition √ and multiplication of real √ numbers. We define ϕ : Z( 2) → Z( 2) by √ ϕ(m + n 2) = m − n 2. Then ϕ is an automorphism. Lemma 1.9. Let ϕ be a homomorphism from the ring R to the ring R0 . Let S be a subring of R, I an ideal of R and J an ideal of R0 . (1) (2) (3) (4) (5)

ϕ(S) = {ϕ(a) | a ∈ S} is a subring of R0 . If ϕ is onto, then ϕ(I) is an ideal of R0 . ϕ−1 (J) = {r ∈ R | ϕ(r) ∈ J} is an ideal of R. If R is commutative then ϕ(R) is commutative. If R has a unit element 1 and ϕ is onto, then ϕ(1) is the unit element of R0 .

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(6) If ϕ is an isomorphism from R to R0 , then ϕ−1 is an isomorphism from R0 to R. Proof.

It is straightforward.



Now, we introduce an important ideal that is intimately related to the image of a homomorphism. Definition 1.22. If ϕ is a ring homomorphism of R into R0 , then the kernel of ϕ is defined by {x ∈ R | ϕ(x) = 0}. Corollary 1.8. If ϕ is a ring homomorphism from R to R0 , then kerϕ is an ideal of R. Theorem 1.12. A ring homomorphism ϕ from R to R0 is one to one if and only if kerϕ = {0}. Proof.

It is straightforward.



We are in a position to establish an important connection between homomorphisms and quotient rings. Many authors prefer to call the next theorem the fundamental theorem of ring isomorphism. Theorem 1.13. (First isomorphism theorem). Let ϕ : R → R0 be a homomorphism from R to R0 . Then R/kerϕ ∼ = ϕ(R); in fact, the mapping ψ : R/kerϕ → ϕ(R) defined by ψ(a + kerϕ) = ϕ(a) defines an isomorphism from R/kerϕ onto ϕ(R). Moreover there is a one to one correspondence between the set of ideals of R0 and the set of ideals of R which contain kerϕ. This correspondence can be achieved by associating with an ideal J in R0 , the ideal I in R defined by I = {x ∈ R | ϕ(x) ∈ J}. With I so defined, R/I is isomorphic to R0 /J. We go on to the next isomorphism theorem. Theorem 1.14. (Second isomorphism theorem). Let I and J be two ideals of a ring R. Then (I + J)/I ∼ = J/(I ∩ J). Finally, we come to the last of the isomorphism theorem that we wish to state. Theorem 1.15. (Third isomorphism theorem). Let I and J be two ideals of a ring R such that J ⊆ I. Then R/I ∼ = (R/J)/(I/J).

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Example 1.19. (1) Z/nZ ∼ = Zn .  n a b  o (2) Let R = | a, b ∈ R . We define ψ : R → C by −b a  a b  ψ = a + bi. Then ψ is an isomorphism and so R is −b a isomorphic to the field of complex numbers. (3) Let R be the ring of all real valued continuous functions defined on the closed unit interval. Then I = {f ∈ R | f ( 21 ) = 0} is an ideal of R. One can shows that R/I is isomorphic to the real field. Lemma 1.10. Let R be a ring with unit element 1. The mapping ϕ : Z → R given by ϕ(n) = n1 is a ring homomorphism. Proposition 1.6. If R is a ring with unit element and the characteristic of R is n > 0, then R contains a subring isomorphic to Zn . If the characteristic of R is 0, then R contains a subring isomorphic to Z. Proof. The set S = {n1 | n ∈ Z} is a subring of R. Lemma 1.10 shows that the mapping ϕ from Z onto S given by ϕ(n) = n1 is a homomorphism, and by the first isomorphism theorem, we have Z/kerϕ ∼ = S. But, clearly ∼ ∼ ∼ kerϕ = nZ. So S = Zn if n > 0, whereas S = Z/h0i = Z if n = 0.  Proposition 1.7. If F is a field of characteristic p, then F contains a subfield isomorphic to Zp . If F is a field of characteristic 0, then F contains a subfield isomorphic to Q. Proof. By Proposition 1.6, F contains a subring isomorphic to Zp if F has characteristic p and F has a subring S isomorphic to Z if F has characteristic 0. In the latter case, let K = {ab−1 | a, b ∈ S, b 6= 0}. Then K is isomorphic to Q.  Now, we define some special ideals of a ring and we give some important results about them. Firstly, we begin with the definition of maximal ideal of a ring. Definition 1.23. A proper ideal M of R is said to be a maximal ideal of R if whenever U is an ideal of R and M ⊆ U ⊆ R then U = M or U = R. Example 1.20. Examples of maximal ideals. (1) In a division ring, < 0 > is a maximal ideal.

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(2) In the ring of even integers, < 4 > is a maximal ideal. (3) In the ring of integers, an ideal nZ is maximal if and only if n is a prime number.

 (4) The ideal x2 + 1 is maximal in R[x]. (5) Let R be the ring of continuous functions from R to R. The set M = {f ∈ R | f (0) = 0} is a maximal ideal of R. Zorn’s lemma is a form of the axiom of choice which is technically very useful for proving existence theorems. For instance, from Zorn’s lemma it follows directly that every ring has a maximal ideal. Theorem 1.16. If R is a commutative ring with unit element and M is an ideal of R, then M is a maximal ideal of R if and only if R/M is a field. Proof. Suppose that M is a maximal ideal and let a ∈ R but a 6∈ M . It suffices to show that a + M has a multiplicative inverse. Consider U = {ar + b | r ∈ R, b ∈ M }. This is an ideal of R that contains M properly. Since M is maximal, we have U = R. Thus 1 ∈ U , so there exist c ∈ R and d ∈ M such that 1 = ac + d. Then 1 + M = ac + d + M = ac + M = (a + M )(c + M ). Now, suppose that R/M is a field and U is an ideal of R that contains M properly. Let a ∈ U but a 6∈ M . Then a + M is a nonzero element of R/M and so there exists an element b + M such that (a + M )(b + M ) = 1 + M . Since a ∈ U , we have ab ∈ U . Also, we have 1 − ab ∈ M ⊆ U . So 1 = (1 − ab) + ab ∈ U which implies that U = R.  The motivation for the definition of a prime ideal comes from the integers. Definition 1.24. An ideal P in a ring R is said to be prime if P 6= R and for any ideals A, B in R, AB ⊆ P ⇒ A ⊆ P or B ⊆ P. The definition of a prime ideal excludes the ideal R for both historical and technical reasons. The following corollary is a very useful characterization of prime ideals. Corollary 1.9. Let R be a commutative ring. An ideal P of R is prime if P 6= R and for any a, b ∈ R, ab ∈ P ⇒ a ∈ P or b ∈ P.

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Example 1.21. Examples of prime ideals. (1) A positive integer n is a prime number if and only if the ideal nZ is a prime ideal in Z. (2) In the ring Z[x] of all polynomials with integer coefficients, the ideal generated by 2 and x is a prime ideal. (3) The prime ideals of Z × Z are {0} × Z, Z × {0}, pZ × Z, Z × qZ, where p and q are primes. (4) If R denotes the ring C[x, y] of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial y 2 −x3 −x−1 is a prime ideal. Also the ideals h0i ⊆ hy − x − 1i ⊆ hx − 2, y − 3i are all prime. (5) In Z[x, y, z], the ideals hxi ⊆ hx, yi ⊆ hx, y, zi are all prime, but none is maximal. Theorem 1.17. If R is a commutative ring with unit element and P is an ideal of R, then P is a prime ideal of R if and only if R/P is an integral domain. Proof. Suppose that R/P is an integral domain and ab ∈ P . Then, (a + P )(b + P ) = ab + P = P . So either a + P = P or b + P = P ; that is either a ∈ P or b ∈ P . Hence P is prime. Now, suppose that P is prime and (a + P )(b + P ) = 0 + P = P . Then ab ∈ P and therefore a ∈ P or b ∈ P . Thus one of a + P or b + P is zero.  Theorem 1.18. Let R be a commutative ring with unit element. Each maximal ideal of R is a prime ideal. Proof. Suppose that M is maximal in R but not prime, so there exist a, b ∈ R such that a 6∈ M , b 6∈ M but ab ∈ M . Then each of the ideals M + hai and M + hbi contains M properly. By maximality we obtain M + hai = R = M + hbi. Therefore, R2 = (M + hai)(M + hbi) ⊆ M 2 + haiM + M hbi + haihbi ⊆ M ⊆ R. This is a contradiction.



Definition 1.25. The radical of an ideal I in a commutative ring R, denoted by Rad(I) is defined as Rad(I) = {r ∈ R | rn ∈ I for some positive integer n}.

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Intuitively, one can think that the radical of I is obtained by taking all the possible roots of elements of I. Rad(I) turns out to be an ideal itself, containing I. The above definition is equivalent to: The radical of an ideal I in a commutative ring R is T Rad(I) = P, P ∈ Spec(R) I⊆P

where Spec(R) is the set of all prime ideals of R. Lemma 1.11. If J, I1 , ..., In are ideals in a commutative ring R, then (1) Rad(Rad(J)) = Rad(J), T  n n T (2) Rad(I1 ...In ) = Rad Ii = Rad(Ii ). i=1

i=1

Example 1.22. In the ring of integers (1) Rad(12Z) = 2Z ∩ 3Z = 6Z, (2) let n = pk11 ...pkr r , where pi ’s are distinct prime numbers. Then we have Rad(nZ) = hp1 , ..., pr i. The concept of a maximal ideal in a commutative ring leads immediately to the very important notion of a Jacobson radical of that ring. Definition 1.26. Let R be a commutative ring. We define the Jacobson radical of R, denoted by Jac(R), as the intersection of all the maximal ideals of R. We can provide a characterization for the Jacobson radical of a commutative ring. Lemma 1.12. (Nakayama’s lemma) Let R be a commutative ring, and let r ∈ R. Then r ∈ Jac(R) if and only if for every a ∈ R, the element 1 − ra is an invertible element of R. 1.3

Modules

Modules over a ring are generalization of abelian groups (which are modules over Z). Definition 1.27. Let R be a ring. A non-empty subset M is said to be an R-module (or, a a module over R) if M is an abelian group under an

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operation + such that for every r ∈ R and m ∈ M there exists an element rm in M subject to (1) r(a + b) = ra + rb; (2) r(sa) = (rs)a; (3) (r + s)a = ra + sa; for all a, b ∈ M and r, s ∈ R. If R has a unit element 1, and if 1m = m for every element m in M , then M is called a unitary R-module. Example 1.23. (1) Every abelian group G is a module over the ring of integers. Addition is carried out according to the group structure of G; the key point is that we can multiply x ∈ G by the integer n. If n > 0, then nx = x + x + ... + x (n times); if n < 0, then nx = −x − x − ... − x (|n| times). (2) Let R be a ring and M be left ideal of R. For r ∈ R, m ∈ M , let rm be the product of these elements as elements in R. The definition of left ideal implies that rm ∈ M , while the axioms defining a ring insure us that M is an R-module. (3) The special case in which M + R, any ring R is an R-module over itself. (4) If S is a ring and R is a subring, then S is an R-module with rm (r ∈ R, m ∈ S) being multiplication in S. In particular, R[x1 , ..., xn ] and R[[x]] are R-modules. (5) Let R be a ring and I be a left ideal of R. Let M consists of all cosets, a+I, where a ∈ R. In M define (a+I)+(b+I) = (a+b)+I and r(a + I) = ra + I. Then, M is an R-module. M is usually written as R/I and is called the quotient module of R by I. (6) Let A be an abelian group and End(A) its endomorphism ring. Then A is a End(A)-module, with f a defined to be f (a), for a ∈ A, f ∈ End(A). (7) Let M = Mmn (R) be the set of all m × n matrices with entries in R. Then, M is an R-module, where addition is ordinary matrix addition, and multiplication of the scalar c by the matrix A means multiplication of each entry of A by c. M = {0} is called zero module and is often written simply as 0.

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Let M be an R-module. The results given for rings in Lemma 1.7 can be applied to establish the following results, which hold for any x ∈ M and r ∈ R. We distinguish the vector 0M from the zero scalar 0R . (1) (2) (3) (4)

r0M = 0M ; 0R x = 0M ; (−r)x = r(−x) = −(rx); If R is a field, or more generally a division ring, then rx = 0M implies that either r = 0R or x = 0M .

Definition 1.28. An additive subgroup A of the R-module M is called a submodule of M if whenever r ∈ R and a ∈ A, then ra ∈ A. Example 1.24. (1) Let I be a left ideal of the ring R, M be an R-module and S a non-empty subset of M . Then, nP o n IS = rr ai | ri ∈ I, ai ∈ S, n ∈ N i=1

is a submodule of M . Similarly, if a ∈ M , then Ia = {ra | r ∈ I} is a submodule of M . (2) If {Ni | i ∈ I} is a family of submodules of an R-module M , then T Ni is a submodule of M . i∈I

Definition 1.29. If X is a subset of an R-module M , then the intersection of all submodules of M containing X is called the submodule generated by X. If X consists of a single element, X = {a}, then the submodule generated by X is called the cyclic submodule generated by a. An R-module M is cyclic if it is generated by a single element a. Finally, if {Ni | i ∈ I} is a family of submodules of an R-module M , then the submodule generated S by X = is called the sum of the modules Ni . If the index set I is finite, i∈I

the sum of N1 , ..., Nn is denoted by N1 + ... + Nn . Theorem 1.19. Let R be a ring, M be an R-module, X be a subset of M , {Ni | i ∈ I} is a family of submodules M , and a ∈ M . (1) Let Ra = {ra | r ∈ R}. Then, Ra is a submodule of M and the map R → Ra given by r 7→ ra is an R-module epimorphism.

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(2) The cyclic submodule A generated by a is {ra + na | r ∈ R, n ∈ Z}. If R has an identity and A is unitary, then A = Ra. (3) The submodule N generated by X is nP o t s P ri ai + nj bj |s, t ∈ N, ai , bj ∈ X, ri ∈ R, nj ∈ Z . i=1

j=1

If R has an identity and M is unitary, then nP o s N = RX = ri ai |s ∈ N, ai ∈ X, ri ∈ R . i=1

(4) The sum of the family {Ni | i ∈ I} consists of all finite sums ai1 + ... + ain with aik ∈ Nik . Proof. It is straightforward. Note that if R has a unit 1R and M is unitary, then n1R ∈ R, for all n ∈ Z and na = (n1R )a, for all a ∈ M .  Given an R-module M and a submodule A, we could constract the quotient module M/A in a manner similar to the way we constructed quotient group and quotient ring. One could also talk about homomorhisms of one R-module into another, and prove the appropriate homomorphism theorems. Theorem 1.20. Let A be a submodule of an R-module M . Then, the quotient group M/A is an R-module with the following external operation r(m + A) = rm + A, for all r ∈ R and m ∈ M. Proof. Since M is an additive abelian group, A is a normal subgroup, and M/A is well defined abelian group. If m+A = m0 +A, then m−m0 ∈ A. Since A is a submodule, it follows that rm − rm0 = r(m − m0 ) ∈ A, for all r ∈ R. Thus, rm + A = rm0 + A and the external operation of R on M/A is well defined. The remainder of the proof is now easy.  Definition 1.30. Let M and M 0 be two R-modules. A function f : M → M 0 is an R-module homomorphism provided that for all a, b ∈ M and r ∈ R, f (a + b) = f (a) + f (b) and f (ra) = rf (a). When the context is clear R-module homomorphisms are called simply homomorphism. An R-module homomorphism f : M → M 0 is called an epimorphism in case it is onto (surjective). It is called a monomorphism in case it is one to one (injective).

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Observe that an R-module homomorphism f : M → M 0 is necessary a homomorphism of additive abelian groups. Consequently, the same terminology is used. f is an R-module isomorphism if it is one to one and onto too. The kernel of f is its kernel as a homomorphism of abelian groups, namely kerf = {m ∈ M | f (a) = 0}. Similarly, the image of f is the set Imf = {y ∈ M 0 | y = f (x) for some x ∈ M }. Finally, we conclude that (1) f is one to one if and only if kerf = {0}. (2) f : M → M 0 is an R-module isomorphism if and only if there is an R-module homomorphism g : M 0 → M such that gf = 1M and f g = 1M 0 . Example 1.25. For any modules the zero map 0 : M → M 0 given by a 7→ 0 (for a ∈ M ) is a module homomorphism. Every homomorphism of abelian groups is a Z-module homomorphism. If R is a ring, the map R[x] → R[x] given by f 7→ xf is an R-module homomorphism, but not a ring homomorphism. A homomorphism f : M → N that is the composite of homomorphisms f = gh is said to factor through g and h. The following result essentially says that a homomorphism f factors uniquely through every epimorphism whole kernel is contained in that of f and through every monomorphism whose image contains the image of f . Theorem 1.21. Let M, M 0 , N and N 0 be R-modules and let f : M → N be an R-module homomorphism. (1) If g : M → M 0 is an epimorphism with kerg ⊆ kerf , then there exists a unique homomorphism h : M 0 → N such that f = hg. Moreover, kerh = g(Kerf ) and Imh = Imf , so that h is monomorphism if and only if kerg = kerf and h is epimorphism if and only if f is epimorphism. (2) If g : N 0 → N is a monomorphism with Imf ⊆ g, then there exists a unique homomorphism h : M → N 0 such that f = gh. Moreover, kerh = kerf and Imh = g −1 (Imf ), so that h is monomorphism if and only if f is monomorphism and h is epimorphism if and only if Img = Imf .

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Proof. (1) Since g : M → M 0 is epimorphism, for each m0 ∈ M 0 there is at least one m ∈ M with g(m) = m0 . Also, if l ∈ M with g(l) = m0 , then clearly m − l ∈ kerg. But since kerg ⊆ kerf , we have that f (m) = f (l). Thus, there is a well defined function h : M 0 → N such that f = hg. To see that h is actually an R-module homomorphism, let x0 , y 0 ∈ M 0 and x, y ∈ M with g(x) = x0 , g(y) = y 0 . Then, for each r ∈ R, g(rx + y) = rx0 + y 0 , so that h(rx0 + y 0 ) = f (rx + y) = rf (x) + f (y) = rh(x0 ) + h(y 0 ). The uniqueness of h with these properties is assured, since g is epimorphism. The final assertion is trivial. (2) For each m ∈ M , f (m) ∈ Imf ⊆ Img. So since g is monomorphism, there is a unique n0 ∈ N 0 such that g(n0 ) = f (m). Therefore, there is a function h : M → N 0 with m 7→ n0 such that f = gh. The rest of the proof is also easy.  In view of the preceding results it is not surprising that the various isomorphism theorems for groups are valid for modules. One need only check at each stage of the proof to see that every subgroup or homomorphism is in fact a submodule or module homomorphism. For convenience we list these results here. Theorem 1.22. Let R be a ring, M and M 0 be two R-modules. If f : M → M 0 is an R-module homomorphism and N is a submodule of kerf , then there is a unique R-module homomorphism φ : M/N → M 0 such that φ(a + N ) = f (a) for all a ∈ M ; Imφ = Imf and kerφ = kerf /N . Moreover, φ is an R-module isomorphism if and only if f is an onto Rmodule homomorphism and N = kerf . In particular, M/kerf ∼ = Imf . Theorem 1.23. Let R be a ring, M and M 0 be two R-modules. If N is a submodule of M , N 0 is a submodule of M 0 and f : M → M 0 is an R-module homomorphism such that f (N ) ⊆ N 0 , then f induces an Rmodule isomorphism φ : M/N → M 0 /N 0 given by a + N 7→ f (a) + N 0 . Moreover, φ is an R-module isomorphism if and only if Imf + N 0 = M 0 and f −1 (N 0 ) ⊆ N . In particular, if f is an onto R-module homomorphism such that f (N ) = N 0 and kerf ⊆ N , then φ is an R-module isomorphism. Theorem 1.24. Let A and B be two submodules of an R-module M . (1) There is an R-module isomorphism A/(A ∩ B) ∼ = (A + B)/B.

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(2) If B ⊆ A, then A/B is a submodule of M/B, and there is an R-module isomorphism (M/B)/(A/B) ∼ = M/A. Proof. (1) Define a map f : A → M/B by f (x) = x + B. Then, f is an R-module homomorphism whose kernel is A ∩ B and whose image is {x + B | x ∈ A} = (A + B)/B. The first isomorphism theorem for modules gives the desired result. (2) Define f : M/B → M/A by f (x + B) = x + A. Then, f is an R-module homomorphism whose kernel is {x + B | x ∈ A} = A/B, and the image is {x + A | x ∈ M } = M/A. The result follows from the first isomorphism theorem for modules.  A pair of R-module homomorphisms f

g

M 0 → M → M 00 is said to be exact at M in case Imf = Kerg. Also, a sequence (finite or infinite) of R-module homomorphisms fn−1

fn

fn+1

... −→ Mn−1 −→ Mn −→ Mn+1 −→ ... is exact in case it is exact at each Mn , i.e., in case for each successive pair fn and fn+1 , Imfn = kerfn+1 . Immediate from the definition is the following set of special cases. Proposition 1.8. Given modules M and N and an R-module homomorphism f : M → N , the sequence f

(1) 0→M → N is exact if and only if f is a monomorphism; f

(2) M → N →0 is exact if and only if f is an epimorphism; f

(3) 0→M → N → 0 is exact if and only if f is an isomorphism. Proof.

1.4

It is straightforward.



Vector space

Note that if the ring R is a field, a unital R-module is nothing more than a vector space over R. Indeed: Definition 1.31. A non-empty set V is said to be a vector space over a field F if V is an abelian group under an operation which we denote by +,

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and if for every α ∈ F , v ∈ V there is defined an element, written αv, in V subject to (1) (2) (3) (4)

α(v + w) = αv + αw; (α + β)v = αv + βv; α(βv) = (αβ)v; 1v = v;

for all α, β ∈ F , v, w ∈ V (where 1 represents the unit element of F under multiplication). Example 1.26. (1) Let F be a field and K be a field which contains F as a subfield. We consider K as a vector space over F , using as the + of the vector space the addition of elements of K, and by defining, for α ∈ F , v ∈ V , αv to be the products of α and v as elements in the field K. Axioms (1), (2), (3) for a vector space are then consequence of the right distributive law, left distributive law, and associative law, respectively, which hold for K as a field. (2) Let F be a field and let V be the totality of all ordered ntuples (α1 , ..., αn ) where αi ∈ F . Two elements (α1 , ..., αn ) and (β1 , ..., βn ) are declared to be equal if and only if αi = βi for each i = 1, ..., n. We now introduce the requisite operations in V to make of it a vector space by defining: (α1 , ..., αn ) + (β1 , ..., βn ) = (α1 + β1 , ..., αn + βn ), γ(α1 , ..., αn ) = (γα1 , ..., γαn ), for γ ∈ F. It is easy to verify that with these operations, V is a vector space over F . Since it will keep reappearing, we assign a symbol to it, namely F n . If V is a vector space over F and if W ⊆ V , then W is a subspace of V if under the operations of V , W , itself, forms a vector space over F . Equivalently, W is a subspace of V whenever α ∈ F , v, w ∈ W implies that αv + w ∈ W . As in our previous models, a homomorphism is a mapping preserving all the algebraic structure of our system. Definition 1.32. If V is a vector space over F and if v1 , ..., vn ∈ V then any element of the form α1 v1 + α2 v2 + ... + αn vn , where αi ∈ F , is a linear combination over F of v1 , ..., vn .

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Definition 1.33. The vector space V is said to be finite dimensional over F , if there is a finite subset X in V such that V = hXi. Note that F n is finite dimensional over F , for if X consists of the n vectors (1, 0, ..., 0), (0, 1, 0, ..., 0), ..., (0, 0, ..., 1), then V = hXi. Definition 1.34. If V is a vector space and if v1 , ..., vn are in V , we say that they are linearly dependent over F if there exist elements c1 , ..., cn in F , not all of them 0, such that c1 v1 + c2 v2 + ... + cn vn = 0. If the vectors v1 , ..., vn are not linearly dependent over F , they are said to be linearly independent over F . Note that if v1 , ..., vn are linearly independent then none of them can be 0, for if v1 = 0, say, then αv1 + 0v2 + ... + 0vn = 0 for any α 6= 0 in F . Lemma 1.13. If v1 , ..., vn ∈ V are linearly independent, then every element in hv1 , ..., vn i has a unique representation in the form c1 v1 + ... + cn vn with ci ∈ F . Proof.

It is straightforward.



Lemma 1.14. If v1 , ..., vn ∈ V are in V , then either they are linearly independent or some vk is a linear combination of the preceding ones, v1 , ..., vk−1 . Proof.

It is straightforward.



Definition 1.35. A subset X of a vector space V is called a basis of V if S consists of linearly independent elements (that is, any finite number of elements in X is linearly independent) and V = hXi.

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Chapter 2

Algebraic hyperstructures

2.1

Semihypergroup

The concept of a semihypergroup is a generalization of the concept of a semigroup. Many authors studied different aspects of semihypergroups. A hypergroupoid (H, ?) is a non-empty set H together with a map ? : H × H → P ∗ (H) called (binary) hyperoperation, where P ∗ (H) denotes the set of all non-empty subsets of H. The image of the pair (x, y) is denoted by x ? y. If A, B are non-empty subsets of H and x ∈ H, then by A ? B, A ? x and x ? B we mean S A?B = a ? b, A ? x = A ? {x} and x ? B = {x} ? B. a∈A b∈B

Definition 2.1. A hypergroupoid (H, ?) is called a semihypergroup if (x ? y) ? z = x ? (y ? z), for all x, y, z ∈ H. This means that S S u?z = x ? v. u∈x?y

v∈y?z

A semihypergroup H is finite if it has only a finitely many elements. A semihypergroup H is commutative if it satisfies x ? y = y ? x, for all x, y ∈ H. Remark 2.1. Every semigroup is a semihypergroup. 43

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Remark 2.2. The associativity for semihypergroups can be applied for subsets, i.e., if (H, ?) is a semihypergroup, then for all non-empty subsets A, B, C of H, we have (A ? B) ? C = A ? (B ? C). The element a ∈ H is called scalar if |a ? x| = |x ? a| = 1, for all x ∈ H. An element e in a semihypergroup (H, ?) is called scalar identity if x ? e = e ? x = {x}, for all x ∈ H. An element e in a semihypergroup (H, ?) is called identity if x ∈ e ? x ∩ x ? e, for all x ∈ H. An element a0 ∈ H is called an inverse of a ∈ H if there exists an identity e ∈ H such that e ∈ a ? a0 ∩ a0 ? a. An element 0 in a semihypergroup (H, ?) is called zero element if x ? 0 = 0 ? x = {0}, for all x ∈ H. Let (H, ?) be a hypergroupoid. The element a ∈ H is called right simplifiable element (respectively, left) if for all x, y ∈ H the following is valid. x ? a = y ? a ⇒ x = y (respectively, a ? x = a ? y ⇒ x = y). Moreover, if x ∈ x ? y (respectively, x ∈ y ? x ) for all y ∈ H, then x is called left absorbing-like element (respectively, right absorbing-like element). Similar to semigroups, we can describe the hyperoperation on a semihypergroup by Cayley table. Example 2.1. (1) Let H = {a, b, c, d}. Define the hyperoperation ? on H by the following table. ? a b c d

a a a a a

b {a, b} {a, b} b b

Then, (H, ?) is a semihypergroup.

c {a, c} {a, c} c c

d {a, d} {a, d} d d

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(2) Let H = {a, b, c, d, e}. Define the hyperoperation ? on H by the following table. ? a b c d e

a a a a a a

b {a, b, d} b {a, b, d} {a, b, d} {a, b, d}

c a a {a, c} a {a, c}

d {a, b, d} {a, b, d} {a, b, d} {a, b, d} {a, b, d}

e {a, b, d} {a, b, d} {a, b, c, d, e} {a, b, d} {a, b, c, d, e}

Then, (H, ?) is a semihypergroup. (3) Let H be the unit interval [0, 1]. For every x, y ∈ H, we define h xy i x ? y = 0, . 2 Then, (H, ?) is a semihypergroup. (4) Let N be the set of non-negative integers. We define the following hyperoperation on N, x ? y = {z ∈ N | z ≥ max{x, y}}, for all x, y ∈ N. Then, (N, ?) is a semihypergroup. (5) Let (S, ·) be a semigroup and K be any subsemigroup of S. Then, the set S/K = {x · K | x ∈ S} becomes a semihypergroup, where the hyperoperation is defined in a usual manner x?y = {z | z ∈ x·y} with x = x · K. (6) The set of real numbers R with the following hyperoperation   (a, b) if a < b a ? b = (b, a) if b < a  {a} if a = b, for all a, b ∈ R is a semihypergroup, where (a, b) is the open interval {x | a < x < b}. (7) Let (S, ·) be a semigroup and P a non-empty subset of S. We define the following hyperoperation on S, x ?P y = x · P · y, for all x, y ∈ S. Then, (S, ?P ) is a semihypergroup. The hyperoperation ?P is called P -hyperoperation. This hyperoperation is introduced in [112]. (8) Let (S, ·) be a semigroup and for all x, y ∈ S, hx, yi denotes the subsemigroup generated by x and y. We define x?y = hx, yi. Then, (S, ?) is a semihypergroup.

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(9) Let (H, ?) and (H 0 , ?) be two semihypergroups. Then, the Cartesian product of these two semihypergroups is a semihypergroup with the following hyperoperation (x, y) ⊗ (x0 , y 0 ) = {(a, b) | a ∈ x ? x0 , b ∈ y ? y 0 }, for all (x, y), (x0 , y 0 ) ∈ H × H 0 .

2.2

Hypergroups

A hypergroupoid (H, ?) is called a quasihypergroup if for all a of H we have a ? H = H ? a = H. This condition is also called the reproduction axiom. Definition 2.2. A semihypergroup (H, ?) is called a hypergroup if a ? H = H ? a = H, for all a ∈ H. A hypergroup is called regular if it has at least one identity and each element has at least one inverse. A hypergroup for which the hyperproduct of any two elements has exactly one element is a group. Indeed, let (H, ?) be a hypergroup, such that for all x, y of H, we have |x ? y| = 1. Then, (H, ?) is a semigroup, such that for all a, b in H, there exist x and y for which we have a = b ? x and a = y ? b. It follows that (H, ?) is a group. Now, we look at some examples of hypergroups. Example 2.2. (1) If H is a non-empty set and for all x, y of H, we define x ? y = H, then (H, ?) is a hypergroup, called the total hypergroup. (2) Let (S, ·) be a semigroup and P be a non-empty subset of S. For all x, y of S, we define x?y = xP y. Then, (S, ?) is a semihypergroup. If (S, ·) is a group, then (S, ?) is a hypergroup, called a P -hypergroup. (3) If G is a group and for all x, y of G, < x, y > denotes the subgroup generated by x and y, then we define x ? y = hx, yi. We obtain that (G, ?) is a hypergroup. (4) If (G, ·) is a group, H is a normal subgroup of G and for all x, y of G, we define x ? y = xyH, then (G, ?) is a hypergroup. (5) Let (G, ·) be a group and let H be a non-normal subgroup of it. If we denote G/H = {xH | x ∈ G}, then (G/H, ?) is a hypergroup, where for all xH, yH of G/H, we have xH ?yH = {zH | z ∈ xHy}.

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(6) If (G, +) is an abelian group, ρ is an equivalence relation in G, which has classes x = {x, −x}, then for all x, y of G/ρ, we define x ? y = {x + y, x − y}. We obtain that (G/ρ, ?) is a hypergroup. (7) Let D be an integral domain and let F be its field of fractions. If we denote by U the group of the invertible elements of D, then we define the following hyperoperation on F/U : for all x, y of F/U , we have x ? y = {z | ∃(u, v) ∈ U 2 such that z = ux + vy}. We obtain that (F/U, ?) is a hypergroup. (8) Let (L, ∧, ∨) be a lattice with a minimum element 0. If for all a ∈ L, F (a) denotes the principal filter generated from a, then we obtain a hypergroup (L, ?), where for all a, b of L, we have a ? b = F (a ∧ b). (9) Let (L, ∧, ∨) be a modular lattice. If for all x, y of L, we define x ? y = {z ∈ L | z ∨ x = x ∨ y = y ∨ z}, then (L, ?) is a hypergroup. (10) Let (L, ∧, ∨) be a distributive lattice. If for all x, y of L, we define x ? y = {z ∈ L | x ∧ y ≤ z ≤ x ∨ y}, then (L, ?) is a hypergroup. (11) Let H be a non-empty set and µ : H → [0, 1] be a function. If for all x, y of H we define x ? y = {z ∈ L | µ(x) ∧ µ(y) ≤ µ(z) ≤ µ(x) ∨ µ(y)}, then (H, ?) is a hypergroup. (12) Let H be a non-empty set and µ, λ be two functions from H to [0, 1]. For all x, y of H we define x ? y = {u ∈ H | µ(x) ∧ λ(x) ∧ µ(y) ∧ λ(y) ≤ µ(u) ∧ λ(u) and µ(u) ∨ λ(u) ≤ µ(x) ∨ λ(x) ∨ µ(y) ∨ λ(y)}. Then, the hyperstructure (H, ?) is a commutative hypergroup. (13) Define the following hyperoperation on the real set R: for all x ∈ R, x ? x = x and for all different real elements x, y, x ? y is the open interval between x and y. Then, (R, ?) is a hypergroup. A non-empty subset K of a semihypergroup (H, ?) is called a subsemihypergroup if it is a semihypergroup. In other words, a non-empty subset K of a semihypergroup (H, ?) is a subsemihypergroup if K ? K ⊆ K. Definition 2.3. A non-empty subset K of a hypergroup (H, ?) is called a subhypergroup if it is a hypergroup. Hence, a non-empty subset K of a hypergroup (H, ?) is a subhypergroup if for all a of K we have a ? K = K ? a = K. There are several kinds of subhypergroups. In what follows, we introduce closed, invertible, ultraclosed and conjugable subhypergroups and some connections among them.

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Definition 2.4. Let (H, ?) be a hypergroup and (K, ?) be a subhypergroup of it. We say that K is: • closed on the left (on the right) if for all k1 , k2 of K and x of H, from k1 ∈ x ? k2 (k1 ∈ k2 ? x, respectively), it follows that x ∈ K; • invertible on the left (on the right) if for all x, y of H, from x ∈ K ?y (x ∈ y ? K), it follows that y ∈ K ? x (y ∈ x ? K, respectively); • ultraclosed on the left (on the right) if for all x of H, we have K ? x ∩ (H\K) ? x = ∅ (x ? K ∩ x ? (H\K) = ∅); • conjugable on the right if it is closed on the right and for all x ∈ H, there exists x0 ∈ H such that x0 ? x ⊆ K. Similarly, we can define the notion of conjugable on the left. We say that K is closed (invertible, ultraclosed, conjugable) if it is closed (invertible, ultraclosed, conjugable, respectively) on the left and on the right. Example 2.3. (1) Let (A, ?) be a hypergroup, H = A ∪ T , where T is a set with at least three elements and A ∩ T = ∅. We define the hyperoperation ⊗ on H, as follows: if (x, y) ∈ A2 , then x ⊗ y = x ? y; if (x, t) ∈ A × T , then x ⊗ t = t ⊗ x = t; if (t1 , t2 ) ∈ T × T , then t1 ⊗ t2 = t2 ⊗ t1 = A ∪ (T \ {t1 , t2 }). Then, (H, ⊗) is a hypergroup and (A, ⊗) is an ultraclosed, nonconjugable subhypergroup of H. (2) Let (A, ?) be a total hypergroup, with at least two elements and let T = {ti }i∈N such that A ∩ T = ∅ and ti 6= tj for i 6= j. We define the hyperoperation ⊗ on H = A ∪ T as follows: if (x, y) ∈ A2 , then x ⊗ y = A; if (x, t) ∈ A × T , then x ⊗ t = t ⊗ x = (A \ {x}) ∪ T ; if (ti , tj ) ∈ T × T , then ti ⊗ tj = tj ⊗ ti = A ∪ {ti+j }. Then, (H, ⊗) is a hypergroup and (A, ⊗) is a non-closed subhypergroup of H. (3) Let us consider the group (Z, +) and the subgroups Si = 2i Z, where i is a non-negative integer. For any x ∈ Z \ {0}, there exists a unique integer n(x), such that x ∈ Sn(x) \ Sn(x)+1 . Define the following commutative hyperoperation on Z \ {0}: if n(x) < n(y), then x ? y = x + Sn(y) ; if n(x) = n(y), then x ? y = Sn(x) \ {0};

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if n(x) > n(y), then x ? y = y + Sn(x) . Notice that if n(x) < n(y), then n(x + y) = n(x). Then, (Z \ {0}, ?) is a hypergroup and for all i ∈ N, (Si \ {0}, ?) is an invertible subhypergroup of Z \ {0}. Lemma 2.1. A subhypergroup K is invertible on the right if and only if {x ? K}x∈H is a partition of H. Proof. If K is invertible on the right and z ∈ x ? K ∩ y ? K, then x, y ∈ z ? K, whence x ? K ⊆ z ? K and y ? K ⊆ z ? K. It follows that x ? K = z ? K = y ? K. Conversely, if {x ? K}x∈H is a partition of H and x ∈ y ? K, then x ? K ⊆ y ? K, whence x ? K = y ? K and so we have x ∈ y ? K = x ? K. Hence, for all x of H we have x ∈ x ? K. From here, we obtain that y ∈ y ? K = x ? K.  Similar to Lemma 2.1, we can give a necessary and sufficient condition for invertible subhypergroups on the left. The following theorems present some connections among the above types of subhypergroups. If A and B are subsets of H such that we have H = A∪B and A∩B = ∅, then we denote H = A ⊕ B. Theorem 2.1. If a subhypergroup K of a hypergroup (H, ?) is ultraclosed, then it is closed and invertible. Proof. First we check that K is closed. For x ∈ K, we have K ∩ x ? (H \ K)=∅ and from H = x?K ∪x?(H \K), we obtain x?(H \K) = H \K, which means that K ?(H \K) = H \K. Similarly, we obtain (H \K)?K = H \K, hence K is closed. Now, we show that {x ? K}x∈H is a partition of H. Let y ∈ x ? K ∩ z ? K. It follows that y ? K ⊆ x ? K and y ? (H \ K) ⊆ x?K?(H \K) = x?(H \K). From H = x?K⊕x?(H \K) = y?K⊕y?(H \K), we obtain x?K = y?K. Similarly, we have z?K = y?K. Hence, {x?K}x∈H is a partition of H, and according to the above lemma, it follows that K is invertible on the right. Similarly, we can show that K is invertible on the left.  Theorem 2.2. If a subhypergroup K of a hypergroup (H, ?) is invertible, then it is closed. Proof. Let k1 , k2 ∈ K. If k1 ∈ x ? k2 ⊆ x ? K, then x ∈ k1 ? K ⊆ K. Similarly, from k1 ∈ k2 ? x, we obtain x ∈ K. 

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We denote the set {e ∈ H | ∃x ∈ H, such that x ∈ x ? e ∪ e ? x} by Ip and we call it the set of partial identities of H. Theorem 2.3. A subhypergroup K of a hypergroup (H, ?) is ultraclosed if and only if K is closed and Ip ⊆ K. Proof. Suppose that K is closed and Ip ⊆ K. First, we show that K is invertible on the left. Suppose there are x, y of H such that x ∈ K ? y and y 6∈ K ? x. Hence, y ∈ (H \ K) ? x, whence x ∈ K ? (H \ K) ? x ⊆ (H \ K) ? x, since K is closed. We obtain that Ip ∩(H \K) 6= ∅, which is a contradiction. Hence, K is invertible on the left. Now, we check that K is ultraclosed on the left. Suppose that there are a and x in H such that a ∈ K?x∩(H \K)?x. It follows that x ∈ K ? a, since K is invertible on the left. We obtain a ∈ (H \ K) ? x ⊆ (H \ K) ? K ? a ⊆ (H \ K) ? a, since K is closed. This means that Ip ∩ (H \ K) 6= ∅, which is a contradiction. Therefore, K is ultraclosed on the left and similarly it is ultraclosed on the right. For the converse, suppose that K is ultraclosed. According to Theorem 2.3.5, K is closed. Now, suppose that Ip ∩ (H \ K) 6= ∅, which means that there is e ∈ H \ K and there is x ∈ H, such that x ∈ e ? x, for instance. We obtain x ∈ (H \ K) ? x, whence K ? x ⊆ (H \ K) ? x, which contradicts that K is ultraclosed. Hence, Ip ⊆ K.  Theorem 2.4. If a subypergroup K of a hypergroup (H, ?) is conjugable, then it is ultraclosed. Proof. Let x ∈ H. Denote B = x?K ∩x?(H \K). Since K is conjugable it follows that K is closed and there exists x0 ∈ H, such that x0 ? x ⊆ K. We obtain x0 ? B = x0 ? (x ? K ∩ x ? (H \ K)) ⊆ K ∩ x0 ? x ? (H \ K) ⊆ K ∩ K ? (H \ K) ⊆ K ∩ (H \ K) = ∅. Hence, B = ∅, which means that K is ultraclosed on the right. Similarly, we check that K is ultraclosed on the left.  Definition 2.5. Let (H1 , ◦) and (H2 , ?) be two hypergroups. A map f : H1 → H2 is called a homomorphism or a good homomorphism if f (x ◦ y) = f (x) ? f (y), for all x, y ∈ H1 . f is called an inclusion homomorphism if f (x ◦ y) ⊆ f (x) ? f (y), for all x, y ∈ H1 .

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By using a certain type of equivalence relations, we can connect semihypergroups to semigroups and hypergroups to groups. These equivalence relations are called strong regular relations. More exactly, by a given (semi)hypergroup and by using a strong regular relation on it, we can construct a (semi)group structure on the quotient set. A natural question arises: Do they also exist regular relations? The answer is positive, regular relations provide us new (semi)hypergroup structures on the quotient sets. Let us define these notions. First, we do some notations. Let (H, ?) be a semihypergroup and R be an equivalence relation on H. If A and B are non-empty subsets of H, then ARB means that ∀a ∈ A, ∃b ∈ B such that aRb and ∀b0 ∈ B, ∃a0 ∈ A such that a0 Rb0 ; ARB means that ∀a ∈ A, ∀b ∈ B, we have aRb. Definition 2.6. The equivalence relation R is called (1) regular on the right (on the left) if for all x of H, from aRb, it follows that (a ? x)R(b ? x) ((x ? a)R(x ? b) respectively); (2) strongly regular on the right (on the left) if for all x of H, from aRb, it follows that (a ? x)R(b ? x) ((x ? a)R(x ? b) respectively); (3) R is called regular (strongly regular) if it is regular (strongly regular) on the right and on the left. Theorem 2.5. Let (H, ?) be a semihypergroup and R be an equivalence relation on H. (1) If R is regular, then H/R is a semihypergroup, with respect to the following hyperoperation: x ⊗ y = {z | z ∈ x ? y}; (2) If the above hyperoperation is well defined on H/R, then R is regular. Proof. (1) First, we check that the hyperoperation ⊗ is well defined on H/R. Consider x = x1 and y = y1 . We check that x ⊗ y = x1 ⊗ y1 . We have xRx1 and yRy1 . Since R is regular, it follows that (x ? y)R(x1 ? y), (x1 ? y)R(x1 ? y1 ) whence (x ? y)R(x1 ? y1 ). Hence, for all z ∈ x ? y, there exists z1 ∈ x1 ? y1 such that zRz1 , which means that z = z1 . It follows that x ⊗ y ⊆ x1 ⊗ y1 and similarly we obtain the converse inclusion. Now, we check the associativity of ⊗. Let x, y, z be arbitrary elements in H/R and u ∈ (x ⊗ y) ⊗ z. This means that there exists v ∈ x ⊗ y such that u ∈ v ⊗ z. In other words, there exist v1 ∈ x ? y and u1 ∈ v ? z, such that vRv1 and uRu1 . Since R is regular, it follows that there exists u2 ∈ v1 ? z ⊆ x ? (y ? z)

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such that u1 Ru2 . From here, we obtain that there exists u3 ∈ y ? z such that u2 ∈ x ? u3 . We have u = u1 = u2 ∈ x ⊗ u3 ⊆ x ⊗ (y ⊗ z). It follows that (x ⊗ y) ⊗ z ⊆ x ⊗ (y ⊗ z). Similarly, we obtain the converse inclusion. (2) Let aRb and x be an arbitrary element of H. If u ∈ a ? x, then u ∈ a ⊗ x = b ⊗ x = {v | v ∈ b ? x}. Hence, there exists v ∈ b ? x such that uRv, whence (a ? x)R(b ? x). Similarly we obtain that R is regular on the left.  Corollary 2.1. If (H, ?) is a hypergroup and R is an equivalence relation on H, then R is regular if and only if (H/R, ⊗) is a hypergroup. Proof. If H is a hypergroup, then for all x of H we have H ? x = x ? H = H, whence we obtain H/R ⊗ x = x ⊗ H/R = H/R. According to the above theorem, it follows that (H/R, ⊗) is a hypergroup.  Notice that if R is regular on a (semi)hypergroup H, then the canonical projection π : H → H/R is a good epimorphism. Indeed, for all x, y of H and z ∈ π(x ? y), there exists z 0 ∈ x ? y such that z = z 0 . We have z = z 0 ∈ x ⊗ y = π(x) ⊗ π(y). Conversely, if z ∈ π(x) ⊗ π(y) = x ⊗ y, then there exists z1 ∈ x ? y such that z = z1 ∈ π(x ? y). Theorem 2.6. If (H, ?) and (K, ∗) are semihypergroups and f : H → K is a good homomorphism, then the equivalence ρf associated with f , that is xρf y ⇔ f (x) = f (y), is regular and ϕ : f (H) → H/ρf , defined by ϕ(f (x)) = x, is an isomorphism. Proof. then

Let h1 ρf h2 and a be an arbitrary element of H. If u ∈ h1 ? a, f (u) ∈ f (h1 ? a) = f (h1 ) ∗ f (a) = f (h2 ) ∗ f (a) = f (h2 ? a).

Then, there exists v ∈ h2 ? a such that f (u) = f (v), which means that uρf v. Hence, ρf is regular on the right. Similarly, it can be shown that ρf is regular on the left. On the other hand, for all f (x), f (y) of f (H), we have ϕ(f (x) ∗ f (y)) = ϕ(f (x ? y)) = {z | z ∈ x ? y} = x ⊗ y = ϕ(f (x)) ⊗ ϕ(f (y)). Moreover, if ϕ(f (x)) = ϕ(f (y)), then xρf y, so ϕ is injective and clearly, it is also surjective. Finally, for all x, y of H/ρf we have ϕ−1 (x ⊗ y) = ϕ−1 ({z | z ∈ x ? y}) = {f (z) | z ∈ x ? y} = f (x ? y) = f (x) ∗ f (y) = ϕ−1 (x) ∗ ϕ−1 (y). Therefore, ϕ is an isomorphism.



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Theorem 2.7. Let (H, ?) be a semihypergroup and R be an equivalence relation on H. (1) If R is strongly regular, then H/R is a semigroup, with respect to the following operation: x ⊗ y = z, for all z ∈ x ? y; (2) If the above operation is well defined on H/R, then R is strongly regular. Proof. (1) For all x, y of H, we have (x ? y)R(x ? y). Hence, x ⊗ y = {z | z ∈ x ? y} = {z}, which means that x ⊗ y has exactly one element. Therefore, (H/R, ⊗) is a semigroup. (2) If aRb and x is an arbitrary element of H, we check that (a ? x)R(b ? x). Indeed, for all u ∈ a ? x and all v ∈ b ? x we have u = a ⊗ x = b ⊗ x = v, which means that uRv. Hence, R is strongly regular on the right and similarly, it can be shown that it is strongly regular on the left.  Corollary 2.2. If (H, ?) is a hypergroup and R is an equivalence relation on H, then R is strongly regular if and only if (H/R, ⊗) is a group. Theorem 2.8. If (H, ?) is a semihypergroup, (S, ∗) is a semigroup and f : H → S is a homomorphism, then the equivalence ρf associated with f is strongly regular. Proof.

Let aρf b, x ∈ H and u ∈ a ? x. It follows that f (u) = f (a) ∗ f (x) = f (b) ∗ f (x) = f (b ? x).

Hence, for all v ∈ b?x, we have f (u) = f (v), which means that uρf v. Hence, ρf is strongly regular on the right and similarly, it is strongly regular on the left.  The fundamental relation has an important role in the study of semihypergroups and especially of hypergroups. Definition 2.7. For all n > 1, we define the relation βn on a semihypergroup H, as follows: n Q a βn b ⇔ ∃(x1 , . . . , xn ) ∈ H n : {a, b} ⊆ xi , i=1

and β =

S

βn , where β1 = {(x, x) | x ∈ H} is the diagonal relation on

n≥1

H. Clearly, the relation β is reflexive and symmetric. Denote by β ∗ the transitive closure of β. Theorem 2.9. β ∗ is the smallest strongly regular relation on H.

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Proof.

We show that

(1) β ∗ is a strongly regular relation on H; (2) If R is a strongly regular relation on H, then β ∗ ⊆ R. (1) Let a β ∗ b and x be an arbitrary element of H. It follows that there exist x0 = a, x1 , . . . , xn = b such that for all i ∈ {0, 1, . . . , n − 1} we have xi β xi+1 . Let u1 ∈ a ? x and u2 ∈ b ? x. We check that u1 β ∗ u2 . From xi β xi+1 it follows that there exists a hyperproduct Pi , such that {xi , xi+1 } ⊆ Pi and so xi ? x ⊆ Pi ? x and xi+1 ? x ⊆ Pi ? x, which means that xi ? xβxi+1 ? x. Hence, for all i ∈ {0, 1, . . . , n − 1} and for all si ∈ xi ? x we have si β si+1 . If we consider s0 = u1 and sn = u2 , then we obtain u1 β ∗ u2 . Then, β ∗ is strongly regular on the right and similarly, it is strongly regular on the left. (2) We have β1 = {(x, x) | x ∈ H} ⊆ R, since R is reflexive. Suppose that βn−1 ⊆ R and show that βn ⊆ R. If aβn b, then there exist x1 , . . . , xn n n−1 Q Q in H, such that {a, b} ⊆ xi . Hence, there exists u, v in xi , such that i=1

i=1

a ∈ u ? xn and b ∈ v ? xn . We have uβn−1 v and according to the hypothesis, we obtain uRv. Since R is strongly regular, it follows that aRb. Hence, βn ⊆ R. By induction, it follows that β ⊆ R, whence β ∗ ⊆ R.  Hence, the relation β ∗ is the smallest equivalence relation on H, such that the quotient H/β ∗ is a group. β ∗ is called the fundamental equivalence relation on H and H/β ∗ is called the fundamental group. If H is a hypergroup, then β = β ∗ . Consider the canonical projection ϕH : H → H/β ∗ . The heart of H is the set ωH = {x ∈ H | ϕH (x) = 1}, where 1 is the identity of the group H/β ∗ . This relation was introduced by Koskas and studied mainly by Corsini, Davvaz, Freni, Leoreanu-Fotea, Vougiouklis and many others. 2.3

Hyperrings

The more general structure that satisfies the ring-like axioms is the hyperring in the general sense: (R, +, ·) is a hyperring if + and · are two hyperoperations such that (R, +) is a hypergroup and · is an associative hyperoperation, which is distributive with respect to +. There are different types of hyperrings. If only the addition + is a hyperoperation and the multiplication · is a usual operation, then we say that R is an additive hyperring. A special case of this type is the hyperring introduced by Krasner.

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Definition 2.8. A Krasner hyperring is an algebraic structure (R, +, ·) which satisfies the following axioms: (1) (R, +) is a canonical hypergroup, i.e., for every x, y, z ∈ R, x + (y + z) = (x + y) + z, for every x, y ∈ R, x + y = y + x, there exists 0 ∈ R such that 0 + x = {x} for every x ∈ R, for every x ∈ R there exists a unique element x0 ∈ R such that 0 ∈ x + x0 ; (We shall write −x for x0 and we call it the opposite of x.) (e) z ∈ x + y implies y ∈ −x + z and x ∈ z − y.

(a) (b) (c) (d)

(2) (R, ·) is a semigroup having zero as a bilateral absorbing element, i.e., x · 0 = 0 · x = 0. (3) The multiplication is distributive with respect to the hyperoperation +. The following elementary facts follow easily from the axioms: −(−x) = x and −(x + y) = −x − y, where −A = {−a | a ∈ A}. Also, for all a, b, c, d ∈ R we have (a + b) · (c + d) ⊆ a · c + b · c + a · d + b · d. In Definition 2.8, for simplicity of notations we write sometimes xy instead of x · y and in (c), 0 + x = x instead of 0 + x = {x}. A Krasner hyperring (R, +, ·) is called commutative (with unit element) if (R, ·) is a commutative semigroup (with unit element). Example 2.4. (1) Let R = {0, 1, 2} be a set with the hyperoperation + and the binary operation · defined as follow: + 0 1 2

0 0 1 2

1 1 1 R

2 2 R 2

and

· 0 1 2

0 0 0 0

1 0 1 1

2 0 2 2

Then, (R, +, ·) is a Krasner hyperring. (2) The first construction of a hyperring appeared in Krasner’s paper and it is the following one: Consider (F, +, ·) a field, G a subgroup of (F ∗ , ·) and take F/G = {aG | a ∈ F } with the hyperaddition and the multiplication given by aG ⊕ bG = {cG | c ∈ aG + bG}, aG bG = abG.

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Then, (F/G, ⊕, ) is a hyperring. If (F, +, ·) is a unitary ring and G is a subgroup of the monoid (F ∗ , ·) such that xG = Gx, for all x ∈ F , then (F/G, ⊕, ) is a Krasner hyperring with identity. (3) Let (A, +, ·) be a ring and N a normal subgroup of its multiplicative semigroup. Then, the multiplicative classes x = xN (x ∈ A) form a partition of R, and let A = A/N be the set of these classes. If for all x, y ∈ A, we define x ⊕ y = {z | z ∈ x + y}, and x ∗ y = x · y, then the obtained structure is a Krasner hyperring. (4) Let R be a commutative ring with identity. We set R = {x = {x, −x} | x ∈ R}. Then, R becomes a Krasner hyperring with respect to the hyperoperation x ⊕ y = {x + y, x − y} and multiplication x ⊗ y = x · y. Definition 2.9. Let (R, +, ·) be a Krasner hyperring and A be a non-empty subset of R. Then, A is said to be a subhyperring of R if (A, +, ·) is itself a Krasner hyperring. The subhyperring A of R is normal in R if and only if x + A − x ⊆ A for all x ∈ R. Definition 2.10. A subhyperring A of a Krasner hyperring R is a left (right) hyperideal of R if r · a ∈ A (a · r ∈ A) for all r ∈ R, a ∈ A. A is called a hyperideal if A is both a left and a right hyperideal. Lemma 2.2. A non-empty subset A of a Krasner hyperring R is a left (right) hyperideal if and only if (1) a, b ∈ A implies a − b ⊆ A, (2) a ∈ A, r ∈ R imply r · a ∈ A (a · r ∈ A). Proof.

It is straightforward.



Definition 2.11. Let A and B be nonempty subsets of a Krasner hyperring R. • The sum A + B is defined by A + B = {x | x ∈ a + b for some a ∈ A, b ∈ B}. • The product AB is defined by n o n P AB = x | x ∈ ai bi , ai ∈ A, bi ∈ B, n ∈ Z+ . i=1

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If A and B are hyperideals of R, then A + B and AB are also hyperideals of R. The second type of a hyperring was introduced by Rota. The multiplication is a hyperoperation, while the addition is an operation, that is why she called it a multiplicative hyperring. Definition 2.12. A triple (R, +, ·) is called a multiplicative hyperring if (1) (R, +) is an abelian group. (2) (R, ·) is a semihypergroup. (3) For all a, b, c ∈ R, we have a · (b + c) ⊆ a · b + a · c and (b + c) · a ⊆ b · a + c · a. (4) For all a, b ∈ R, we have a · (−b) = (−a) · b = −(a · b). If in (3) we have equalities instead of inclusions, then we say that the multiplicative hyperring is strongly distributive. An element e in R, such that for all a ∈ R, we have a ∈ a · e ∩ e · a, is called a weak identity of R. Example 2.5. (1) Let (R, +, ·) be a ring and I be an ideal of it. We define the following hyperoperation on R: For all a, b ∈ R, a ∗ b = a · b + I. Then, (R, +, ∗) is a strongly distributive hyperring. Indeed, first of all, (R, +) is an abelian group. Then, for all a, b, c ∈ R, we have S S a∗(b∗c) = a∗(b·c+I) = a∗(b·c+h) = a·(b·c+h)+I = a·b·c+I h∈I

h∈I

and similarly, we have (a ∗ b) ∗ c = a · b · c + I. Moreover, for all a, b, c ∈ R, we have a∗(b+c) = a·(b+c)+I = a·b+a·c+I = a∗b+a∗c and similarly, we have (b + c) ∗ a = b ∗ a + c ∗ a. Finally, for all a, b ∈ R, we have a ∗ (−b) = a · (−b) + I = (−a) · b + I = (−a) ∗ b and −(a ∗ b) = (−a · b) + I = (−a) · b + I = a ∗ (−b). (2) Let (R, +, ·) be a non-zero ring. For all a, b ∈ R we define the hyperoperation a ∗ b = {a · b, 2a · b, 3a · b, . . .}. Then, (R, +, ∗) is a multiplicative hyperring, which is not strongly distributive. Notice that for all a ∈ R, we have a ∗ 0 = 0 ∗ a = {0}. Proposition 2.1. If (R, +, ·) is a multiplicative hyperring, then for all a, b, c ∈ R, a · (b − c) ⊆ a · b − a · c and (b − c) · a ⊆ b · a − c · a.

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If (R, +, ·) is a strongly distributive, then for all a, b, c ∈ R, a · (b − c) = a · b − a · c and (b − c) · a = b · a − c · a. Proof. The statement follows from the conditions (3) and (4) of Definition 2.12.  Proposition 2.2. In a strong distributive hyperring (R, +, ·), we have 0 ∈ a · 0 and 0 ∈ 0 · a, for all a ∈ R. Proof. b = c.

The statement follows from the above proposition, by considering 

Theorem 2.10. For a strongly distributive hyperring (R, +, ·), the following statements are equivalent: (1) (2) (3) (4) (5)

There exists a ∈ R such that |0 · a| = 1. There exists a ∈ R such that |a · 0| = 1. |0 · 0| = 1. |a · b| = 1, for all a, b ∈ R. (R, +, ·) is a ring.

Proof. (2⇒3): Suppose a 6= 0. For all a ∈ R we have 0 · 0 = (a − a) · 0 = a · 0 − a · 0 and so by (2), it follows that 0 · 0 = {0}, whence we obtain (3). (3⇒4): For all a ∈ R, we have 0 · 0 = a · 0 − a · 0 and so by (3) it follows that |a · 0| = 1, otherwise if we suppose that there exist x 6= y elements of a · 0, then 0 · 0 would contain x − y 6= 0 and 0, a contradiction. On the other hand, for all a, b ∈ R we have a · 0 = a · (b − b) = a · b − a · b, whence it follows that a · b contains only an element. The other implications (4⇒5) and (5⇒2) are immediate. Similarly, the condition (1) is equivalent to (3), (4) and (5).  Corollary 2.3. A strongly distributive hyperring (R, +, ·) is a ring if and only if there exist a0 , b0 ∈ R such that |a0 · b0 | = 1. Proof. According to the above theorem, it is sufficient to check that |a0 · 0|=1. We have a0 · 0 = a0 · (b0 − b0 ) = a0 · b0 − a0 · b0 , whence we obtain that a0 · 0 contains only 0.  Notice that there exist multiplicative hyperrings, which are not strongly distributive and for which we have a ∗ 0 = {0} for all a ∈ R. Definition 2.13. A hyperring (R, +, ·) is called unitary if it contains an element u, such that a · u = u · a = {a} for all a ∈ R.

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We obtain the following result. Theorem 2.11. Every unitary strongly distributive hyperring (R, +, ·) is a ring. Proof. If u is the unit element, then we have u · u = {u} and according to the above corollary, it follows that R is a ring.  Theorem 2.12. In any multiplicative hyperring (R, +, ·), if there are a, b ∈ R such that |a · b| = 1, then 0 · 0 = {0}. Proof. We have a · 0 = a · (b − b) ⊆ a · b − a · b = {0}. On the other hand, 0 · 0 = (a − a) · 0 ⊆ a · 0 − a · 0. But this must also be {0}, since a · 0 is a singleton.  Corollary 2.4. In any unitary multiplicative hyperring (R, +, ·), we have 0 · 0 = {0}. Definition 2.14. Let (R, +, ·) be a multiplicative hyperring and H be a non-empty subset of R. We say that H is a subhyperring of (R, +, ·) if (H, +, ·) is a multiplicative hyperring. In other words, H is a subhyperring of (R, +, ·) if H − H ⊆ H and for all x, y ∈ H, x · y ⊆ H. Definition 2.15. We say that H is a hyperideal of (R, +, ·) if H − H ⊆ H and for all x, y ∈ H, r ∈ R, x · r ∪ r · x ⊆ H. The intersection of two subhyperrings of a multiplicative hyperring (R, +, ·) is a subhyperring of R. The intersection of two hyperideals of a multiplicative hyperring (R, +, ·) is a hyperideal of R. Moreover, any intersection of subhyperrings of a multiplicative hyperring is a subhyperring, while any intersection of hyperideals of a multiplicative hyperring is a hyperideal. In this manner, we can consider the hyperideal generated by any subset S of (R, +, ·), which is the intersection of all hyperideals of R, which contain S. For each multiplicative hyperring (R, +, ·), the zero hyperideal is the hyperideal generated by the additive identity 0. Contrary to what happens in ring theory, the zero hyperideal can contain other elements than 0. If we denote the zero hyperideal of R by < 0 >, then we have nP P P h0i = xi + yj + zk | each sum is finite and for each i, j, k there i j k o exist ri , sj , tk , uk ∈ R such that xi ∈ ri · 0, yj ∈ 0 · sj , zk ∈ tk · 0 · uk .

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Denote by H ⊕ K the hyperideal generated by H ∪ K, where H and K are hyperideals of (R, +, ·). Theorem 2.13. If H and K are hyperideals of R, then H ⊕ K = {h + k | h ∈ H, k ∈ K}. Proof. Denote the set {h + k | h ∈ H, k ∈ K} by I. Then, I is a hyperideal of R, which contains H and K. Moreover, if J is a hyperideal of R, containing H and K, then I ⊆ J. Hence, we have I = H ⊕ K.  Notice that the above theorem can be extended to an whichever family of hyperideals. If we denote by I the set of all hyperideals of a multiplicative hyperring (R, +, ·), then (I, ⊆) is a complete lattice. The infimum of any family of hyperideals is their intersection, while the supremum is the hyperideal generated by their union. Definition 2.16. A homomorphism (good homomorphism) between two multiplicative hyperrings (R, +, ◦) and (R0 , +0 , ◦0 ) is a map f : R → R0 such that for all x, y of R, we have f (x + y) = f (x) +0 f (y) and f (x ◦ y) ⊆ f (x) ◦0 f (y) (f (x ◦ y) = f (x) ◦0 f (y) respectively). The following definition introduces a hyperring in general form. Both addition and multiplication are hyperoperations, that satisfy a set of conditions. Definition 2.17. A hyperringoid (H, ⊕, ) is called a hyperring if the following conditions are satisfied. (1) (H, ⊕) is a commutative hypergroup. (2) (H, ) is a semihypergroup. (3) For all x, y, z ∈ H, (x ⊕ y) z = (x z) ⊕ (y z), z (x ⊕ y) = (z x) ⊕ (z y). (4) For all x ∈ H and all u ∈ ω(H,⊕) , x u ⊆ ω(H,⊕) ⊇ u x. Definition 2.18. Let R be a hyperring. We define the relation γ as follows: xγy ⇔ ∃n ∈ N, ∃ki ∈ N, ∃(xi1 , . . . , xiki ) ∈ Rki , 1 ≤ i ≤ n, such that {x, y} ⊆

ki n  Q P i=1

j=1

 xij .

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Definition 2.19. Let R be a hyperring. We consider the relation α as follows: x α y ⇔ ∃n ∈ N, ∃(k1 , . . . , kn ) ∈ Nn , ∃σ ∈ Sn and [∃(xi1 , . . . , xiki ) ∈ Rki , ∃σi ∈ Ski , (i = 1, . . . , n)] such that  ki n  Q n P P x∈ xij and y ∈ Aσ(i) , i=1

where Ai =

ki Q j=1

j=1

i=1

xiσi (j) .

The relation α and γ are reflexive and symmetric. Let α∗ and γ ∗ be the transitive closure of α and γ. Theorem 2.14. Let (R, +, ·) be a hyperring. (1) γ ∗ is the smallest equivalence relation on R such that the quotient R/γ ∗ is a ring. R/γ ∗ is called the fundamental ring. (2) α∗ is the smallest equivalence relation on R such that the quotient R/α∗ is a commutative ring. R/α∗ is called the commutative fundamental ring. 2.4

Hypermodules

Definition 2.20. A non-empty set M is a hypermodule over a hyperring R (R-hypermodule) if (M, +) is a canonical hypergroup and there exists a map · : R × M → P ∗ (M ) by (r, m) 7→ rm such that for all r1 , r2 ∈ R and m1 , m2 ∈ M , we have (1) r1 (m1 + m2 ) = r1 m1 + r2 m2 , (2) (r1 + r2 )m1 = r1 m1 + r2 m1 , (3) (r1 r2 )m1 = r1 (r2 m1 ). Example 2.6. (1) Let A be an ordinary ring and M be an R-module. If M 0 is a submodule of M and one defines the following scalar hyperoperation ∀(a, x) ∈ A × M, a ◦ x = ax + M 0 then M is an R-hypermodule. (2) Let M = R2 with the following hyperoperation a · (x, y) = {(u, v) ∈ R2 | xv = yv}

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for every a ∈ R and for every (x, y) ∈ R2 − {(0, 0)} and a · (0, 0) = {(0, 0)}. Then, R2 is an R-hypermodule. Let A be a non-empty subset of an R-hypermodule M . Then A is called a subhypermodule of M if A is itself a hypermodule. A subhypermodule A of M is normal in M if x + A − x ⊆ A for all x ∈ M . Example 2.7. Let R = {0, 1, 2} be a set with hyperoperation + and binary operation · as follow: + 0 1 2

0 0 1 2

1 1 1 R

2 2 R 2

· 0 1 2

and

0 0 0 0

1 1 1 1

2 2 2 2

Then (R, +, ·) is a hyperring. Let M = {0, 1, 2, 3, 4, 5, 6, 7, 8} be a set with the hyperoperation as follows: ⊕ 0 1 2 3 4 5 6 7 8

0 0 1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 {0,1,2} 4 4 {3,4,5} 7 7 {6,7,8} {0,1,2} 2 5 {3,4,5} 5 8 {6,7,8} 8 4 5 3 4 5 {0,3,6} {1,4,7} {2,5,8} 4 {3,4,5} 4 4 {3,4,5} {1,4,7} {1,4,7} M {3,4,5} 5 5 {3,4,5} 5 {2,5,8} M {2,5,8} 7 8 {0,3,6} {1,4,7} {2,5,8} 6 7 8 7 {6,7,8} {1,4,7} {1,4,7} M 7 7 {6,7,8} {6,7,8} 8 {2,5,8} M {2,5,8} 8 {6,7,8} 8

Then (M, ⊕) is a canonical hypergroup. Now, we define the external product from R × M → M as follows: ⊗ 0 1 2

0 0 0 0

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

5 5 5 5

6 6 6 6

7 7 7 7

8 8 8 8

Then (M, ⊕, ⊗) is an R-hypermodule. Clearly A = {0, 3, 6} is a normal subhypermodule of M .

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Let A be a subhypermodule of an R-hypermodule M . Then the quotient hypergroup M/A = {m + A | m ∈ M }, with external composition R × M/A −→ M/A, (r, m + A) −→ rm + A is an R-hypermodule, and it is called the quotient R-hypermodule of M by A. Many authors worked on hypermodules. Here, we present some results from [49]. Let M be a hypermodule and N be a non-empty subset of M . Then, N is called a subhypermodule of M if (N, +) is a canonical subhypergroup of (M, +) and for every r ∈ R and n ∈ N , r · n ⊆ N . A subhypermodule N is called normal if for every m ∈ M , m + N − m ⊆ N . Let X be a subset of a hypermodule of M and {Mi | i ∈ I} T be the family of all subhypermodule of M which contain X. Then, Mi i∈I

is called the hypermodule generated by X. This hypermodule is denoted by hXi. If X = {m1 , m2 , . . . , mn }, then the hypermodule hXi is denoted by hm1 , m2 , . . . , mn i. Let M be an R-hypermodule, R1 and M1 , M2 be non-empty subsets of R and M , respectively. We define n o n P R1 · M1 = x ∈ M | x ∈ ri · mi , ri ∈ R1 , mi ∈ M1 , n ∈ N , i=1 n o M1 + M2 = x ∈ M | x ∈ m1 + m2 , m1 ∈ M1 , m2 ∈ M2 , n o n P ZX = m ∈ M | m ∈ ni xi , ni ∈ Z, xi ∈ X . i=1

Proposition 2.3. Let M be an R-hypermodule and X ⊆ M . Then, hXi = ZX + R · X. Definition 2.21. Let M be an R-hypermodule such that (M, +) be an abelian group. Then, M is called multiplicative hypermodule. If N is a subhypermodule of a hypermodule M , then we define the relation m1 ≡ m2 if and only if m1 ∈ m2 + N, for every m1 , m2 ∈ M . This relation is denoted by m1 N ∗ m2 . Proposition 2.4. Let N be a subhypermodule of hypermodule M . Then, N ∗ is an equivalence relation. Proof. Suppose that m ∈ M . Since 0 is neutral element and 0 ∈ N , it follows that m = m + 0 ∈ m + N , the relation N ∗ is reflexive. Let m1 , m2 ∈ M and m1 N ∗ m2 . Then, m1 ∈ m2 + n, for some n ∈ N . Hence, m2 ∈ m1 − n ∈ m1 + N . So this relation is symmetric. Let m1 , m2 , m3 ∈ M such that m1 N ∗ m2 and m2 N ∗ m3 . Then, for some n1 , n2 ∈ N , m1 ∈ m2 + n1 , m2 ∈ m3 + n2 . So, m1 ∈ m2 + n1 ⊆ m3 + n1 + n2 ⊆ m3 + N . Therefore, m1 N ∗ m3 . This completes the proof. 

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If I is a hyperideal of a hyperring R, then we define the relation with the following hyperoperations: x ≡ y if and only if x ∈ y + I. This relation is denoted by xI ∗ y. Proposition 2.5. Let I be a hyperideal of R. Then, [R : I ∗ ] is a hyperring with the following hyperoperations: I ∗ (x) ⊕ I ∗ (y) = {I ∗ (z) | z ∈ I ∗ (x) + I ∗ (y)}, I ∗ (x) I ∗ (y) = {I ∗ (z) | z ∈ I ∗ (x) · I ∗ (y)}. Proof.

The proof is straightforward.



Theorem 2.15. Let M be an R-hypermodule, I be an ideal of R and N be a subhypermodule of M . Then, [M : N ∗ ] is a [R : I ∗ ] hypermodule with the following hyperoperations: N ∗ (m1 ) ⊕ N ∗ (m2 ) = {N ∗ (m) | m ∈ N ∗ (m1 ) + N ∗ (m2 )}, I ∗ (r) N ∗ (m) = {N ∗ (m) | m ∈ I ∗ (r) · N ∗ (m)}, and [M : N ∗ ] is R-hypermodule with the following hyperoperations: N ∗ (m1 ) ⊕ N ∗ (m2 ) = {N ∗ (m) | m ∈ N ∗ (m1 ) + N ∗ (m2 )}, r N ∗ (m) = {N ∗ (m) | m ∈ r · N ∗ (m)}. Proof.

The proof is straightforward.



Let M be an R-hypermodule and N be a subhypermodule of M . Then, the zero element of [M : N ∗ ] is {N } and |h{N }i| = 1. Proposition 2.6. Let N be a normal subhypermodule of hypermodule M . Then, for every m1 , m2 ∈ M the following are equivalent: (1) m2 ∈ m1 + N , (2) m1 − m2 ⊆ N , (3) (m1 − m2 ) ∩ N 6= ∅. Proof. Suppose that (m1 − m2 ) ∩ N 6= ∅. Then there exists m ∈ (m1 − m2 ) ∩ N . So −m2 + m1 ⊆ −m2 + m + m2 ⊆ N . If x ∈ −m2 + m1 , then x ∈ N . Hence, −m2 ∈ x − m1 and m2 ∈ m1 − x ⊆ m1 + N . Therefore, (3) implies (1). It is easy to see that (1) implies (2) and (2) implies (3).  Definition 2.22. Let M be an R-hypermodule and N be a subhypermodule of M . We denote Ω(N ) = {m ∈ M | m − m ⊆ N }. Proposition 2.7. Let M be an R-hypermodule and N be a subhypermodule of M . Then, Ω(N ) is a subhypermodule of M and N ⊆ Ω(N ).

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Proof. Since N 6= ∅, the set Ω(N ) is non-empty. Let m1 , m2 , m ∈ Ω(N ), r ∈ R, x ∈ m1 − m2 and y ∈ r · m. Then, x − x ⊆ (m1 − m2 ) − (m1 − m2 ) = (m1 − m1 ) + (m2 − m2 ) ⊆ N + N = N, y − y ⊆ r · m − r · m = r · (m − m) ⊆ N. Hence, m1 − m2 ⊆ Ω(N ) and r · m ⊆ Ω(N ). Moreover, for every n ∈ N , since N is a subhypermodule of M , n − n ⊆ N . Therefore, Ω(N ) is a subhypermodule of M containing N .  Proposition 2.8. Let M be an R-hypermodule and m1 , m2 ∈ Ω({0}). Then, m1 + m2 is a singleton set. Proof.

The proof is straightforward.



Proposition 2.9. Let M be an hypermodule. Then, Ω({0}) is an abelian group and for every submodule M1 of M , M1 ⊆ Ω({0}). Proof.

Suppose that m1 , m2 ∈ Ω({0}) and x, y ∈ m1 + m2 . Then

x − y ⊆ (m1 + m2 ) − (m1 + m2 ) = (m1 − m2 ) − (m1 − m2 ) = 0. This implies that m1 + m2 is a singleton and Ω({0}) is a subgroup. Let M1 be any subgroup of M and x ∈ M1 . Then, x − x = {0}. Hence x ∈ Ω({0}) and M1 ⊆ Ω({0}). This completes the proof.  Corollary 2.5. Let M be an R-hypermodule and N be a subhypermodule of M . Then, N is normal if and only if Ω(N ) = M . Moreover, (M, +) is an abelian group if and only if Ω({0}) = M . Let H(M ) = {x | x ∈ m − m, for all m ∈ M }. Proposition 2.10. Let M be an R-hypermodule and N be a subhypermodule of M . Then, N is normal if and only if H(M ) ⊆ N . Proof. Suppose that N be a subhypermodule and H(M ) ⊆ N . Then for every m ∈ M and n ∈ N we have m + n − m = m − m + n ⊆ H(M ) + n ⊆ N + N = N. Hence, N is normal. Let N be a normal subhypermodule and m ∈ M . This implies that m + 0 − m ⊆ m + N − m ⊆ N . Hence m − m ⊆ N , for every m ∈ M . Therefore, H(M ) ⊆ N . This completes the proof.  Corollary 2.6. Let M be an R-hypermodule. Then, H(M ) is the smallest normal subhypermodule of M .

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Corollary 2.7. Let N1 and N2 be subhypermodules of M such that N1 ⊆ N2 and N1 be normal subhypermodule. Then, N2 is also normal. Corollary 2.8. Let M be an R-hypermodule such that {0} is normal. Then, all subhypermodules of M are normal. Theorem 2.16. Let M be an R-hypermodule. Then, (M, +) is abelian group if and only if H(M ) = {0}. Proof. We know that (M, +) is abelian group if and only if Ω({0}) = M . Moreover, Ω({0}) = M if and only if {0} is a normal subhypermodule. Hence, (M, +) is an abelian group if and only if {0} is a normal subhypermodule of M . Since H(M ) is a smallest subhypermodule of M , then {0} is normal if and only if H(M ) = {0}. This completes the proof.  Corollary 2.9. Since H(M ) is the smallest normal subhypermodule of M , it follows that M is a module if and only if all subhypemodules of M are normal. Corollary 2.10. Let N be a normal subhypermodule of hypermodule M . Then, the equivalence relation defined in Proposition 2.4, is a strongly regular relation. Hence [M : N ∗ ] is abelian group. Theorem 2.17. Let M be an R-hypermodule and N be a normal subhypermodule of M . Then, [M : N ∗ ] is a multiplicative hypermodule. Proof. Suppose that N is a normal subhypermodule of M . The zero element of this quotient hypermodule is {N }. Moreover, {N } is normal. By Theorem 2.16, [M : N ∗ ] is a multiplicative hypermodule.  Theorem 2.18. Let M be a multiplicative hypermodule. Then, the following statements are equivalent: (1) (2) (3) (4)

there exists m ∈ M such that |0 · m| = 1, there exists r ∈ R, such that |r · 0| = 1, |0 · 0| = 1, for all r ∈ R, m ∈ M , we have |r · m| = 1.

Proof. (2⇒3) Suppose that r ∈ R. We have 0 · 0 = (r − r) · 0 = r · 0 − r · 0 and by (2), it follows that 0 · 0 = {0}, whence we obtain (3). (3⇒4) Let r 6= 0 be an element of R. We have 0·0 = (r−r)·0 = r·0−r·0. If there exists x 6= y elements of r · 0, then 0 · 0 would contain x − y 6= 0 and 0, and it is a contradiction. On the other hand, for every r ∈ R and

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m ∈ M , r · (m − m) = r · m − r · m, whence it follows that r · m contains only an element. The other implications (4⇒1) are immediate. Similarly, the condition (1) is equivalent to (3) and (4).  Proposition 2.11. Let M be a multiplicative hypermodule. Then, (1) 0 ∈ r · 0, for every r ∈ R, (2) 0 ∈ 0 · m, for every m ∈ M , (3) if there exist r0 ∈ R and m0 ∈ M such that |r0 · m0 | = 1, then |0 · 0| = 1, (4) if N is a subhypermodule of M , then for any element N ∗ (m) ∈ [M : N ∗ ], we have |N ∗ (m) N ∗ (0)| = 1. Proof.

By Theorem 2.18, we obtain (1), (2), (3) and (4).



Proposition 2.12. Let M be a multiplicative hypermodule over hyperring R. Then, the external hyperoperation R × M → P ? (M ) is operation if and only if there exist r0 ∈ R and m0 ∈ M such that |r0 · m0 | = 1. Proof. have

By Theorem 2.18, it is sufficient to check that |r0 · 0| = 1. We r0 · 0 = r0 · (m0 − m0 ) = r0 · m0 − r0 · m0 ,

whence we obtain that r0 · 0 contain only 0.



Corollary 2.11. Let M be an R-hypermodule and N be a subhypermodule of M . Then, [M : N ∗ ] is also an R-hypermodule. Moreover, if N is a normal subhypermodule of M , then by the Corollary 2.10, [M : N ∗ ] is a multiplicative hypermodule and by Theorem 2.18, the external hyperoperation in this quotient is operation. Definition 2.23. A mapping f : M → M 0 is called a homomorphism if for all a, b ∈ M and r ∈ R, we have f (a + b) = f (a) + f (b), f (ra) = rf (a) and f (0) = 0. Clearly, a homomorphism f is an isomorphism if f is both injective and surjective. We now write M ∼ = M 0 if M is isomorphic to M 0 . Let R be a hyperring and M be a hypermodule over R. We define the relation ε on M as follows: xεy ⇔ x, y ∈

n P i=1

m0i ; m0i = mi or m0i =

ni k ij P Q ( xijk )zi , j=1 k=1

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mi ∈ M, xijk ∈ R, zi ∈ M. The fundamental relation ε∗ on M can be defined as the smallest equivalence relation such that the quotient M/∗ be a module over the corresponding fundamental ring.

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Chapter 3

Hv -groups

Hv -structures were introduced by Vougiouklis at the Fourth AHA congress (1990) [118]. The concept of an Hv -structure constitutes a generalization of the well-known algebraic hyperstructures (hypergroup, hyperring, hypermodule and so on). Actually some axioms concerning the above hyperstructures such as the associative law, the distributive law and so on are replaced by their corresponding weak axioms. Since the quotients of the Hv -structures with respect to the fundamental equivalence relations (β ∗ , γ ∗ , ∗ , etc.) are always ordinary structures. The quotients of Hv -structure theory has been pursued in many diˇ rections by Vougiouklis, Davvaz, Spartalis, Dramalidis, Leoreanu-Fotea, S. Hoˇskov´ a, and others. We invite the reader to consult the references for an in depth exposition of the theory and its applications.

3.1

Hv -groups and some examples

Definition 3.1. Let H be a non-empty set and ◦ : H × H → P ∗ (H) be a hyperoperation. The “ ◦ ” in H is called weak associative if x ◦ (y ◦ z) ∩ (x ◦ y) ◦ z 6= ∅, for all x, y, z ∈ H. The “ ◦ ” is called weak commutative if x ◦ y ∩ y ◦ x 6= ∅, for all x, y ∈ H. The “ ◦ ” is called strongly commutative if x ◦ y = y ◦ x, for all x, y ∈ H. The hyperstructure (H, ◦) is called an Hv -semigroup if “ ◦ ” is weak associative. An Hv -semigroup is called an Hv -group if a ◦ H = H ◦ a = H, for all a ∈ H. 69

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In an obvious way, the Hv -subgroup of an Hv -group is defined. All the weak properties for hyperstructures can be applied for subsets. For example, if (H, ◦) is a weak commutative Hv -group, then for all nonempty subsets A, B and C of H, we have (A ◦ B) ∩ (B ◦ A) 6= ∅ and A ◦ (B ◦ C) ∩ (A ◦ B) ◦ C 6= ∅. To prove this, one has simply to take one element of each set. A motivation to study the above structures is given by the following examples. Example 3.1. Let (G, ·) be a group and ρ be an equivalence relation on G. In G/ρ, the set of quotient, consider the hyperoperation defined by ρ(x) ρ(y) = {ρ(z) | z ∈ ρ(x) · ρ(y)}, where ρ(x) denotes the equivalence class of the element x. Then, (G/ρ, ) is an Hv -group which is not always a hypergroup. Example 3.2. On the set Zmn consider the hyperoperation ⊕ defined by setting 0 ⊕ m = {0, m} and x ⊕ y = x + y for all (x, y) ∈ Z2mn − {(0, m)}. Then (Zmn , ⊕) is an Hv -group. ⊕ is weak associative but not associative, since taking k 6∈ mZ we have (0 ⊕ m) ⊕ k = {0, m} ⊕ k = {k, m + k}, 0 ⊕ (m ⊕ k) = 0 ⊕ (m + k) = {m + k}. Moreover, it is weak commutative but not commutative. Example 3.3. Consider the group (Zn , +) and take m1 , ..., mn ∈ N. We define a hyperoperation ⊕ in Zn as follows: (m1 , 0, ..., 0) ⊕ (0, 0, ..., 0) = {(m1 , 0, ..., 0), (0, 0, ..., 0)}, (0, m2 , ..., 0) ⊕ (0, 0, ..., 0) = {(0, m1 , ..., 0), (0, 0, ..., 0)}, (0, 0, ..., mn ) ⊕ (0, 0, ..., 0) = {(0, 0, ..., mn ), (0, 0, ..., 0)}, and ⊕ = + in the remaining cases. Then, (Zn , ⊕) is an Hv -group. Definition 3.2. Let (H1 , ◦) and (H2 , ?) be two Hv -groups. A map f : H1 → H2 is called an Hv -homomorphism or a weak homomorphism if f (x ◦ y) ∩ f (x) ? f (y) 6= ∅, for all x, y ∈ H1 . f is called an inclusion homomorphism if f (x ◦ y) ⊆ f (x) ? f (y), for all x, y ∈ H1 .

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Finally, f is called a strong homomorphism if f (x ◦ y) = f (x) ? f (y), for all x, y ∈ H1 . If f is onto, one to one and strong homomorphism, then it is called an isomorphism. Moreover, if the domain and the range of f are the same Hv -group, then the isomorphism is called an automorphism. We can easily verify that the set of all automorphisms of H, defined by AutH, is a group. Several Hv -structures can be defined on a set H. A partial order on these hyperstructures can be introduced, as follows. Definition 3.3. Let (H, ◦) and (H, ?) be two Hv -groups defined on the same set H. We say that ◦ smaller than ?, and ? greater than ◦, and we write ◦ ≤ ?, if there is f ∈ Aut(H, ?) such that x ◦ y ⊆ f (x ? y), for all x, y ∈ H. If a hyperoperation is weak associative, then every greater hyperoperation, defined on the same set is also weak associative. In [122], the set of all Hv -groups with a scalar unit defined on a set with three elements is determined using this property. The nth power of an element h, denoted hs , is defined to be the union of all expressions of n times of h, in which the parentheses are put in all possible ways. An Hv -group (H, ?) is called cyclic with finite period respect to h ∈ H, if there exists a positive integer s such that H = h1 ∪ h2 ∪ ... ∪ hs . The minimum of such s is called the period of the generator h. If all generators have the same period, then H is cyclic with period. If there exists h ∈ H and s positive integer, the minimum one, such that H = hs , then H is called single-power cyclic and h is a generator with single-power period s. The cyclicity in the infinite case is defined similarly. Thus, for example, the Hv -group (H, ?) is called single-power cyclic with infinite period with generator h if every element of H belongs to a power of h and there exists s0 ≥ 1 such that for all s ≥ s0 we have h1 ∪ h2 ∪ ... ∪ hs−1 ⊂ hs . An element a ∈ H is called idempotent element if a2 = a. 3.2

Enumeration of Hv -groups

In [121], it is proved that all the quasi-hypergroups with two elements are Hv -groups. It is also proved that up to the isomorphism there are exactly 20 different Hv -groups. Theorem 3.1. All quasihypergroups with two elements are Hv -groups.

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Proof. One can see that on a set with two elements H = {a, b}, exactly eight hypergroups, up to isomorphism, can be defined. Besides the eight hypergroups, there are twelve more quasihypergroups, i.e., the reproduction axiom is valid. It is easy to see that the weak associativity is also valid for them, so they are Hv -groups. Therefore, the twenty Hv -groups, up to isomorphism, defined on a set H = {a, b} are given in the following table. H1 a b b a H11 b H a b

H2 H3 H4 H H H a H a a a H b b b H12 H H b a

H13 H b H a

H14 H a a H

H5 a H H b

H6 H H H a

H7 H H H b

H8 H9 H10 H H a H b H H b b H a a

H15 H a b H

H16 a H H a

H17 H H a H

H18 H H b H

H19 H b a H

H20 b H H a

In this table in each column the Hv -groups H1 , H2 , . . . , H20 are presented. The hyperproduct a ◦ a, a ◦ b, b ◦ a and b ◦ b are presented on the first, second, third and forth row, respectively. In the above table, the first eight Hv -groups are hypergroups. Moreover, all the above, expect the H15 and H19 are weak commutative.  Definition 3.4. An Hv -group is called Hb -group if it contains an operation which defines a group. In the reverse case, given a group, every greater hyperoperation defines an Hv -group or more precisely an Hb -group. Example 3.4. Examples 3.2 and 3.3 are examples of Hb -groups. In a given hypergroupoid, it is laborious to check the weak associativity. In the following, some properties are presented to reduce the cases of the triples of elements for which one needs to check the weak associativity. Proposition 3.1. Let (H, ◦) be a weak commutative hypergroupoid. Then, a ◦ (b ◦ a) ∩ (a ◦ b) ◦ a 6= ∅, for all a, b ∈ H. Proof. Since (H, ◦) is weak commutative, consider z ∈ a ◦ b ∩ b ◦ a. Then, z ◦ a ∩ a ◦ z 6= ∅. Therefore a ◦ (b ◦ a) ∩ (a ◦ b) ◦ a 6= ∅. 

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Proposition 3.2. Let (H, ◦) be a commutative (strong) hypergroupoid. Then, for all a, b ∈ H we have a ◦ (a ◦ b) ∩ (a ◦ a) ◦ b 6= ∅ implies b ◦ (a ◦ a) ∩ (b ◦ a) ◦ a 6= ∅. Proof. It is straightforward. Since b ◦ (a ◦ a) = (a ◦ a) ◦ b and (b ◦ a) ◦ a = a ◦ (b ◦ a) = a ◦ (a ◦ b), it follows that b ◦ (a ◦ a) ∩ (b ◦ a) ◦ a 6= ∅.  Remark 3.1. From Propositions 3.1 and 3.2 we see that the cases one has to check with respect to the weak associativity are: (1) If (H, ◦) is weak commutative, then for a, b ∈ H with a 6= b, check the triples (a, a, b), (a, b, b), (b, a, a), (b, b, a) and every triple with three different elements. (2) If (H, ◦) is commutative (strong), then for a, b ∈ H with a 6= b, check the triples (a, a, b), (b, b, a) and every triple with three different elements. Recall that an element e of a hypergroupoid (H, ◦), is called a unit if h ∈ e ◦ h ∩ h ◦ e, for all h ∈ H. If h = e ◦ h = h ◦ e, for all h ∈ H, then e is called a scalar unit. Proposition 3.3. Let e be a unit of a hypergroupoid (H, ◦). If x, y, z ∈ H, such that e ∈ {x, y, z}, then (x ◦ y) ◦ z ∩ x ◦ (y ◦ z) 6= ∅. Proof. Suppose that x = e. Then, we have y ◦ z ⊆ (e ◦ y) ◦ z and y ◦ z ⊆ e ◦ (y ◦ z). Thus, we obtain (e ◦ y) ◦ z ∩ e ◦ (y ◦ z) 6= ∅. Similarly, one can check the other cases.  In this paragraph we find all Hv -groups with three elements which contain a scalar unit element e. Therefore, the set of all Hv -groups defined on the set H = {e, a, b} up to the isomorphism is determined. Notice that the number of all hypergroupoids (non-degenerate) with a unit element, on a set with three elements, is (23 − 1)4 = 2401. Problem 3.1. Find all Hv -groups with three elements which contain a scalar unit. Therefore, for H = {e, a, b}, consider the multiplication table ◦ e a b e e a b a a 1 2 b b 3 4

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and then find all the fours, of non-empty subsets of H, in the place of the four (1, 2, 3, 4), such that the reproduction axiom and the weak associativity are valid. In the set H only one automorphism can be defined: f : e ↔ e, a ↔ b. So, two Hv -groups, (H, ◦) and (H, ?), are isomorphic if a ? a = f (b ◦ b), a ? b = f (b ◦ a), b ? a = f (a ◦ b) and b ? b = f (a ◦ a). Thus, the fours (a ◦ a, a ◦ b, b ◦ a, b ◦ b) and (f (b ◦ b), f (b ◦ a), f (a ◦ b), f (a ◦ a)) are isomorphic. In other words, instead of the table 12 43 , take the table 34 21 and then replace a by b and b by a. The result is the isomorphic Hv -group to the first one. Theorem 3.2. Let H = {e, a, b} be a set with three elements. Consider the set U of all quasihypergroups (H, ◦) defined in H, with e to be the scalar unit. Then the subset of all with all not weak associative quasihypergroups is described as follows. Denote by a four (a ◦ a, a ◦ b, b ◦ a, b ◦ b) the only one product needed to be defined. Then the fours which are not weak associative, with the corresponding isomorphic to the fours, are the following: (e, {e, b}, {e, b}, a) ∼ = (b, {e, a}, {e, a}, e), ∼ (b, H, {e, a}, e), (e, {e, b}, H, a) = (e, H, {e, b}, a) ∼ = (b, {e, a}, H, e), ({e, b}, e, a, {e, a}) ∼ = ({e, b}, b, e, {e, a}), ({e, b}, e, a, H) ∼ = (H, b, e, {e, a}), ∼ ({e, b}, {a, b}, e, {e, a}), ({e, b}, e, {a, b}, {e, a}) = ({e, b}, e, {a, b}, H) ∼ (H, {a, b}, e, {e, a}), = ∼ ({e, b}, {e, b}, a, {e, a}) = ({e, b}, b, {e, a}, {e, a}), ({e, b}, {e, b}, a, H) ∼ = (H, b, {e, a}, {e, a}), ({e, b}, e, b, {e, a}) ∼ = ({e, b}, a, e, {e, a}), ({e, b}, e, b, H) ∼ = (H, a, e, {e, a}), ({e, b}, {e, a}, b, {e, a}) ∼ = ({e, b}, a, {e, b}, {e, a}), ({e, b}, a, b, {e, a}) ({e, b}, a, b, H) ∼ = (H, a, b, {e, a}), ({e, b}, a, {e, b}, H) ∼ = (H, {e, a}, b, {e, a}), ({e, b}, b, a, {e, a}) ({e, b}, b, a, H) ∼ = (H, b, a, {e, a}).

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Proof. To prove this theorem the following algorithm was applied: In position 1 of the four (1, 2, 3, 4), the sets {e}, {a}, {b}, {e, a}, {e, b}, {a, b} and H, were consequently put. Then for each case, positions 2, 3, 4 were completed in order to obtain small hyperoperations such that the reproduction axiom is valid. In each case it was checked if the weak associativity was valid taking into account Remark 3.1. If the weak associativity was valid, then the result was considered a “minimal” Hv -group. If the weak associativity was not valid, then the sets in positions 2, 3, 4 were enlarged by adding new elements. Again the weak associativity was checked and this algorithm was continued. The procedure was stopped if the weak associativity was valid. In each result, whether it was weak associative or not, the isomorphic, reproductive, hyperoperation was obtained and they were used in the next steps. In each enlargement of the hyperoperations it was checked if a weak associative hyperoperation from the ones obtained previously, was already contained. The results are the ones written in the theorem.  Within the same procedure the following, “complement”, theorem is proved. Theorem 3.3. The set of all Hv -groups, with a scalar unit e, defined on a set with three H = {e, a, b} is the following: Every greater four (a ◦ a, a ◦ b, b ◦ a, b ◦ b) than the ones, with the corresponding isomorphic ones, defines an Hv -group in H. M1 : (e, b, b, {e, a}) ∼ = ({e, b}, a, a, e), M2 : (e, {e, b}, {e, b}, {a, b}) ∼ = ({a, b}, {e, a}, {e, a}, e), M3 : (e, H, H, a) ∼ = (b, H, H, e), M4 : (e, {a, b}, {a, b}, e), M5 : (e, H, H, b) ∼ = (a, H, H, e), M6 : (a, {e, b}, {e, b}, a) ∼ = (b, {e, a}, {e, a}, b), M7 : (a, H, H, b), M8 : (b, e, e, a), M9 : ({e, b}, {e, a}, b, H) ∼ = (H, a, {e, b}, {e, a}), M10 : ({e, b}, b, {e, a}, H) ∼ = (H, {e, b}, a, {e, a}). An Hv -group is called minimal if it contains no other Hv -group defined on the same set H. For a given Hv -group, not minimal, one can remove an element from one product x ◦ y, with x, y ∈ H, and then to check if the remaining hyperproduct gives an Hv -group. So, a series can be obtained, and if H is finite, leads to a minimal Hv -group. Therefore, the above set of

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fours, is the set of minimal ones of all Hv -groups with three elements which have a scalar unit. Remark 3.2. One can classify all the minimals according to their fundamental group H/β ∗ . For this classification we obtain the following: The Hv -group M8 has H/β ∗ = {{e}, {a}, {b}}, so it has three elements in the fundamental group. The Hv -group M1 has H/β ∗ = {{e, a}, {b}}, so it has two elements in the fundamental groups. The rest of the minimals have one element in the fundamental group. Remark 3.3. The Hv -group M1 is a very thin hypergroup. There is also the very thin Hv -group of the second kind. This has the four (e, b, b, H) and contains the M1 . The other very thin Hv -groups can be obtained from the group M8 by setting in one of the positions 1, 2, 3 or 4 greater subsets. These are Hv -groups and, under the isomorphism, we have the fours: ({e, b}, e, e, a), ({a, b}, e, e, a), (H, e, e, a), (b, {e, a}, e, a), (b, {e, b}, e, a), (b, H, e, a). Therefore, in the set of all Hv -groups on H = {e, a, b}, there are exactly 8 very thin hypergroups. Remark 3.4. From the minimals only the M9 and M10 are not commutative. Enlarging them, one can obtain only two more non-commutative in the set H = {e, a, b}. Therefore, in the set of all Hv -groups in H, with e scalar unit, there are only four non-commutative Hv -groups: the M9 , M10 and (H, {e, a}, b, H), (H, b, {e, a}, H).

3.3

Fundamental relation on Hv -groups

Let (H, ◦) be an Hv -group. The relation β ∗ is the smallest equivalence relation on H such that the quotient H/β ∗ is a group. β ∗ is called the fundamental equivalence relation on H. The relation β ∗ was introduced on hypergroups by Koskas [64] and studied mainly by Corsini [16] and Davvaz [30, 29]. Vougiouklis in [120] studied the relation β ∗ on Hv -groups. If U denotes the set of all finite products of elements of H, then a relation β can be defined on H whose transitive closure is the fundamental relation β ∗ . The relation β is defined as follows: for x and y in H we write xβy if and only if {x, y} ⊆ u for some u ∈ U. The following theorem was proved by Vougiouklis [120].

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Theorem 3.4. The fundamental relation β ∗ is the transitive closure of the relation β. ˆ Proof. Suppose that βˆ is the transitive closure of β and β(a) is the class of the element a ∈ H. First, we prove that the quotient set H/βˆ is a group. The product in H/βˆ is defined in the usual manner ˆ ˆ = {β(c) ˆ ˆ ˆ β(a) β(b) | c ∈ β(a) ◦ β(b)}, ˆ ˆ for all a, b ∈ H. Take a0 ∈ β(a) and b0 ∈ β(b). Then, we have ˆ ⇔ there exist x1 , ..., xm+1 with x1 = a0 , xm+1 = a and u1 , ..., um ∈ U a0 βa such that {xi , xi+1 } ⊆ ui , for i = 1, ..., m and ˆ ⇔ there exist y1 , ..., yn+1 with y1 = b0 , yn+1 = b and v1 , ..., vn ∈ U b0 βb such that {yj , yj+1 } ⊆ vj , for j = 1, ..., n. Thus, we obtain {xi , xi+1 } ◦ y1 ⊆ ui ◦ v1 , for i = 1, ..., m − 1,

(3.1)

xm+1 ◦ {yj , yj+1 } ⊆ um ◦ vj , for j = 1, ..., n.

(3.2)

Therefore, we obtain ui ◦v1 = ti ∈ U, for i = 1, ..., m−1 and um ◦vj = tm+j−1 ∈ U, for j = 1, ..., n. So, tk ∈ U, for all k ∈ {1, ..., m + n − 1}. Now, pick up any elements z1 , ..., zm+n such that zi ∈ xi ◦ y1 , for i = 1, ..., m and zm+j ∈ xm+1 ◦ yj+1 , for j = 1, ..., n. By using Equations 3.1 and 3.2, we have {zk , zk+1 } ⊆ tk , for k = 1, ..., m + n − 1. So, every element z1 ∈ x1 ◦ y1 = a0 ◦ b0 is βˆ equivalent to every element ˆ β(b) ˆ is singleton. Hence, we zm+n ∈ xm+1 ◦ yn+1 = a ◦ b. Therefore, β(a) can write ˆ ˆ = β(c), ˆ ˆ ˆ β(a) β(b) for all c ∈ β(a) ◦ β(b). Moreover, since ◦ is weak associative, it follows that is associative, and consequently, H/βˆ is a group. Now, let σ be an equivalence relation in H such that H/σ is a group. Denote σ(a) the class of a. Then, σ(a) σ(b) is singleton for all a, b ∈ H, i.e., σ(a) σ(b) = σ(c), for all c ∈ σ(a) ◦ σ(b).

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But also, for every a, b ∈ H and A ⊆ σ(a), B ⊆ σ(b), we have σ(a) σ(b) = σ(a ◦ b) = σ(A ◦ B). So, the above relation is valid for all finite products. That means that σ(x) = σ(u), for all u ∈ U and x ∈ u. Thus, for every a ∈ H, we obtain x ∈ β(a) ⇒ x ∈ σ(a). But σ is transitively closed, so we have ˆ x ∈ β(a) ⇒ x ∈ σ(a). Thus, the relation βˆ is the smallest equivalence relation in H such that H/βˆ is a group, i.e., βˆ = β ∗ .  We can rewrite the definition of β ∗ on H as follows. aβ ∗ b if and only if there exist z1 , ..., zn+1 ∈ H with z1 = a, zn+1 = b and u1 , ..., un ∈ U such that {zi , zi+1 } ⊆ ui (i = 1, ..., n). Suppose that β ∗ (a) is the equivalence class containing a ∈ H. Then the product on H/β ∗ is defined as follows: β ∗ (a) β ∗ (b) = β ∗ (c) for all c ∈ β ∗ (a) ◦ β ∗ (b). Since H/β ∗ is a group, one can say that (H, ) is by virtue a group. This is why the hyperstructure H is called Hv -group. Problem 3.2. Freni in [55] proved that for hypergroups β ∗ = β. We do not know yet if this equality is valid for Hv -groups. Example 3.5. Consider the Hv -group defined in Example 3.2. From the sum [. . . (0 ⊕ m) ⊕ . . . ⊕ m] ⊕ m = {0, m, 2m, . . . , (n − 1)m}, | {z } n−1 times

it is obtained that β(0) = {0, m, 2m, . . . , (n − 1)m}. Similarly, for every 0 < k < n − 1 we have k ⊕ [. . . (0 ⊕ m) ⊕ . . . ⊕ m] ⊕ m = {k, k + m, k + 2m, . . . , k + (n − 1)m}. {z } | n−1 times

So β(k) = k + mZ. That means that β ∗ = β and Zmn /β ∗ ∼ = Zm .

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Example 3.6. Consider the Hv -group defined in Example 3.3. We have Zn /β ∗ ∼ = Zm × Zm × . . . × Zm . 1

2

n

Let (H, ◦) be an Hv -group. An element x ∈ H is called single if its fundamental class is singleton, i.e., β ∗ (x) = {x}. Denote by SH the set of all single elements of H. Theorem 3.5. Let (H, ◦) be an Hv -group and x ∈ SH . Let a ∈ H and take any element v ∈ H such that x ∈ a ◦ v. Then, β ∗ (a) = {h ∈ H | h ◦ v = x}. Proof. We have x ∈ a ◦ v. So, x = a ◦ v which means x = β ∗ (a) ◦ β ∗ (v). Thus, for all h ∈ β ∗ (a) we have h ◦ v = x. Conversely, let x = h ◦ v. Then, x = β ∗ (h) ◦ β ∗ (v). Since H/β ∗ is a group, we have β ∗ (h) = x ◦ (β ∗ (v))−1 = β ∗ (a), so h ∈ β ∗ (a).  Theorem 3.6. Let (H, ◦) be an Hv -group and x ∈ SH . Then, the core of H is ωH = {u | u ◦ x = x} = {u | x ◦ u = x}. Proof.

It is obvious.



Theorem 3.7. Let (H, ◦) be an Hv -group and x ∈ SH . Then, x ◦ y = β ∗ (x ◦ y) and y ◦ x = β ∗ (y ◦ x), for all y ∈ H. Proof. Suppose that for some y there exist t ∈ x ◦ y and t0 ∈ β ∗ (t) such that t0 6∈ x ◦ y. From the reproductivity, there exists v 6= x in H such that t0 ∈ v ◦ y. So, we have β ∗ (v) ◦ β ∗ (y) = β ∗ (t0 ). On the other hand, β ∗ (x) ◦ β ∗ (y) = β ∗ (t). Thus, β ∗ (v) ◦ β ∗ (y) = β ∗ (x) ◦ β ∗ (y). Hence, β ∗ (v) = β ∗ (x). Since x ∈ SH , β ∗ (x) = {x}. Thus, v = x which is a contradiction.  The previous theorem proves that the product of a single element with any arbitrary element is always a whole fundamental class. Suppose that (H, ◦) is an Hv -group such that SH is non-empty. Then, the only greater hyperoperations ◦ < ? for which the Hv -groups (H, ?) contain single elements are the ones with the same fundamental group, since the fundamental classes are determined from the products of a single element with the elements of the group. On the other hand, a less hyperoperation  < ◦ can have the same set SH if only in the products of non-single elements the  is less than ◦. Finally, if ρ and σ are equivalence relations with ρ < σ such that H/ρ and H/σ are non-equal groups, then they can not have both single elements.

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Corollary 3.1. Let (H, ◦) be an Hv -group. If SH is non-empty, then β ∗ = β. Proof. It is obvious, since all the β ∗ -classes can be obtained as products of two elements one of which is single.  Using the fundamental equivalence relation, one can define semidirect hyperproducts of Hv -groups, see [120]. Theorem 3.8. Let (A, ◦) and (B, ?) be two Hv -groups, β ∗ be the fundamental relation on B and let’s take the group AutA. Consider any homomorphism b: B/β ∗ → AutA ∗ (b). β ∗ (b) 7→ β[ ∗ (b) = b We denote β[ b. In A × B, we define a hyperproduct as follows: (a, b) · (c, d) = {(x, y) | x ∈ a ◦ bb(c), y ∈ b ? d} = (a ◦ bb(c), b ? d). Then, the set A × B becomes an Hv -group.

From the fundamental property, for all x, y ∈ B we have x[ ?y =x byb = zb, for all z ∈ x ? y. Now, suppose that (a1 , b1 ), (a2 , b2 ), (a3 , b3 ) ∈ A × B. Then,   (a1 , b1 ) · (a2 , b2 ) · (a3 , b3 ) S = (x, y) · (a3 , b3 )

Proof.

x∈a1 ◦bc 1 (a2 ) y∈b1 ?b2

=

S x∈a1 ◦bc 1 (a2 ) y∈b1 ?b2

{(z, w) | z ∈ x ◦ yb(a3 ), w ∈ y ? b3 }

b \ = {(z,  w) | z ∈ (a1 ◦ b1 (a2 )) ◦ b1 ? b2 (a3 ), w∈ (b1 ? b2 ) ? b3 } = (a1 ◦ bb1 (a2 )) ◦ b\ 1 ? b2 (a3 ), (b1 ? b2 ) ? b3 . On the other hand, we have   (a1 , b1 ) · (a2 , b2 ) · (a3 , b3 ) S = (a1 , b1 ) · (x, y) x∈a2 ◦bc 2 (a3 ) y∈b2 ?b3

=

S

{(z, w) | z ∈ a1 ◦ bb1 (x), w ∈ b1 ? y}

x∈a2 ◦bc 2 (a3 ) y∈b2 ?b3

= {(z, w) | z ∈ a1 ◦ bb1 (a2 ◦ bb2 (a3 )), w ∈ b1 ? (b2 ? b3 )} = {(z, w) | z ∈ a1 ◦ (bb1 (a2 ) ◦ bb1 (bb2 (a3 ))), w ∈ b1 ? (b2 ? b3 )} b \ = {(z,  w) | z ∈ a1 ◦ (b1 (a2 ) ◦ b1 ? b2 (a3 )), w ∈ b1 ? (b2 ? b3 )} b = a1 ◦ (b1 (a2 ) ◦ b\ 1 ? b2 (a3 )), w ∈ b1 ? (b2 ? b3 )

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But A and B are Hv -groups, so     b \ a1 ◦ bb1 (a2 ) ◦ b\ 1 ? b2 (a3 ) ∩ a1 ◦ b1 (a2 ) ◦ b1 ? b2 (a3 ) 6= ∅ and (b1 ? b2 ) ? b3 ∩ b1 ? (b2 ? b3 ) 6= ∅. Therefore, we obtain     (a1 , b1 ) · (a2 , b2 ) · (a3 , b3 ) ∩ (a1 , b1 ) · (a2 , b2 ) · (a3 , b3 ) 6= ∅, i.e., the weak associativity is valid. It is also easy to see that the reproduction axiom is valid too.



The Hv -group defined in Theorem 3.8 is called semidirect hyperproduct of b A and B corresponding to b and it is denoted by A×B. Lemma 3.1. Let (A, ◦) and (B, ?) be two Hv -groups, β ∗ the fundamental relation on B and b: B/β ∗ → AutA. Then, bb(β ∗ (a)) = β ∗ (bb(a)), for all a ∈ A, b ∈ B. Proof. 3.4

It is straightforward.



Reversible Hv -groups

Let (H, ◦) be an Hv -group with (left, right) identity elements. Then, H is called (left, right) reversible in itself when any relation c ∈ a ◦ b implies the existence of a left inverse a0 of a and a right inverse b0 of b such that b ∈ a0 ◦ c and a ∈ c ◦ b0 . An Hv -group (H, ◦) is called feebly quasi-canonical if it is regular, reversible and satisfies the following conditions: For each a ∈ H, if a0 , a00 are inverses of a, then for each x ∈ H, we have a0 ◦ x = a00 ◦ x and x ◦ a0 = x ◦ a00 . A feebly quasi-canonical Hv -group H is called feebly canonical if it is strongly commutative. In the rest of this section, we study a wide class of reversible Hv -groups that investigated by Spartalis [102]. Let (H, ◦) be an Hv -group with left or right identity elements. We denote by El (respectively, Er ) the set of left (respectively, right) identities. Hence, the set of all identities of the Hv -group H is E = El ∪ Er . We also denote by il (x, e) the set of all left inverses of an element x with respect to the identity e of E, i.e., il (x, e) = {x0 ∈ H | e ∈ x0 ◦ x}. Consequently,

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il (x) =

S

il (x, e) is the set of all left inverses of the element x. If A is a S non-empty subset of H, then il (A) = il (a). Similar notations hold for e∈E

a∈A

the right inverses, too. Definition 3.5. Let (H, ◦) be an Hv -group. Then, H is called left completely reversible in itself if, for all a, b ∈ H, it satisfies the following condition: (1) c ∈ a ◦ b implies b ∈ u ◦ c, for all u ∈ il (a). Similarly, H is called right completely reversible in itself if (2) c ∈ a ◦ b implies a ∈ c ◦ v, for all v ∈ ir (b). Lemma 3.2. If H is left completely reversible, then for each el ∈ El , er ∈ Er , e ∈ E = El ∪ Er and a ∈ H, we have (1) il (el ) = El ; (2) il (a) = il (a, er ); (3) il (il (il (a, e))) ⊆ il (a, e). Proof. (1) Suppose that el ∈ El . Clearly, El ⊆ il (el , el ) ⊆ il (el ). Moreover, let u ∈ il (el ). Since for all x ∈ H, x ∈ el ◦ x, it follows that x ∈ u ◦ x. Therefore, u ∈ El , i.e., il (el ) ⊆ El and so il (el ) = El . (2) Suppose that a ∈ H and er ∈ Er . Obviously, il (a, er ) ⊆ il (a). Conversely, assume that u ∈ il (a). From the relation a ∈ a ◦ er it follows that er ∈ u ◦ a, that is u ∈ il (a, er ) and so il (a) = il (a, er ). (3) Suppose that a ∈ H, e ∈ E and u ∈ il (a, e). Since e ∈ u ◦ a, we have that a ∈ w ◦ e for all w ∈ il (u) and e ∈ v ◦ a for all v ∈ il (il (u)). Consequently, v ∈ il (a, e) and so il (il (il (a, e))) ⊆ il (a, e).  Proposition 3.4. Let H be left completely reversible, a ∈ H and e ∈ E such that a ∈ il (il (a, e)). Then, the following conditions hold: (1) il (a, e) = il (a) = il (il (il (a, e))); (2) If u is an inverse of a with respect to e for which the hypothesis holds, i.e., a ∈ il (u), then il (il (a)) = il (u) and, moreover, for all x ∈ H, v ∈ il (a) we have u ◦ x ⊆ v ◦ x. Proof. (1) Suppose that a ∈ il (il (a, e)). Then, il (a) ⊆ il (il (il (a, e))). According to Lemma 3.2(3), we have il (a) ⊆ il (il (il (a, e))) ⊆ il (a, e) ⊆ il (a).

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Thus, il (a, e) = il (a) = il (il (il (a, e))). (2) Suppose that u is an inverse of the element a with respect to e such that a ∈ il (u). Then, there exists an identity e0 ∈ E such that a ∈ il (u, e0 ). Therefore, u ∈ il (a) ⊆ il (il (u, e0 )). According to (1), we have il (u, e0 ) = il (u) = il (il (il (u))). Consequently, u ∈ il (a) ⊆ il (il (u)). So, il (u) ⊆ il (il (a)) ⊆ il (u). Hence, il (il (a)) = il (u). Finally, let x ∈ H and y ∈ u ◦ x. From the reversibility of H, it follows that x ∈ a ◦ y and y ∈ v ◦ x, for all v ∈ il (a). Thus, u ◦ x ⊆ v ◦ x.  Notice that if a ∈ H, a ∈ E and a ∈ il (a, e), then the following relation is satisfied: il (a, e) = il (a) = il (il (il (a, e))) = il (il (a)). Proposition 3.5. Let H be left completely reversible and a ∈ H such that T a∈ il (u). Then, for all x ∈ H, u, v ∈ il (a) and er ∈ Er we have u∈il (a)

(1) u ◦ x = v ◦ x; (2) If Er is non-empty, then il (a) ⊆ u ◦ er ⊆ ir (a, er ). Proof. (1) We observe that the assumption of Proposition 3.33 are satisfied for all e ∈ E and for all u ∈ il (a). Therefore, for all x ∈ H and u, v ∈ il (a), we have u ◦ x ⊆ v ◦ x and so u ◦ x = v ◦ x. S (2) Suppose that er ∈ Er . Then, we have il (a) ◦ er = u ◦ er and u∈il (a)

because of (1), it follows that il (a) ◦ er = u ◦ er , for all u ∈ il (a). Thus, il (a) ⊆ u ◦ er . Moreover, let y ∈ u ◦ er . Since a ∈ il (u), it follows that er ∈ a ◦ y, that is, y ∈ ir (a, er ).  If (H, ◦) is a strongly commutative Hv -group, then from the first condition of previous proposition we conclude that the concepts of left (respectively, right) completely reversible Hv -group and the feebly canonical Hv -group are identical. Theorem 3.9. If H is completely reversible, then it is a feebly quasicanonical Hv -group. Proof. At the first, we prove that all identities and inverses in H are two sided. Let a ∈ H, e ∈ E, u ∈ il (a, e) and v ∈ ir (a, e). Since e ∈ a ◦ v, it follows that v ∈ u ◦ e. Since e ∈ ir (e), u ∈ v ◦ e. Since a ∈ il (v, e), we have e ∈ a ◦ u. Therefore, il (a, e) ⊆ ir (a, e). Similarly, we obtain ir (a, e) ⊆ il (a, e). Hence, il (a, e) = ir (a, e). Now, suppose that el ∈ El

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and a ∈ H. Then, there exists u ∈ H such that el ∈ u ◦ a ∩ a ◦ u and so a ∈ a ◦ el . Thus, El ⊆ Er . In the same manner, every right identity is also a left identity and hence El = Er . Consequently, H is regular and reversible. Moreover, by Proposition 3.5(1) for the left (right) completely reversible Hv -groups, we have that, for a ∈ H, if u, v are inverses of a, then for all x ∈ H, u ◦ x = v ◦ x and x ◦ u = x ◦ v. Therefore, H is a feebly quasi-canonical Hv -group.  In what follows, we consider Hv -groups with only two sided identities and inverses. The following relation introduced by De Salvo [50] and studied by Spartalis [102]. Consider the binary relation ∼ on H as follows: x ∼ y ⇔ there exists z ∈ H such that {x, y} ⊆ i(z), for all x, y ∈ H. If H is left completely reversible, then by using Proposition 3.5(1) we obtain that ∼ is an equivalence relation. In the quotient set ˆ = H/ ∼ we define the following hyperoperation between classes in the H usual manner x ˆ yˆ = {w ˆ |w∈x ˆ ◦ yˆ}, for all x, y ∈ H, where x ˆ is the equivalence class containing x. According to Proposition 3.5(1), this hyperoperation is equivalent to the following one x ˆ yˆ = {w ˆ | w ∈ x ◦ y}. ˆ ) is an Hv -group. Lemma 3.3. (H, Proof.

It is straightforward.



ˆ Proposition 3.6. Let H be left completely reversible and e ∈ E. Then, H ˆ is a left reversible in itself Hv -group and eˆ = E is the unique identity of H. ˆ ˆ Moreover, eˆ is a right scalar, H is regular and each x ˆ ∈ H has a unique inverse. Proof. Suppose that a ∈ E. By Lemma 3.2 (1) we have i(e) = E. So, E ⊆ eˆ. Further, if x ∈ eˆ, then there exists y ∈ H such that {x, e} ⊆ i(y). Since y ∈ i(e), it follows that x ∈ E. Hence, eˆ ⊆ E. Thus, eˆ = E. ˆ x Obviously, for all x ˆ ∈ H, ˆ∈x ˆ eˆ ∩ eˆ x ˆ. Now, suppose that sˆ is a left ˆ Then, eˆ ∈ sˆ eˆ∩ eˆ tˆ and so there exist ˆ identity or t is a right identity of H. e0 ∈ E and t0 ∈ tˆ such that e ∈ s ◦ e0 ∩ e ◦ t0 . Therefore, s, t0 ∈ i(e) = i(e0 ) ˆ and hence sˆ = eˆ = tˆ. Finally, for all x ˆ ∈ H, x ˆ eˆ = {w ˆ | w ∈ x ◦ e = x ◦ E}.

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If w ∈ x◦e0 and e0 ∈ E, then for all u ∈ i(x), e0 ∈ u◦w. Thus, {w, x} ⊆ i(u) and so x ∼ w. Consequently, x ˆ eˆ = {ˆ x}. ˆ has a unique inverse, i.e., i(ˆ It is easy to prove that each x ˆ∈H x) = {ˆ a}, ˆ is a regular Hv -group. Finally, we show that where a ∈ i(x). Moreover, H ˆ is left reversible in itself. Suppose that x ˆ and zˆ ∈ x H ˆ, yˆ ∈ H ˆ yˆ. Then, 0 0 there exists y ∈ yˆ such that z ∈ x ◦ y and hence for all a ∈ i(x), y 0 ∈ a ◦ z. Since i(ˆ x) = {ˆ a}, it follows that yˆ ∈ a ˆ zˆ.  Now, let K be an Hv -subgroup of H. Let the left coset expansion H/K = {x ◦ K | x ∈ H} satisfies the following conditions: (1? ) for all x ∈ H, x ∈ x ◦ K, (2? ) for all x, y ∈ H, x ◦ K ∩ y ◦ K 6= ∅ implies x ◦ K = y ◦ K. It is easy to see that H/K becomes an Hv -group with respect to the usual hyperoperation: x ◦ K y ◦ K = {z ◦ K | z ∈ (x ◦ K) ◦ (y ◦ K)}, for all x, y ∈ H. A similar remark holds for the right coset expansion. Moreover, for the right coset expansion, we have the following proposition. Proposition 3.7. Let H be left completely reversible and K be an Hv subgroup of H. If K ∩ E 6= ∅ and for all x ∈ H, K ◦ (K ◦ x) ⊆ K ◦ x, then H/K = {K ◦ x | x ∈ H} is an Hv -group. Proof.

It suffices to prove the following conditions:

(1) for all x ∈ H, x ∈ K ◦ x, (2) for all x, y ∈ H, K ◦ x ∩ K ◦ y 6= ∅ implies K ◦ x = K ◦ y. Obviously, for all x ∈ H, x ∈ K ◦ x. Moreover, suppose that x, y ∈ H and z ∈ K ◦ x ∩ K ◦ y. Then, K ◦ z ⊆ K ◦ (K ◦ x) ⊆ K ◦ x. From z ∈ K ◦ x, it follows that z ∈ u ◦ x, where u ∈ K. Therefore, for all a ∈ i(u), x ∈ a ◦ z. Since i(u) ∩ K 6= ∅, we have x ∈ K ◦ z. Thus, K ◦ x ⊆ K ◦ z. Consequently, K ◦ z = K ◦ x. Similarly, K ◦ z = K ◦ y.  Theorem 3.10. Let H be left completely reversible and E be the set of identities and H/E be the left coset expansion of H with respect to E. Then the following conditions hold: (1) E is a total Hv -subgroup of H;

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(2) for all x ∈ H, x◦E = x ˆ, that is H/E is identical with the Hv -group ˆ H; (3) E is the smallest of the Hv -subgroup K of H such that H/K satisfies (1? ) and (2? ). Proof. By using Proposition 3.5(2) and Lemma 3.2(2), for all a ∈ H, e ∈ E and u ∈ i(a), we obtain i(a) = u ◦ a = i(a, e).

(3.3)

(1) Suppose that e ∈ E. Then, from the previous relation, we obtain i(e) = u ◦ e0 , for all u ∈ i(e) and e0 ∈ E. Moreover, according to Lemma 3.2(1), i(e) = E and hence E = e00 ◦ e0 , for all e0 , e00 ∈ E. Therefore, E is a total Hv -subgroup of H. S (2) By hypothesis H/E = {x ◦ E | x ∈ H} and x ◦ E = x ◦ e. e∈E

Suppose that x ∈ H, e ∈ E and u ∈ i(x). Then, applying (3.3) we obtain x ◦ E = i(u) = x ◦ e and hence H/E = {x ◦ e | x ∈ H}. Furthermore, for all x, y ∈ H we have the following x ∼ y ⇒ there exists z ∈ H such that {x, y} ⊆ i(z) ⇔ there exists z ∈ H, e ∈ E such that y ∈ x ◦ e = i(z) ⇔ y ∈ x ◦ e. ˆ Consequently, H/E is identical with the Hv -group H. (3) Suppose that K is an Hv -subgroup of H such that the left coset expansion H/K satisfies (1). Let e ∈ E. Then, e ∈ e ◦ K and so e ∈ e ◦ u, for some u ∈ K. Therefore, u ∈ i(e) and since i(e) = E we have K ∩ E 6= ∅. Finally, since E is a total Hv -subgroup of H, it follows that E ⊆ K.  3.5

A sequence of finite Hv -groups

This section deals with a sequence of Hv -structures. The results are obtained by Antampoufis and Dramalidis [7]. Firstly, they defined a hyperoperation on a set and studied, in the general case, the hyperstructure resulting. The hyperstructure is an Hv -group. The hyperoperation is defined in every finite hyperstructure using indices of the cyclic group Zn . The case of infinite order is separately studied with indices in N. They also studied the existence of identities, inverses elements and powers of the elements of the Hv -group. The hyperstructures of small order, provided with the particular hyperoperation are groups, those of greater order are

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hypergroups and then those of the greatest order lead to Hv -groups. A sequence of finite Hv -groups with common properties is created. Finally, they presented the motivating example. The hyperoperation is defined in a geometrical figure on R2 [6, 51], which is partitioned into a finite or infinite number of parts. The hyperoperation is defined in the sense of the boundary among the parts. A lot of properties of this geometrical Hv -structures are studied. Consider the set H = {α} ∪ {z | z = ar , r ∈ Zn }. On H we define the hyperoperation ◦ as follows: Definition 3.6. For every x, y ∈ H define ◦ : H × H → P ∗ (H): (x, y) 7→ x ◦ y such that x ◦ y = {α, ap−1 , ap+1 , ak−1 , ak+1 }, x = ap , y = ak , for all p, k ∈ Zn x ◦ α = α ◦ x = x, for all x ∈ H.

Some properties of the hyperoperation ◦: (1) Obviously, ◦ is commutative, i.e., x ◦ y = y ◦ x, for all x, y ∈ H. (2) According to Definition 3.6, α ◦ α = α and α ◦ ap = ap ◦ α = ap , for all p ∈ Zn . That means that α ◦ x = x ◦ α = x, for all x ∈ H, so the element α is scalar unit element. (3) αm = α, m ∈ N∗ . (4) Since α2 = α, it follows that the element α is also an idempotent element and obviously I◦ (α, α) = α. (5) |H| = n + 1. (6) H/β ∗ = H. (7) x · y = x2 ∪ y 2 , for all x, y ∈ H with (x, y) 6= (α, y) and (x, y) 6= (x, α). Proposition 3.8. I◦ (ap , α) = H \ {α}, for all p ∈ Zn . Proof. Suppose that ak ∈ I◦ (ap , α), where k, p ∈ Zn . Then, α ∈ ak ◦ ap and α ∈ ap ◦ ak . So, α ∈ {α, ak−1 , ak+1 , ap−1 , ap+1 } and α ∈ {α, ap−1 , ap+1 , ak−1 , ak+1 }. Since the previous relations are both true for every k, p ∈ Zn , we obtain I◦ (ap , α) = H \ {α}, for all p ∈ Zn .  S 2 S 2 Proposition 3.9. x = ak = H. x∈H

k∈Zn

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Proof. S k∈Zn

Notice that S a2k = {α, ak−1 , ak+1 } = {α, a−1 , a1 , a0 , a2 , a3 , . . . , an−1 , an } k∈Zn

= {α, a0 , a1 , a2 , a3 , . . . , an−1 } = H, since a−1 = an−1 and an = a0 . S 2 S 2 Also, since α2 = α, it follows that x = ak = H. x∈H

k∈Zn



Proposition 3.10. a2k ⊆ ap ◦ ak , for all k, p ∈ Zn . Proof.

For k, p ∈ Zn ,

ap ◦ ak = {α, ap−1 , ap+1 , ak−1 , ak+1 } ⊃ {α, ak−1 , ak+1 } = a2k . When p = k we get that ak ◦ ak = {α, ak−1 , ak+1 } = a2k . So, generally, a2k j ap ◦ ak , for all k, p ∈ Zn .  Proposition 3.11. The hyperstructure (H, ◦) is a commutative Hv -group. Proof.

For the reproduction axiom, if x = α, then S S x◦H =α◦H = (α ◦ h) = h = H = H ◦ α = H ◦ x. h∈H

h∈H

If x = ak , k ∈ Zn , then S S x ◦ H = ak ◦ H = (ak ◦ h) = (ak ◦ α) ∪ [ (ak ◦ ap )] h∈H

k∈Zn

= {ak } ∪ {α, ak−1 , ak+1 , a−1 , a0 , a1 , a2 , a3 , . . . , ak−1 , ak+1 , . . . , an−2 , an } = {α, a0 , a1 , a2 , a3 , . . . , an−2 , an−1 } = H, since a−1 = an−1 and an = a0 . Obviously H ◦ ak = H ◦ x = H, then x ◦ H = H ◦ x = H, for all x ∈ H. Since ◦ is commutative, we have to check only the following cases for the associativity: For p, k ∈ Zn , • α ◦ (α ◦ ak ) = α ◦ ak = ak and (α ◦ α) ◦ ak = α ◦ ak = ak . • α◦(ap ◦ak ) = {α, ap−1 , ap+1 , ak−1 , ak+1 } = ap ◦ak and (α◦ap )◦ak = ap ◦ ak . • ak ◦ (ak ◦ α) = ak ◦ ak = {α, ak−1 , ak+1 } and (ak ◦ ak ) ◦ α = {α, ak−1 , ak+1 }. So far, notice that if one or two elements α appear in the triples (x, y, z), then the equality appears for the associativity, i.e. x ◦ (y ◦ z) = (x ◦ y) ◦ z. Furthermore, we have to check the following two cases: For p, k, m ∈ Zn ,

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• ak ◦ (ak ◦ ap ) = ak ◦ {α, ak−1 , ak+1 , ap−1 , ap+1 } = {ak , ak−1 , ak+1 , ak−2 , ak+2 , ap , ap−2 , ap+2 } and (ak ◦ ak ) ◦ ap = {α, ak−1 , ak+1 } ◦ ap = {ap , ak−2 , ak , ap−1 , ap+1 , ak+2 }. So, ak ◦ (ak ◦ ap ) ∩ (ak ◦ ak ) ◦ ap = {ap , ak , ak−2 , ak+2 } = 6 ∅. • ak ◦ (ap ◦ am ) = {ak , ak−1 , ak+1 , ap , ap−2 , ap+2 , am , am−2 , am+2 } and (ak ◦ ap ) ◦ am = {am , ak−2 , ak , am−1 , am+1 , ak+2 , ap , ap−2 , ap+2 }. Thus, ak ◦ (ap ◦ am ) ∩ (ak ◦ ap ) ◦ am = {ap , ak , am , ap−2 , ap+2 } = 6 ∅. Therefore, generally x ◦ (y ◦ z) ∩ (x ◦ y) ◦ z 6= ∅, for all x, y ∈ H.



Proposition 3.12. apk ⊆ am k , for all k ∈ Zn , p, m ∈ N, p, m > 1 ⇔ m = p. Proof. a2k

Since ◦ is commutative, for k ∈ Zn ,

= {α, ak−1 , ak+1 };

a3k = a2k ◦ ak = {α, ak−2 , ak−1 , ak , ak+1 , ak+2 }; a4k = a3k ◦ ak ∪ a2k ◦ a2k = {α, ak−3 , ak−2 , ak−1 , ak , ak+1 , ak+2 , ak+3 } ∪ {α, ak−2 , ak−1 , ak , ak+1 , ak+2 } = {α, ak−3 , ak−2 , ak−1 , ak , ak+1 , ak+2 , ak+3 }. So, by induction, for p, m ∈ N, p, m > 1 we obtain apk = {α, ak−p+1 , ak−p+2 , . . . , ak+p−2 , ak+p−1 } am k = {α, ak−m+1 , ak−m+2 , . . . , ak+m−2 , ak+m−1 }. Let apk ⊆ am k ⇒ k − m + 1 5 k − p + 1 and k + m − 1 = k + p − 1 ⇒ m = p. Now, let m = p ⇒ k − m + 1 5 k − p + 1 and k + m − 1 = k + p − 1, k ∈ Zn . Since k − p + 1, k + p − 1, k − m + 1, k + m − 1 ∈ Zn , it follows that {ak−p+1 , ak−p+2 , . . . , ak+p−1 } ⊆ {ak−m+1 , ak−m+2 , . . . , ak+m−1 }. Hence, we have {α, ak−p+1 , ak−p+2 , . . . , ak+p−1 } ⊆ {α, ak−m+1 , ak−m+2 , . . . , ak+m−1 } So, apk ⊆ am k .



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m−1 Corollary 3.2. am ◦ ak , for all k ∈ Zn , m ∈ N, m > 1. k = ak

Proof. For k ∈ Zn , m ∈ N, m > 1, since (◦) is commutative: m/2 m/2 m−1 am ◦ ak ∪ am−2 ◦ a2k ∪ . . . ∪ ak ◦ ak , if m = 2p, p ∈ N; k = ak k m+1/2 m−1/2 m−1 am ◦ ak ∪ am−2 ◦ a2k ∪ . . . ∪ ak ◦ ak , if m = 2p + 1, p ∈ N k = ak k According to Proposition 3.12, the power m − 1 is the greatest one in both cases, so m−1 am ◦ ak , for all k ∈ Zn , m ∈ N, m > 1. k = ak



Proposition 3.13. The (H, ◦) is a single-power cyclic Hv -group and each element ak ∈ H, k ∈ Zn , n = 2, 3 is a generator with period 3. Proof.

Let x = ak , k ∈ Zn be any element of H. Then, a2k = {α, ak−1 , ak+1 }, a3k = a2k ◦ ak = {α, ak−2 , ak−1 , ak , ak+1 , ak+2 }.

For n = 2, H = {α, a0 , a1 }: Notice that |H| = 3 and |a2k | = 3 but ak ∈ / a2k , so a2k 6= H. Since |a3k | = 6 and ak ∈ a3k , it follows that a3k = H. That means that ({α, a0 , a1 }, ◦) is single-power cyclic Hv -group and each element ak ∈ H, k ∈ Z2 is generator with period 3. For n = 3, H = {α, a0 , a1 , a2 }: Notice that |H| = 4, |a3k | = 6 and ak ∈ a3k ⇒ a3k = H. That means that ({α, a0 , a1 , a2 }, ◦) is single-power cyclic Hv -group and each element ak ∈ H, k ∈ Z3 is generator with period 3.  Proposition 3.14. The (H, ◦) is a single-power cyclic Hv -group and each element ak ∈ H, k ∈ Zn is a generator with period the minimum m ∈ N, m > 1 such that m = n+1 2 , n = 4. Proof. Let x = ak , k ∈ Zn , n = 4 be any element of H. Then, from Proposition 3.12, am k = {α, ak−m+1 , ak−m+2 , . . . , ak+m−2 , ak+m−1 }, m ∈ N, m > 1. Notice that |am k | = 1 + [(k + m − 1) − (k − m + 1) + 1] = 2m. Since |H| = n + 1, we obtain am k = H ⇒ 2m = n + 1 ⇒ m =

n+1 2 .



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Consider now the infinite set H 0 = {α} ∪ {z/z = am , m ∈ N}. Then (H , ◦) is also a commutative Hv -group. 0

Proposition 3.15. The (H 0 , ◦) is a single-power cyclic Hv -group with infinite period and each element am ∈ H 0 , m ∈ N is a generator. Proof. Let x = am , am ∈ H 0 , m ∈ N be any element of H 0 . Then, a1m = am , a2m = {α, am−1 , am+1 }, a3m = {α, am−2 , am−1 , am , am+1 , am+2 }, a4m = {α, am−3 , am−2 , am−1 , am , am+1 am+2 , am+3 }, ... = {α, am−n+2 , am−n+3 , . . . , am+n−3 , am+n−2 }, n ∈ N∗ , an−1 m anm = {α, am−n+1 , am−n+2 , . . . , am+n−2 , am+n−1 }, n ∈ N∗ , ... So, every element of H 0 belongs to a power of am and there exists n0 = 1 such that for every n = n0 : ⊂ anm . a1m ∪ a2m ∪ a3m ∪ . . . ∪ an−1 m



Let us consider the geometrical shape of the figure below. It is partitioned into n + 1 parts such that:

a0

a1 α

a2

an–1 a...

(1) There is a part, denoted by α, which borders all the rest. (2) In addition, each of the remaining parts, denoted by ai , i ∈ Zn borders the two others (its adjacent ones). Thus, the set H of all parts of the figure consists of one central and n peripheral parts. Definition 3.7. On H we introduce a hyperoperation (∗) such that

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(1) the hyperproduct between the central part α and any other x is equal to x, i.e. α ∗ x = x ∗ α = x. (2) The hyperproduct of two peripheral parts is the set of all parts border to them (their adjacent ones), i.e., ai ∗ aj = {α, ai−1 , ai+1 , aj−1 , aj+1 }, i, j ∈ Zn . We call ∗, boundary hyperoperation. Notice that ∗ ≡ ◦, for example α∗α = α, α∗a2 = a2 , a1 ∗a2 = {α, a0 , a1 , a2 , a3 }, a2 ∗a5 = {α, a1 , a3 , a4 , a6 }. We study below some cases, depending on n, of the boundary hyperoperation, n = 1, 2, 3, 4, 5 then a sequence of hyperstructures is created and in each one we present the corresponding figure, the Cayley table of the hyperoperation and the kind of the hyperstructure resulting.

Case 1 If n = 1 then H = {α, a0 } ∗ α a0

α α a0

a0 a0 α

α

a0

Notice that ({α, a0 }, ∗) is a group and H ∼ = Z2 . Case 2 If n = 2 then H = {α, a0 , a1 } ∗ α a0 a1

α α a0 a1

a0 a0 α, a1 H

a1 a1 H α, a0

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a1

α

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a0

Notice that ({α, a0 , a1 }, ∗) is a hypergroup. It is also a Hb -group, greater than a group, isomorphic to Z3 . We say that H contains Z3 up to isomorphism.

Case 3 If n = 3 then H = {α, a0 , a1 , a2 } ∗ α a0 a1 a2

α α a0 a1 a2

a0 a0 α, a1 , a2 H H

a1 a1 H α, a0 , a2 H

a1

a2 a2 H H α, a0 , a1

a0 α

a2

Notice that ({α, a0 , a1 , a2 }, ∗) is a hypergroup. It is also a Hb -group which contains Z4 up to isomorphism. Case 4 If n = 4 then H = {α, a0 , a1 , a2 , a3 } ∗ α a0 a1 a2 a3

α α a0 a1 a2 a3

a0 a0 α, a1 , a3 H α, a1 , a3 H

a1 a1 H α, a0 , a2 H α, a0 , a2

a2 a2 α, a1 , a3 H α, a1 , a3 H

a3 a3 H α, a0 , a2 H α, a0 , a2

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a1 a2

a0 α

a3

Notice that ({α, a0 , a1 , a2 , a3 }, ∗) is a Hv -group. Case 5 If n = 5 then H = {α, a0 , a1 , a2 , a3 , a4 } ∗ α a0

α α a0

a0 a0 α, a1 , a4

a1

a1

a2

a2

a3

a3

a4

a4

α, a0 , a1 a2 , a4 α, a1 , a3 a4 α, a1 , a2 a4 α, a0 , a1 a3 , a4

a1 a1 α, a0 , a1 a2 , a4 α, a0 , a2

a2 a2 α, a1 , a3 a4 α, a0 , a1 a2 , a3 α, a1 , a3

α, a0 , a1 a2 , a3 α, a0 , a2 a4 α, a0 , a2 a3

α, a1 , a2 a3 , a4 α, a0 , a1 a3

α, a0 , a2 a3 , a4

a4 a4 α, a0 , a1 a3 , a4 α, a0 , a2 a3 α, a0 , a1 a3 α, a0 , a2 a3 , a4 α, a0 , a3

a0

a1

a4

α

a2

a3 a3 α, a1 , a2 a4 α, a0 , a2 a4 α, a1 , a2 a3 , a4 α, a2 , a4

a3

Notice that ({α, a0 , a1 , a2 , a3 , a4 }, ∗) is a Hv -group. 3.6

Fuzzy Hv -groups

It is well known that the concept of fuzzy sets, introduced by Zadeh [141], has been extensively applied to many scientific fields. Definition 3.8. Let X be a set. A fuzzy subset A in X is characterized by a membership function µA : X → [0, 1] which associates with each point x ∈ X its grade or degree of membership µA (x) ∈ [0, 1].

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Let A and B be fuzzy sets in X. Then • • • •

A = B if and only if µA (x) = µB (x) for all x ∈ X, A ⊆ B if and only if µA (x) ≤ µB (x) for all x ∈ X, C = A∪B if and only if µC (x) = max{µA (x), µB (x)} for all x ∈ X, D = A∩B if and only if µD (x) = min{µA (x), µB (x)} for all x ∈ X.

The complement of A, denoted by Ac , is defined by µAc (x) = 1 − µA (x) for all x ∈ X. For the sake of simplicity, we shall show every fuzzy set by its membership function. Definition 3.9. Let f be a mapping from a set X to a set Y . Let µ be a fuzzy subset of X and λ be a fuzzy subset of Y . Then the inverse image f −1 (λ) of λ is the fuzzy subset of X defined by f −1 (λ)(x) = λ(f (x)) for all x ∈ X. The image f (µ) of µ is the fuzzy subset of Y defined by  sup{µ(t) | t ∈ f −1 (y)} if f −1 (y) 6= ∅ f (µ)(y) = 0 otherwise for all y ∈ Y . It is not difficult to see that the following assertions hold: (1) If {λi }i∈I be a family of fuzzy subsets of Y , then S S −1 T T −1 f −1 ( λi ) = f (λi ) and f −1 ( λi ) = f (λi ). i∈I

i∈I

i∈I

i∈I

−1

(2) If µ is a fuzzy subset of X, then µ ⊆ f (f (µ)). Moreover, if f is one to one, then f −1 (f (µ)) = µ. (3) If λ is a fuzzy subset of Y , then f (f −1 (λ)) ⊆ λ. Moreover, if f is onto, then f (f −1 (λ)) = λ. The concept of a fuzzy subgroup of a group (G, ·) is introduced in [89]. If G is a group and µ : G → [0, 1] is a fuzzy subset of G, then µ is called a fuzzy subgroup if it satisfies, (1) min{µ(x), µ(y)} ≤ µ(x · y), for all x, y ∈ G, (2) µ(x) ≤ µ(x−1 ), for all x ∈ G. In [23, 26], Davvaz applied the concept of fuzzy sets to the theory of algebraic hyperstructures and defined fuzzy subhypergroup (respectively, Hv subgroup) of a hypergroup (resp. Hv -group) which is a generalization of

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the concept of Rosenfeld’s fuzzy subgroup of a group. In this section first we define fuzzy Hv -subgroup of an Hv -group and then we obtain the relation between a fuzzy Hv -subgroup and level Hv -groups. This relation is expressed in terms of a necessary and sufficient condition. Definition 3.10. Let (H, ◦) be an Hv -group and let µ be a fuzzy subset of H. Then, µ is said to be a fuzzy Hv -subgroup of H if the following axioms hold. (1) min{µ(x), µ(y)} ≤ inf {µ(α)}, for all x, y ∈ H. α∈x◦y

(2) For all x, a ∈ H there exists y ∈ H such that x ∈ a ◦ y and min{µ(a), µ(x)} ≤ µ(y). (3) For all x, a ∈ H there exists z ∈ H such that x ∈ z ◦ a and min{µ(a), µ(x)} ≤ µ(z). Condition (2) is called the left fuzzy reproduction axiom, while (3) is called the right fuzzy reproduction axiom. Example 3.7. Let (G, ·) be a group and µ be a fuzzy subgroup of G. If we define the following hyperoperation on G, ? : G × G → P ∗ (G) with x ? y = {t | µ(t) = µ(x · y)}, then (G, ?) is an Hv -group and µ is a fuzzy Hv -subgroup of G. Example 3.8. Suppose that H is a set and µ is a fuzzy subset of H. We define the hyperoperation ? : H × H → P ∗ (H) as follows: Assume that x, y ∈ H, if µ(x) ≤ µ(y), then y ? x = x ? y = {t | t ∈ H, µ(x) ≤ µ(t) ≤ µ(y)}. Then, (H, ?) is a hypergroup as well as a join space. If (H, ·) is a group and µ is a fuzzy subgroup of H, then µ is a subhypergroup of (H, ?). Lemma 3.4. Let (H, ◦) be an Hv -group and µ be a fuzzy Hv -subgroup of H. Then, min{µ(x1 ), . . . , µ(xn )} ≤

inf

α∈(...((x1 ◦x2 )◦x3 )...)◦xn

{µ(α)},

for all x1 , x2 , . . . , xn ∈ H. Proof. We shall prove the validity of this lemma by mathematical induction. First, the lemma is clearly true for n = 2. To complete the proof we assume the validity of the lemma for n = k − 1, that is, we assume that min{µ(x1 ), . . . , µ(xk−1 )} ≤

inf

r∈(...((x1 ◦x2 )◦x3 )...)◦xk−1

{µ(r)},

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for all x1 , x2 , . . . , xk−1 ∈ H. Then, min{µ(x1 ), . . . , µ(xk−1 ), µ(xk )} = min{min{µ(x1 ), . . . , µ(xk−1 )}, µ(xk )} n o ≤ min inf {µ(r), µ(xk )} r∈(...((x1 ◦x2 )◦x3 )...)◦xk−1

= ≤

inf

{min{µ(r), µ(xk )} o inf {µ(α)}

r∈(...((x1 ◦x2 )◦x3 )...)◦xk−1 n

inf

r∈(...((x1 ◦x2 )◦x3 )...)◦xk−1

≤

α∈r◦xk

inf

α∈((...((x1 ◦x2 )◦x3 )...)◦xk−1 )◦xk

{µ(α)}. 

Let (H, ◦) be an Hv -group. An n-ary hyperproduct can be defined, induced by ◦, by inserting n − 2 parentheses in the sequence of elements x1 , . . . , xn in a standard position. Let us denote by p(x1 , . . . , xn ) such a pattern of n − 2 parentheses and by, Pn the set of all such patterns. Corollary 3.3. Let (H, ◦) be an Hv -group and µ be a fuzzy Hv -subgroup of H. Then, S min{µ(x1 ), . . . , µ(xn )} ≤ inf{µ(α) | α ∈ p(x1 , . . . , xn )}. p(x1 ,...,xn )∈Pn (x1 ,...,xn )

for all x1 , x2 , . . . , xk−1 ∈ H. Theorem 3.11. Let (H, ◦) be an Hv -group and µ be a fuzzy subset of H. Then, µ is a fuzzy Hv -subgroup of H if and only if for every t, 0 ≤ t ≤ 1, µt 6= ∅ is an Hv -subgroup of H. Proof. Let µ be a fuzzy Hv -subgroup of H. For every x, y in µt we have min{µ(x), µ(y)} ≥ t and so inf {µ(α)} ≥ t. Therefore, for every α∈x◦y

α ∈ x ◦ y we have α ∈ µt , so x ◦ y ⊆ µt . Hence, for every a ∈ µt we have a ◦ µt ⊆ µt . Now, let x ∈ µt . Then, there exists y ∈ H such that x ∈ a ◦ y and min{µ(a), µ(x)} ≤ µ(y). From x ∈ µt and a ∈ µt we get min{µ(x), µ(a)} ≥ t. So, y ∈ µt , and this proves µt ⊆ a ◦ µt . Conversely, assume that for every t, 0 ≤ t ≤ 1, µt 6= ∅ is an Hv subgroup of H. For every x, y in H we can write µ(x) ≥ min{µ(x), µ(y)} and µ(y) ≥ min{µ(x), µ(y)}. If we put t0 = min{µ(x), µ(y)}, then x ∈ µt0 and y ∈ µt0 , so x ◦ y ⊆ µt0 . Therefore, for every α ∈ x ◦ y we have µ(α) ≥ t0 implying inf {µ(α)} ≥ min{µ(x), µ(y)} and in this way the condition (1) α∈x◦y

of Definition 3.10 is verified. To verify the second condition, if for every a, x ∈ H we put t1 = min{µ(a), µ(x)}, then x ∈ µt1 and a ∈ µt1 . So, there exists y ∈ µt1 such that x ∈ a ◦ y. On the other hand, since y ∈ µt1 ,

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then t1 ≤ µ(y) and hence min{µ(a), µ(x)} ≤ µ(y). The proof of the third condition of Definition 3.10 is similar to the proof of the second condition.  The following two corollaries are exactly obtained from Theorem 3.11. Corollary 3.4. Let (H, ◦) be an Hv -group and µ be a fuzzy Hv -subgroup of H. If 0 ≤ t1 < t2 ≤ 1, then µt1 = µt2 if and only if there is no x in H such that t1 ≤ µ(x) < t2 . Corollary 3.5. Let (H, ◦) be an Hv -group and µ be a fuzzy Hv -subgroup of H. If the range of µ is the finite set {t1 , t2 , . . . , tn }, then the set {µti | 1 ≤ i ≤ n} contains all the level Hv -subgroups of µ. Moreover, if t1 > t2 > . . . > tn , then all the level Hv -subgroups µti form the following chain µt1 ⊆ µt2 ⊆ . . . ⊆ µtn . Proposition 3.16. Let (H, ◦) be an Hv -group and µ be a fuzzy subset of H. Then, µ is a fuzzy Hv -subgroup of H if and only if for every t, 0 ≤ t ≤ 1, (1) µt ◦ µt ⊆ µt , (2) a ◦ (H − µt ) − (H − µt ) ⊆ a ◦ µt , for all a ∈ µt . Proof. Let µ be a fuzzy Hv -subgroup of H. Then, by Theorem 3.11, µt is an Hv -subgroup of H, and then it is clear that the three conditions of Definition 3.10 are valid. Conversely, suppose that the three conditions of Definition 3.10 hold. Then, by Theorem 3.11 it is enough to prove that µt is an Hv -subgroup of H. For the proof of the left reproduction axiom, it is enough to show that µt ⊆ a ◦ µt , for every a ∈ µt . Assume that there exists x ∈ µt such that x 6∈ a◦µt . Since x ∈ µt , then there exists b ∈ H such that x ∈ a◦b. If b ∈ µt , then x ∈ a ◦ b ⊆ a ◦ µt , which is a contradiction. If b ∈ H − µt , then we have x ∈ {x}−(H −µt ) ⊆ a◦b−(H −µt ) ⊆ a◦(H −µt )−(H −µt ) ⊆ a◦µt , again a contradiction. Therefore, µt − a ◦ µt = ∅ which implies that µt ⊆ a ◦ µt . The proof of the right reproduction axiom is similar.  Theorem 3.12. Let (H, ◦) be an Hv -group. Then, every Hv -subgroup of H is a level Hv -subgroup of a fuzzy Hv -subgroup of H. Proof. Let A be an Hv -subgroup of H. For a fixed real number c, 0 < c ≤ 1, the fuzzy subset µ is defined as follows  c if x ∈ A, µ(x) = 0 otherwise.

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We have A = µc and by Theorem 3.11, it is enough to prove that µ is a fuzzy Hv -subgroup. This is straightforward and we omit it.  Corollary 3.6. Let (H, ◦) be an Hv -group and A be a non-empty subset of H. Then, a necessary and sufficient condition for A to be an Hv -subgroup is that A = µt0 , where µ is a fuzzy Hv -subgroup and 0 < t0 ≤ 1. Definition 3.11. Let (H, ◦) be an Hv -group and µ be a fuzzy Hv -subgroup of H. Then, µ is called right fuzzy closed with respect to H if for every a, b in H all the x in b ∈ a ◦ x satisfy min{µ(b), µ(a)} ≤ µ(x). We call µ left fuzzy closed with respect to H if for all a, b in H all the y in b ∈ y ◦ a satisfy min{µ(b), µ(a)} ≤ µ(y). If µ is left and right fuzzy closed, then µ is called fuzzy closed. Theorem 3.13. If the fuzzy Hv -subgroup µ is right fuzzy closed, then µt .(H − µt ) = H − µt . Proof. If b ∈ µt ◦ (H − µt ), then there exists a ∈ µt and x ∈ H − µt such that b ∈ a ◦ x. Therefore, µ(x) < t ≤ µ(a) and since µ is right fuzzy closed we get min{µ(a), µ(b)} ≤ µ(x). Hence, µ(b) ≤ µ(x) < t which implies that b ∈ H − µt . So, we have proved µt ◦ (H − µt ) ⊆ H − µt . On the other hand, if x ∈ H − µt , then for every a ∈ µt by the reproduction axiom there exists y ∈ H such that x ∈ a ◦ y and so it is enough to prove y ∈ H − µt . Since µ is a fuzzy Hv -subgroup of H, by the definition we have min{µ(a), µ(y)} ≤ inf {µ(α)} which implies that α∈a◦y

min{µ(a), µ(y)} ≤ µ(x).

(3.4)

Since µ is right fuzzy closed so min{µ(x), µ(a)} ≤ µ(y).

(3.5)

Now, from x ∈ H − µt we get a ∈ µt and so µ(x) < t ≤ µ(a). Using 3.5 we obtain µ(x) ≤ µ(y). Therefore, µ(x) ≤ min{µ(a), µ(y)} and by 3.4 the relation min{µ(a), µ(y)} = µ(x) is obtained. But µ(x) < µ(a) and hence min{µ(a), µ(y)} = µ(y). So, µ(x) = µ(y). Since x ∈ H − µt we get y ∈ H − µt and the theorem is proved.  Now, we define anti fuzzy Hv -subgroup of an Hv -group and then we present some results in this connection. Definition 3.12. Let (H, ◦) be an Hv -group and let µ be a fuzzy subset of H. Then, µ is said to be an anti fuzzy Hv -subgroup of H if the following axioms hold.

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(1) sup {µ(α)} ≤ max{µ(x), µ(y)}, for all x, y ∈ H. α∈x◦y

(2) For all x, a ∈ H there exists y ∈ H such that x ∈ a ◦ y and µ(y) ≤ max{µ(a), µ(x)}. (3) For all x, a ∈ H there exists z ∈ H such that x ∈ z ◦ a and µ(z) ≤ max{µ(a), µ(x)}. Condition (2) is called the left anti fuzzy reproduction axiom, while (3) is called the right anti fuzzy reproduction axiom. For the sake of similarity, only left reproduction axiom for the Hv -groups is verified throughout this section. Lemma 3.5. Let (H, ◦) be an Hv -group and µ be an anti fuzzy Hv -subgroup of H. Then, (1)

sup

{µ(α)} ≤ max{µ(x1 ), . . . , µ(xn )}, for all

α∈(...((x1 ◦x2 )◦x3 )...)◦xn

x1 , x2 , . . . , xn ∈ H, (2) sup{µ(α) | α ∈

T

p(x1 , . . . , xn )}

≤

p(x1 ,...,xn )∈Pn (x1 ,...,xn )

max{µ(x1 ), . . . , µ(xn )}. Proof.

The proof is similar to the proof of Lemma 3.4.



Proposition 3.17. Let H be an Hv -group and µ be a fuzzy Hv -subgroup of H. Then, the set µ = {x ∈ H | µ(x) = 1} is empty or an Hv -subgroup of H. Proof. Let µ 6= ∅. Then, for all x, y in µ we have 1 = min{µ(x), µ(y)} ≤ inf {µ(α)}. Therefore, for every α ∈ x ◦ y we have µ(α) = 1 which implies

α∈x◦y

that α ∈ µ. So, x ◦ y ⊆ µ implying x ◦ y ∈ P ∗ (µ). Hence, for every a ∈ µ, we have a ◦ µ ⊆ µ and to prove the left reproduction axiom it is enough to prove µ ⊆ a ◦ µ. Since µ is a fuzzy Hv -subgroup of H, for every x ∈ µ there exists y ∈ H such that x ∈ a ◦ y and min{µ(a), µ(x)} ≤ µ(y). Since x ∈ µ and a ∈ µ, we have min{µ(a), µ(x)} = 1. Therefore, µ(y) = 1 which implies that y ∈ µ and the proposition is proved.  Example 3.9. Let (G, ◦) be a group and µ be a fuzzy subset of G. We define the hyperoperation ◦ : G × G → P ∗ (G) as follows: x ◦ y = {t | µ(t) ≤ µ(x ◦ y)}. Then,

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(1) (G, ◦) is an Hv -group. (2) If µ is an anti fuzzy subgroup of (G, ◦), then µ is an anti fuzzy Hv -subgroup of (G, ◦). Theorem 3.14. Let H be an Hv -group and µ be a fuzzy subset of H. Then µ is a fuzzy Hv -subgroup of H if and only if it’s complement µc is an anti fuzzy Hv -subgroup of H. Proof. Let µ be a fuzzy Hv -subgroup of H, for every x, y in H, we have min{µ(x), µ(y)} ≤ inf {µ(α)}, or min{1 − µc (x), 1 − µc (y)} ≤ inf {1 − α∈x◦y

α∈x◦y

µc (α)}, or min{1−µc (x), 1−µc (y)} ≤ 1− sup {µc (α)}, or sup {µc (α)} ≤ α∈x◦y

α∈x◦y

1 − min{1 − µc (x), 1 − µc (y)}, or sup {µc (α)} ≤ max{µc (x), µc (y)}, and α∈x◦y

in this way the first condition is verified for µc . Since µ is a fuzzy Hv -subgroup of H, for every a, x in H, there exists y ∈ H such that x ∈ a◦y and min{µ(a), µ(x)} ≤ µ(y), or min{1−µc (a), 1− µc (x)} ≤ 1 − µc (y), or µc (y) ≤ 1 − min{1 − µc (a), 1 − µc (x)}, or µc (y) ≤ max{µc (a), µc (x)} and the second condition is satisfied. Thus, µc is an anti fuzzy Hv -subgroup. The converse also can be proved similarly.  Now, let H be a non-empty set and µ be a fuzzy subset of H. Then, for 0 ≤ t ≤ 1, the lower level subset of µ is the set µt = {x ∈ H | µ(x) ≤ t}. Clearly µ1 = H and if t1 < t2 , then µt1 ⊆ µt2 . Theorem 3.15. Let H be an Hv -group and µ be a fuzzy subset of H. Then, µ is an anti fuzzy Hv -subgroup of H if and only if for every t, 0 ≤ t ≤ 1, µt 6= ∅ is an Hv -subgroup of H. Proof. Let µ be an anti fuzzy Hv -subgroup of H. For every x, y in µt we have µ(x) ≤ t and µ(y) ≤ t. Hence, max{µ(x), µ(y)} ≤ t and so sup {µ(α)} ≤ t. Therefore, for every α ∈ x ◦ y we have µ(α) ≤ t which α∈x◦y

implies that α ∈ µt , so x ◦ y ⊆ µt implying x ◦ y ∈ P ∗ (µt ). Hence, for every a ∈ µt we have a ◦ µt ⊆ µt and to prove this part of the theorem it is enough to prove that µt ⊆ a ◦ µt . Since µ is an anti fuzzy Hv -subgroup of H, for every x ∈ µt there exists y ∈ H such that x ∈ a ◦ y and µ(y) ≤ max{µ(a), µ(x)}. From x ∈ µt and a ∈ µt we get max{µ(x), µ(a)} ≤ t and so y ∈ µt . Therefore, we have proved that for every x ∈ µt there exists y ∈ µt such that x ∈ a ◦ y implying that x ∈ a ◦ µt and this proves µt ⊆ a ◦ µt . Conversely, assume that for every t, 0 ≤ t ≤ 1, µt 6= ∅ is an Hv subgroup of H. For every x, y in H we can write µ(x) ≤ max{µ(x), µ(y)}

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and µ(y) ≤ max{µ(x), µ(y)} and if we put t0 = max{µ(x), µ(y)}, then x ∈ µt0 and y ∈ µt0 . Since µt0 is an Hv -subgroup, so x◦y ⊆ µto . Therefore, for every α ∈ x ◦ y we have µ(α) ≤ to implying sup {µ(α)} ≤ t0 and α∈x◦y

so sup {µ(α)} ≤ max{µ(x), µ(y)} and in this way the first condition of α∈x◦y

Definition 3.12 is verified. To verify the second condition, if for every a, x ∈ H we put t1 = max{µ(a), µ(x)}, then x ∈ µt1 and a ∈ µt1 . Since a ◦ µt1 = µt1 , so there exists y ∈ µt1 such that x ∈ a ◦ y. On the other hand, since y ∈ µt1 , then µ(y) ≤ t1 and hence µ(y) ≤ max{µ(a), µ(x)} and the second condition of Definition 3.12 is satisfied.  Definition 3.13. Let X be a non-empty set and xt , with t ∈ [0, 1], be a fuzzy point of X characterized by the fuzzy subset µ defined by µ(x) = t and µ(y) = 0, for y ∈ X − {x}. We define the height of xt by hgt(µ) = t. Moreover, ∼

µ= {xt | µ(x) ≥ t, x ∈ X}.

∼

and X the family of all fuzzy points in X. The support of fuzzy subset µ of X is the set supp(µ) = {x ∈ X | µ(x) > 0}. Proposition 3.18. Let H be an Hv -group and µ be a fuzzy Hv -subgroup of H. Then, the set supp(µ) is an Hv -subgroup of H. Proof. For every x, y in supp(µ), we have µ(x) > 0, µ(y) > 0 and so min {µ(x), µ(y)} > 0 which implies that inf > 0. Therefore, for every α∈x◦y

α ∈ x ◦ y we have µ(α) > 0 which implies that α ∈ supp(µ). Hence, for every a ∈ supp(µ), we have a ◦ supp(µ) ⊆ supp(µ) and to prove the left reproduction axiom it is enough to prove supp(µ) ⊆ a ◦ supp(µ). Since µ is a fuzzy Hv -subgroup of H, for every x ∈ supp(µ) there exists y ∈ H such that x ∈ a ◦ y and min {µ(a), µ(x)} ≤ µ(y). Since x ∈ supp(µ) and a ∈ supp(µ) we have min {µ(a), µ(x)} > 0. Therefore, µ(y) > 0 which implies that y ∈ supp(µ) and the proposition is proved.  Definition 3.14. Let H be an Hv -group and µ be a fuzzy Hv -subgroup of ∼ H. We define the following hyperoperation on µ, ∼

∼

∼

◦ :µ × µ→ P ∗ (µ) xt ◦ ys = {αt∧s | α ∈ x ◦ y}, where t ∧ s = min {t, s}. ∼ Suppose that xt , ys ∈ µ.

Then, µ(x) ≥ t, µ(y) ≥ s and so

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min {µ(x), µ(y)} ≥ t ∧ s which implies that inf {µ(α)} ≥ t ∧ s. Therefore, α∈x◦y

for every α ∈ x ◦ y, we have αt∧s

∼

∈ µ.

∼

Lemma 3.6. (µ, ◦) is an Hv -semigroup. Proof.

∼

For every xt , ys , zr ∈ µ, we have (xt ◦ ys ) ◦ zr = {α(t∧s)∧r | α ∈ (x ◦ y) ◦ z}, xt ◦ (ys ◦ zr ) = {αt∧(s∧r) | α ∈ x ◦ (y ◦ z)}.

Since (H, ◦) is weak associative, (H, ◦) is weak associative.



Definition 3.15. Let H1 , H2 be two Hv -groups and µ1 , µ2 be fuzzy Hv ∼ ∼ ∼ subgroups of H1 , H2 , respectively. Let f be a mapping from µ1 into µ2 ∼

∼

∼

such that supp f (xt ) = supp f (xs ), for all xt , xs ∈ µ1 . Then, f is called (1) a strong fuzzy homomorphism if ∼

∼

∼

∼

f (xt ◦ ys ) =f (xt )◦ f (ys ), for all xt , ys ∈ µ1 , (2) an inclusion fuzzy homomorphism if ∼

∼

∼

∼

f (xt ◦ ys ) ⊆f (xt )◦ f (ys ), for all xt , ys ∈ µ1 , (3) a fuzzy Hv -homomorphism if ∼

∼

∼

∼

f (xt ◦ ys )∩ f (xt )◦ f (ys ) 6= ∅, for all xt , ys ∈ µ1 . ∼∼

∼

A mapping f :µ1 →µ2 is called a fuzzy isomorphism if it is bijective and strong fuzzy homomorphism. Two fuzzy Hv -subgroups µ1 and µ2 are said to be fuzzy isomorphic, denoted by µ1 ∼ = µ2 , if there exists a fuzzy isomor∼ ∼ phism from µ1 onto µ2 . Theorem 3.16. Let H1 , H2 be two Hv -groups and µ1 , µ2 be fuzzy Hv ∼ ∼ ∼ subgroups of H1 , H2 , respectively. Let f : µ1 →µ2 be a fuzzy inclusion homomorphism. Then, ∼

∼

(1) hgt f (xt ) = hgt f (yt ). ∼

∼

(2) hgt f (xt ) ≤ hgt f (xs ), whenever t ≤ s. ∼

Proof. (1) For every xt and yt in µ1 , there exists z ∈ H1 such that y ∈ x ◦ z and min {µ1 (x), µ1 (y)} ≤ µ1 (z). Since µ1 (x) ≥ t and µ1 (y) ≥ t,

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we have µ1 (z) ≥ t which implies that zt ∈ µ1 . From yt ∈ xt ◦ zt we get ∼

∼

∼

∼

∼

f (yt ) ∈ f (xt ◦ zt ) or f (yt ) ∈ f (xt )◦ f (zt ) and so ∼

∼

∼

∼

hgt f (yt ) = min {hgt f (xt ), hgt f (zt )} ≤ hgt f (xt ). ∼

∼

∼

Similarly, we obtain hgt f (xt ) ≤ hgt f (yt ). Therefore, hgt f (xt ) = ∼

hgt f (yt ). ∼ (2) Suppose that t ≤ s. If xt ∈ µ1 , then there exists y ∈ H1 such that ∼ ∼ x ∈ x ◦ y and µ1 (x) ≤ µ1 (y), so yt ∈ µ1 . From xt ∈ xt ◦ yt we have f (xt ) ∈ ∼

∼

∼

∼

∼

∼

f (xt ◦ yt ) which implies f (xt ) ∈ f (xs ◦ yt ) or f (xt ) ∈ f (xs )◦ f (yt ). Therefore, ∼

∼

∼

∼

hgt f (xt ) = min {hgt f (xs ), hgt f (yt )} ≤ hgt f (xs ).



Theorem 3.17. Let H1 , H2 be two Hv -groups and µ1 , µ2 be fuzzy Hv ∼ ∼ ∼ subgroups of H1 , H2 , respectively. A mapping f : µ1 →µ2 is a fuzzy strong homomorphism if and only if there exists an ordinary strong homomorphism of Hv -groups f : supp(µ1 ) → supp(µ2 ) and an increasing function ϕ : (0, 1] → (0, 1] such that ∼

∼

f (xt ) = [f (x)]ϕ(t) , for all xt ∈µ1 . ∼ ∼

∼

Proof. Suppose that f : µ1 →µ2 is a fuzzy strong homomorphism. We define a mapping f : supp(µ1 ) → supp(µ2 ) and a function ϕ : (0, 1] → (0, 1] as follows: ∼

f (x) = supp f (xµ1 (x) ), for all x ∈ supp(µ1 ) and ∼

ϕ(t) = hgt f (xt ), for all t ∈ (0, 1]. ∼

∼

∼

Since f is a fuzzy strong homomorphism, then supp f (xt ) = supp f ∼

∼

(xµ1 (x) ) and so supp f (xt ) = f (x) which implies that f (xt ) = [f (x)]ϕ(t) . By definition of ϕ and Theorem 3.16, it is easy to see that ϕ is increasing. Therefore, it remains to show that f is a strong homomorphism from the Hv -group supp(µ1 ) into the Hv -group supp(µ2 ). For every x, y ∈ supp(µ1 ), we put µ1 (x) = t and µ1 (y) = s. Then,

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we have S

[f (x ◦ y)]ϕ(t∧s) =

α∈x◦y

[f (α)]ϕ(t∧s)

S

=

∼

f (αt∧s )

α∈x◦y ∼ S

=f ( ∼

αt∧s )

α∈x◦y

∼

∼

=f (xt ◦ ys ) =f (xt )◦ f (ys ) = [f (x)]ϕ(t) ◦ [f (y)]ϕ(s) S = zϕ(t)∧ϕ(s) z∈f (x)◦f (y)

= [f (x) ◦ f (y)]ϕ(t)∧ϕ(s) . Therefore, f (x ◦ y) = f (x) ◦ f (y), i.e., f is a strong homomorphism. ∼∼ ∼ ∼ Conversely, we consider a mapping f :µ1 →µ2 such that f (xt ) = ∼

[f (x)]ϕ(t) . It is enough to show that f is a strong fuzzy homomorphism. ∼ For every xt , ys ∈µ1 (t ≤ s), we have ∼

∼

S

f (xt ◦ ys ) =

f (αt∧s )

α∈x◦y

S

=

α∈x◦y

S

=

α∈x◦y

[f (α)]ϕ(t∧s) [f (α)]ϕ(t)

= [f (x ◦ y)]ϕ(t) = [f (x) ◦ f (y)]ϕ(t)∧ϕ(s) = [f (x)]ϕ(t) ◦ [f (y)]ϕ(s) ∼

∼

=f (xt )◦ f (ys ).



Let f be a strong homomorphism from H1 into H2 . We can define a map∼ ∼

∼

∼

∼

ping f :H 1 →H 2 as follows: f (xt ) = [f (x)]t . Obviously, f is a strong fuzzy ∼

∼

homomorphism from H 1 into H 2 , where ϕ(λ) = λ, ∀λ ∈ (0, 1]. Therefore, the concept of strong fuzzy homomorphism between two Hv -groups can be seen as an extension of the concept of strong homomorphism between two Hv -groups. Remark 3.5. Let f : X → Y and let ϕ : (0, 1] → (0, 1] be an increasing ∼

∼

mapping. We define the mapping fϕ :X →Y by fϕ (xt ) = [f (x)]ϕ(t) . Then, for every fuzzy subset µ of X we have fϕ (µ)(y) =

sup x∈f −1 (y)

{ϕ(µ(x))}.

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Theorem 3.18. Let ϕ be bijective and f : H1 → H2 be a surjective strong homomorphism and let µ be a fuzzy Hv -subgroup of H1 . Then, fϕ (µ) is a fuzzy Hv -subgroup of H2 . Proof. Let µ be a fuzzy Hv -subgroup of H1 . By Theorem 3.11, for every t, 0 ≤ t ≤ 1, level subset µt (µt 6= ∅) is an Hv -subgroup of H1 and so f (µϕ−1 (t) ) is an Hv -subgroup of H2 . Now, it is enough to show that f (µϕ−1 (t) ) = (fϕ (µ))t . For every y in (fϕ (µ))t we have fϕ (µ)(y) ≥ t which implies that sup

{ϕ(µ(x))} ≥ t.

x∈f −1 (y)

Therefore, there exists x0 ∈ f −1 (y) such that ϕ(µ(x0 )) ≥ t which implies that µ(x0 ) ≥ ϕ−1 (t) or x0 ∈ µϕ−1 (t) and so f (x0 ) ∈ f (µϕ−1 (t) ) implying y ∈ f (µϕ−1 (t) ). Now, for every y in f (µϕ−1 (t) ), there exists x ∈ µϕ−1 (t) such that f (x) = y. Since x ∈ µϕ−1 (t) , we have µ(x) ≥ ϕ−1 (t) or ϕ(µ(x)) ≥ t and so sup {ϕ(µ(x))} ≥ t which implies that fϕ (µ)(y) ≥ t. Therefore, x∈f −1 (y)

y ∈ (fϕ (µ))t . 3.7



Hv -semigroups and noise problem

Noise pollution is displeasing human, animal or machine-created sound that disrupts the activity or balance of human or animal life. The source of most outdoor noise worldwide is not only transportation systems (including motor vehicle noise, aircraft noise and rail noise), but, noise caused by people as well (audio entertainment systems, electric megaphones and loud people) [54]. The fact that noise pollution is also a cause of annoyance, is that, a 2005 study by Spanish researchers found that in urban areas households are willing to pay approximately 4 Euros per decibel per year for noise reduction [11]. Poor urban planning may give rise to noise pollution, since side-by-side industrial and residential buildings can result in noise pollution in the residential area. We set the following problem: The noise pollution that comes from a certain block of flats in urban areas, obviously annoys not only the block of flats itself but possibly neighboring blocks of flats or buildings, as well. If every city is considered as a set Ω with elements the blocks of flats or the buildings, then the above situation could be described with an algebraic hyperstructure and its properties. In this section, the main reference is [53] and we present the right reproductive

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Hv -semigroup, as a tool to study the noise pollution problem in urban areas. Definition 3.16. A hypergroupoid (H, ◦) such that the weak associativity holds and H ◦ x = H, for all x ∈ H, is called right reproductive Hv semigroup. A hypergroupoid (H, ◦) such that the weak associativity holds and x ◦ H = H, for all x ∈ H, is called left reproductive Hv -semigroup. Now we give the following definition. Definition 3.17. Let Ω 6= ∅ and f : Ω → P(Ω) be a map, then we define a hyperoperation rL : Ω × Ω → P(Ω), on Ω as follows: for all x, y ∈ Ω, we set xrL y = f (x) ∪ {x}. We call the hyperoperation rL , noise hyperoperation. Note that the noise hyperoperation, always contains the element x ∈ Ω. That means that the element x ∈ Ω could be considered as the representative of the elements of the set xrL y. So, we symbolize: xrL y = f (x) ∪ {x} = [x]. If x ∈ f (x), for all x ∈ Ω, then the hyperoperation is simplified as xrL y = f (x) = [x]. Therefore, the noise hyperoperation rL depends only on the left element. That means that if one composes an element x, on the left, with any other element y, on the right, then the result is always the same set [x]. Example 3.10. Consider a set Ω = {x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 , x9 } and a map f : Ω → P(Ω) such that f (x1 ) = {x2 }, f (x2 ) = {x2 , x3 }, f (x3 ) = {x2 }, f (x4 ) = {x4 } f (x5 ) = {x5 , x6 , x7 }, f (x6 ) = {x6 , x7 }, f (x7 ) = {x5 }, f (x8 ) = {x8 , x9 }, f (x9 ) = ∅.

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Then, as in the defined above noise hyperoperation, we have [x1 ] = f (x1 ) ∪ {x1 } = {x1 , x2 }, [x2 ] = f (x2 ) = {x2 , x3 }, [x3 ] = f (x3 ) ∪ {x3 } = {x2 , x3 } [x4 ] = f (x4 ) = {x4 }, [x5 ] = f (x5 ) = {x5 , x6 , x7 }, [x6 ] = f (x6 ) = {x6 , x7 } [x7 ] = f (x7 ) ∪ {x7 } = {x5 , x7 }, [x8 ] = f (x8 ) = {x8 , x9 }, [x9 ] = f (x9 ) ∪ {x9 } = {x9 }. Then, the “multiplication” table of (rL ) is given by: rL x1 x2 x3 x4 x5 x6 x7 x8 x9

x1 x2 x3 x1 , x2 x1 , x2 x1 , x2 x2 , x3 x2 , x3 x2 , x3 x2 , x3 x2 , x3 x2 , x3 x4 x4 x4 x5 , x6 , x5 , x6 , x5 , x6 , x7 x7 x7 x7 x6 , x7 x6 , x7 x6 , x7 x5 , x7 x5 , x7 x5 , x7 x8 , x9 x8 , x9 x8 , x9 x9 x9 x9

x4 x5 x6 x1 , x2 x1 , x2 x1 , x2 x2 , x3 x2 , x3 x2 , x3 x2 , x3 x2 , x3 x2 , x3 x4 x4 x4 x5 , x6 , x5 , x6 , x5 , x6 , x7 x7 x7 x6 , x7 x6 , x7 x6 , x7 x5 , x7 x5 , x7 x5 , x7 x8 , x9 x8 , x9 x8 , x9 x9 x9 x9

x7 x8 x9 x1 , x2 x1 , x2 x1 , x2 x2 , x2 x2 , x3 x2 , x3 x2 , x2 x2 , x3 x2 , x3 x4 x4 x4 x5 , x6 , x5 , x6 , x5 , x6 , x7 x7 x7 x6 , x7 x6 , x7 x6 , x7 x5 , x7 x5 , x7 x5 , x7 x8 , x9 x8 , x9 x8 , x9 x9 x9 x9

Example 3.11. Let X 6= ∅ and µ : X → [0, 1] be a fuzzy subset of X. We define the hyperoperation ( ) on X as follows: for all x, y ∈ X, : X × X → P(X) such that x y = {z ∈ X µ(z) = µ(x)}. Then, consider the map f (x) = {z ∈ X | µ(z) = µ(x)}. Since x ∈ f (x), for all x ∈ X, as above we have the hyperoperation rL on X as follows: rL : X × X → P(X), for all x, y ∈ X, such that xrL y = f (x). Some properties of (rL ) are as follows. (1) xrL Ω = [x], for all x ∈ Ω. (2) [x]rL y = [x]rL [y] ⊇ [x], for all y ∈ Ω. (3) x2 = xrL x = [x], for all x ∈ Ω. Proposition 3.19. The hypergroupoid (Ω, rL ) is an Hv -semigroup.

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Proof. We have to prove that the weak associativity holds. Indeed, for all x, y, z ∈ Ω, S S xrL (yrL z) = (xrL v) = (xrL v) = [x], v∈yrL z v∈[y] S S S (xrL y)rL z = (wrL z) = (wrL z) = [w] ⊇ [x]. w∈xrL y

w∈[x]

w∈[x]

Therefore, (xrL y)rL z ⊇ xrL (yrL z). Thus, we have (xrL y)rL z ∩ xrL (yrL z) 6= ∅, for all x, y, z ∈ Ω.



Proposition 3.20. For all x ∈ Ω, ΩrL x = Ω and xrL Ω = [x]. Proof.

For all x ∈ Ω, we have S S ΩrL x = (ωrL x) = [ω] = Ω. ω∈Ω

ω∈Ω

On the other hand, S

xrL Ω =

(xrL ω) = [x].

ω∈Ω



Proposition 3.21. The hypergoupoid (Ω, rL ) is a right reproductive Hv semigroup. Proof.

It is straightforward.



Note that the right reproductive Hv -semigroup (Ω, rL ) is an Hv -group if, for all x ∈ Ω, we have xrL Ω = Ω. Proposition 3.22. The strong associativity of (rL ) is valid if and only if we have S (wrL z) = [x], for all x, z ∈ Ω. w∈[x]

Proof. Then,

Suppose that (x, y, z) ∈ Ω3 such that (xrL y)rL z = xrL (yrL z). (xrL y)rL z = xrL (yrL z) ⇒ [x]rL z = xrL [y] ⇒ [x]rL z = [x] S ⇒ (wrL z) = [x]. w∈[x]

Now, let (x, y, z) ∈ Ω3 such that S

(wrL z) = [x].

w∈[x]

Then, (xrL y)rL z = [x]rL z =

S

(wrL z) = [x],

w∈[x]

xrL (yrL z) = xrL [y] = [x].



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For the hyperoperation rL , we shall check the conditions such that the strong or the weak commutativity is valid. Proposition 3.23. If y ∈ [x] and x ∈ [y], for all x, y ∈ Ω, then the weak commutativity of (rL ) is valid. The strong commutativity of (rL ) is valid, iff [x] = [y], for all x, y ∈ Ω. Proof.

If y ∈ [x] and x ∈ [y], for all x, y ∈ Ω, then y ∈ [x] and x ∈ [y] ⇒ x, y ∈ [x] and x, y ∈ [y] ⇒ [x] ∩ [y] 6= ∅ ⇒ (xrL y) ∩ (yrL x) 6= ∅.

The proof for the strong commutativity is straightforward.



Proposition 3.24. Let (Ω, +, rL ) be an Hv -ring. If xrL Ω = Ω, for all x ∈ Ω then the hyperstructure (Ω, +, rL ) is a dual Hv -ring, i.e. both (Ω, +, rL ) and (Ω, rL , +) are Hv -rings. Proof. We have that the (Ω, rL ) is an Hv -group. For the weak distributivity of (+) with respect to (rL ) we have for all x, y, z ∈ Ω, x + (yrL z) ⊇ x + y and (x + y)rL (x + z) =

S s∈x+y,t∈x+z

(srL t) ⊇

S

s = x + y.

s∈x+y

So, we have [x + (yrL z)] ∩ [(x + y)rL (x + z)] 6= ∅, for all x, y, z ∈ Ω. Similarly, the weak distributivity of + with respect to rL from the right side.  Proposition 3.25. (1) All the elements of Ω are right unit elements with respect to rL . (2) All the elements of Ω are left absorbing-like elements with respect to rL . Proof.

Since x ∈ xrL y, for all x, y ∈ Ω, the proof is straightforward. 

Proposition 3.26. The left scalar elements of the Hv -semigroup (Ω, rL ), are left absorbing elements.

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Proof. Let β ∈ Ω be a left scalar unit element, then βrL x = x, for all x ∈ Ω. But since β ∈ βrL x, for all x ∈ Ω, we obtain that βrL x = β, for all x ∈ Ω.  Proposition 3.27. The right scalar unit elements of the Hv -semigroup (Ω, rL ), are idempotent elements. Proof. If α ∈ Ω is a right scalar unit element, then xrL α = x, for all x ∈ Ω. So, αrL α = α. This implies that α2 = α.  Proposition 3.28. If there exists x ∈ Ω such that f (x) = x or [x] = x, then x is left absorbing element and every element of (Ω, rL ) is right scalar unit of x. Proof. Suppose there exists x ∈ Ω such that [x] = x, then for all y ∈ Ω, we have xrL y = [x] = x. That means that x is left absorbing element and every element of (Ω, rL ) is right scalar unit of x.  Since all the elements of the Hv -semigroup (Ω, rL ) are right unit elements, let us denote by Irl L (x, y) the set of the left inverses of the element x ∈ Ω, associated with the right unit y ∈ Ω, with respect to the hyperoperation (rL ). The set of the right inverses of the element x ∈ Ω, associated with the right unit y ∈ Ω, with respect to the hyperoperation (rL ), is denoted by IrrL (x, y). Proposition 3.29. y ∈ Irl L (x, y). Proof. Suppose that x0 ∈ Ω such that x0 ∈ Irl L (x, y) ⇒ y ∈ x0 rL x. But, for all x ∈ Ω the relation y ∈ yrL x is valid. That means that y ∈ Irl L (x, y).  Proposition 3.30. IrrL (x, y) = Ω if and only if y ∈ [x]. Proof.

Let y ∈ Ω be right unit element and x ∈ Ω, then y ∈ [x] ⇔ y ∈ xrL x0 , for all x0 ∈ Ω ⇔ x0 ∈ IrrL (x, y), for all x0 Ω ⇔ IrrL (x, y) = Ω.

Since x ∈ [x], for all x ∈ Ω, the following corollary is obvious. Corollary 3.7. IrrL (x, x) = Ω.



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Remark 3.6. Notice that, according to Example 3.10, the elements x4 and x9 are idempotent elements, since x24 = x4 and x29 = x9 . They are, also, left absorbing elements, since x4 rL x = x4 and x9 rL x = x9 , for all x ∈ Ω. Also, taking for instance, the element x2 of Ω, notice that Irl L (x, x2 ) = {x1 , x2 , x3 }, for all x ∈ Ω. Even more, since x2 ∈ [x1 ] we obtain that IrrL (x1 , x2 ) = Ω and IrrL (x1 , x1 ) = Ω. Definition 3.18. The inclusion associativity in an Hv -semigroup can be held in two ways, either (xy)z ⊂ x(yz) and then is called inclusion on the right parenthesis or (xy)z ⊇ x(yz) and then is called inclusion on the left parenthesis. Proposition 3.31. Let (H, ◦) be an Hv -semigroup where the inclusion on the left parenthesis holds. Let a1 , a2 , , an be n elements (distinct or not) of H, then we set: p = a1 ◦ a2 ◦, ..., ◦an = [...[(a1 ◦ a2 ) ◦ a3 ]a4 ◦ ...]an . Then, every other possible meaningful expression p0 obtained by inserting parentheses in the finite sequence a1 , a2 , ..., an is subset to p. Remark that for the above Hv -semigroup and for any positive integer n, the nth power an , of an element a is an = [...[(a ◦ a) ◦ a]a ◦ ...]a. Proof.

It is straightforward.



Proposition 3.32. The right reproductive Hv -semigroup (Ω, rL ) is singlepower cyclic with infinite period with generator x ∈ Ω if and only if there exists s0 ≥ 2 such that for all s ≥ s0 , we have ∪[x]s−2 = Ω. Proof.

For all x ∈ Ω, we have x2 = xrL x = [x], x3 = x2 rL x = [x]rL [x] = [x]2 , x4 = [x]3 , ... xs−1 = [x]s−2 .

Suppose that there exists s0 ≥ 2, such that for all s ≥ s0 , we have ∪[x]s−2 = Ω ⇔ [x] ∪ [x]2 ∪ ... ∪ [x]s−2 = Ω ⇔ x2 ∪ x3 ∪ ... ∪ xs−1 = Ω

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and since x ∈ x2 , it follows that there exists s0 ≥ 1 such that for all s ≥ s0 x1 ∪ x2 ∪ x3 ∪ ... ∪ xs−1 = Ω. So, every element of Ω belongs to a power of x. But, since the next one is valid, i.e., x2 = [x] 3 x, x3 = x2 rL x ⊇ x2 and by induction x1 ⊂ x2 ⊂ ... ⊂ xs−1 ⊂ xs , for all x ∈ Ω, then there exists s0 ≥ 1 such that for all s ≥ s0 , xs = Ω. So, there exists s0 ≥ 1 such that for all s ≥ s0 x1 ∪ x2 ∪ x3 ∪ ... ∪ xs−1 ⊂ xs .



As we mentioned above, the noise pollution in urban areas coming from a spot, annoys a certain area in which the noisy spot belongs to. That was the motivation which led to the mathematical expression xrL y = [x], for all x, y ∈ Ω. That means that if a city is considered as a set Ω with elements its buildings (or spots which could produce noise pollution), then every building (or a spot) x, which is a source of noise pollution, together with any other building (or a spot) y of the city, will affect anyhow the noise pollution area [x], where x ∈ [x] and maybe y. It is clear, that the source of the noise pollution x, could not be seen as the center of a cyclic disk, but as any spot of a certain area which is affected by x. We shall try to explain some of the properties of the noise hyperoperation (rL ) developed above, in terms of noise pollution problems in urban areas. The property x ∈ xrL y, for all y ∈ Ω means that the building x, as a source of noise pollution, first of all, annoys the residents of the building x. The property rL [y] = [x] means that the source of noise pollution x together with any region [y] is not only independent on the spots of the region [y] but the noise pollution region remains [x], as well. The property [x]rL y = [x]rL [y] ⊇ [x] means that the noise pollution region that results when either the noise pollution region [x] operates with the spot y or with the region [y], is the same and anyhow this noise pollution region is bigger than [x]. The property xrL y = xrL z means that [x] remains the noise pollution region when x as a source of noise pollution affects any other spot of the city Ω. Continuously, the relation xrL Ω 6= Ω means that, the noise pollution region coming from spot x, can’t affect the whole city Ω. The weak associativity which is expressed by the inclusion on the left parenthesis, i.e., (xrL y)rL z ⊇ xrL (yrL z) actually means that, the noise pollution region coming from the noise pollution region [x] together with any spot, is not only bigger than that one which comes from the noise pollution spot x together with any other region but includes it, as well. An absorbing element, as in the relation αrL x = α,

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could be considered as a spot surrounded by a wall or a forest, which doesn’t annoy any other spot of the city Ω. Since the weak associativity is valid, the concept of transitive closure can be applied here, in order to obtain the fundamental β* classes. The actual meaning of this situation is that the city Ω can be divided, using the noise hyperoperation, in a partition, where every fundamental class does not annoy any other blocks of flats from other fundamental classes. The next example gives an idea: Example 3.12. According to Example 3.10, consider now that Ω is a city where Ω = {x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 , x9 }. From “multiplication” table of (rL ), we obtain that β*(x1 ) = {x1 , x2 , x3 }, β*(x4 ) = {x4 }, β*(x5 ) = {x5 , x6 , x7 }, β*(x8 ) = {x8 , x9 }. So, the fundamental semigroup Ω/β* is  Ω/β* = {x1 , x2 , x3 }, {x4 }, {x5 , x6 , x7 }, {x8 , x9 } and the “multiplication” table is ◦ x1 x4 x5 x8

x1 x1 x4 x5 x8

x4 x1 x4 x5 x8

x5 x1 x4 x5 x8

x8 x1 x4 x5 x8

In other words and beyond the mathematical content of the present example, the city Ω was divided into four regions, where every region (fundamental class) does not annoy any other spot belonging to the rest regions. So, one could consider that among the four regions there exists a green park, full of trees, which absorbs the possible noise pollution caused by any of the four regions. Since β*(x4 ) = {x4 }, the element x4 ∈ Ω (spot or building of the city) is a single element and that means that it doesn’t annoy any other spot of the city Ω , so it can be considered as the remotest spot of the city.

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4.1

Hv -rings and some examples

Definition 4.1. A multi-valued system (R, +, ·) is an Hv -ring if (1) (R, +) is an Hv -group; (2) (R, ·) is an Hv -semigroup; (3) · is weak distributive with respect to +, i.e., for all x, y, z in R we have (x · (y + z)) ∩ (x · y + x · z) 6= ∅ and ((x + y) · z) ∩ (x · z + y · z) 6= ∅. An Hv -ring may be commutative with respect either to + or ·. If H is commutative with respect to both + and ·, then we call it a commutative Hv -ring. If there exists u ∈ R such that x · u = u · x = {x} for all x ∈ R, then u is called the scalar unit of R and it is denoted by 1. Example 4.1. Let (R, +, ·) be a ring and µ : R → [0, 1] be a function. We define the hyperoperations ], ⊗, ∗ on R as follows: x ] y = {t| µ(t) = µ(x + y)}, x ⊗ y = {t| µ(t) = µ(x · y)}, x ? y = y ? x = {t| µ(x) ≤ µ(t) ≤ µ(y)}, (if µ(x) ≤ µ(y)). Then, (R, ?, ?), (R, ?, ⊗), (R, ?, ]), (R, ], ?), and (R, ], ⊗) are Hv -rings. Definition 4.2. An Hv -ring (R, +, ·) is called a dual Hv -ring if (R, ·, +) is an Hv -ring. If both + and · are weak commutative, then R is called a weak commutative dual Hv -ring. Proposition 4.1. If (H, ?) is an Hv -group, then for every hyperoperation ◦ such that {x, y} ⊆ x ◦ y for all x, y ∈ H, the hyperstructure (H, ?, ◦) is a dual Hv -ring. 115

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Proof. First we prove that (H, ?, ◦) is an Hv -ring. For every x, y, z in H, we have {x} ∪ (y ? z) ⊆ x ◦ (y ? z) (x ? x) ∪ (x ? z) ∪ (y ? x) ∪ (y ? z) = {x, y} ? {x, z} ⊆ (x ◦ y) ? (x ◦ z). Thus, y ? z ⊆ (x ◦ (y ? z)) ∩ ((x ◦ y) ? (x ◦ z)) 6= ∅. Therefore, the left and similarly the right weak distributivity are valid and the rest axioms can be easily verified. Now, we prove that (H, ◦, ?) is an Hv -ring. For every x, y, z in H, we have (x ? y) ∪ (x ? z) = x ? {y, z} ⊆ x ? (y ◦ z) (x ? y) ∪ (x ? z) ⊆ (x ? y) ◦ (x ? z). So, we have (x ? y) ∪ (x ? z) ⊆ (x ? (y ◦ z)) ∩ ((x ? y) ◦ (x ? z)) 6= ∅. Thus, ? is a left weak distributive with respect to ◦ and the rest axioms are easily verified.  Proposition 4.2. Let (H, +) be an Hv -group with a scalar zero element 0. Then, for every hyperoperation such that {x, y} ⊆ x y for all x, y in H \ {0}, x 0 = 0 x = 0 for all x in H, the hyperstructure (H, +, ) is an Hv -ring. Proof.

For every non-zero elements x, y, z ∈ H, we have y + z ⊆ (x (y + z)) ∩ ((x y) + (x z)) 6= ∅.

Moreover, if one of the elements x, y, z is zero, then the strong distributivity is valid. The rest of the weak axioms are also valid.  Proposition 4.3. [51] We define the following three hyperoperations on the set Rn , where R is the set of real numbers: x ⊕ y = {r(x + y)| r ∈ [0, 1]}, x y = {x + r(y − x)| r ∈ [0, 1]}, x • y = {x + ry| r ∈ [0, 1]}. Then the hyperstructure (Rn , ?, ◦) is a weak commutative dual Hv -ring where ?, ◦ ∈ {⊕, , •}. Proof.

The associativity:

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(1) We have x ⊕ (y ⊕ z) = {rx + rmy + rmz | r, m ∈ [0, 1]} (x ⊕ y) ⊕ z = {tnx + tny + tz | t, n ∈ [0, 1]}. If r = t = 0 then {0} ⊆ (x ⊕ (y ⊕ z)) ∩ ((x ⊕ y) ⊕ z). If m = n = 1 then {r(x + y + z) | r ∈ [0, 1]} = {t(x + y + z) | t ∈ [0, 1]} ⊆ (x ⊕ (y ⊕ z)) ∩ ((x ⊕ y) ⊕ z). We claim that for all m, n ∈ [0, 1) and for all r, t ∈ (0, 1] the following assertion is valid: x ⊕ (y ⊕ z) 6= (x ⊕ y) ⊕ z. Indeed, if there exist m, n ∈ [0, 1) and r, t ∈ (0, 1] such that we have the equality in the above condition to be hold, then r = tn, rm = tn, rm = t which imply rm = r, tn = t. So m = n = 1, which is a contradiction. (2) We have x (y z) = {(1 − r)x + r(1 − m)y + rmz | r, m ∈ [0, 1]} (x y) z = {(1 − t)(1 − n)x + t(1 − n)y + nz | t, n ∈ [0, 1]}. We claim that the above two sets are equal. Let r, m ∈ [0, 1]. Then we have (1 − n)(1 − t) = 1 − r, t(1 − n) = r(1 − m), n = rm if and only if (1 − rm)(1 − t) = 1 − r, t(1 − rm) = r(1 − m), n = rm. It is obvious that n = rm ∈ [0, 1]. Now, we shall prove that t ∈ [0, 1]. If rm = 1 then r = m = 1, so we obtain n = 1 and 0t = 0 which is valid for all t ∈ [0, 1]. If rm 6= 1 then we have r(1 − m) t= and r 6= 1 or m 6= 1. Let m 6= 1, then 0 ≤ rm < 1, 1 − rm so we have 1 − rm > 0. Now, from r ≤ 1 we obtain r − rm ≤ 1 − rm ⇔

r(1 − m) ≤ 1 ⇔ t ≤ 1. 1 − rm

Obviously, r(1 − m) ≥ 0, so t ≥ 0. Let n, t ∈ [0, 1]. Using the same technique, we can easily show that r, m ∈ [0, 1].

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(3) We have x • (y • z) = {x + ry + rmz | r, m ∈ [0, 1]} (x • y) • z = {x + ny + tz | t, n ∈ [0, 1]}. By setting r = n and rm = t we obtain x • (y • z) ⊆ (x • y) • z. The commutativity: For the first two hyperoperations we have: x ⊕ y = [0, x + y] = [0, y + x] = y ⊕ x and x y = [x, y] = [y, x] = y x. For the third one, we have x • y = {x + ry | r ∈ [0, 1]} and y • x = {mx + y | m ∈ [0, 1]}. The above two sets have common elements for all x, y ∈ Rn , only in the case r = m = 1. The reproduction axiom: We can see easily that, for all x ∈ Rn x ⊕ Rn = Rn ⊕ x = Rn , x Rn = Rn x = Rn , x • Rn = Rn • x = Rn . The distributivity: The following assertions hold for all x, y, z ∈ Rn . (x ⊕ (y ⊕ z)) ∩ ((x ⊕ y) ⊕ (x ⊕ z)) 6= ∅, ((x ⊕ y) ⊕ z) ∩ ((x ⊕ z) ⊕ (y ⊕ z)) 6= ∅, x ⊕ (y z) = (x ⊕ y) (x ⊕ z), (x y) ⊕ z = (x z) ⊕ (y z), (x ⊕ (y • z)) ∩ ((x ⊕ y) • (x ⊕ z)) 6= ∅, ((x • y) ⊕ z) ∩ ((x ⊕ z) • (y ⊕ z)) 6= ∅, x (y z) = (x y) (x z), (x y) z = (x z) (y z), (x (y ⊕ z)) ∩ ((x y) ⊕ (x z)) 6= ∅, ((x ⊕ y) z) ∩ ((x z) ⊕ (y z)) 6= ∅, (x (y • z)) ∩ ((x y) • (x z)) 6= ∅, ((x • y) z) ∩ ((x z) • (y z)) 6= ∅, (x • (y • z)) ∩ ((x • y) • (x • z)) 6= ∅, ((x • y) • z) ∩ ((x • z) • (y • z)) 6= ∅, x • (y z) = (x • y) (x • z), (x y) • z = (x • z) (y • z), (x • (y ⊕ z)) ∩ ((x • y) ⊕ (x • z)) 6= ∅, ((x ⊕ y) • z) ∩ ((x • z) ⊕ (y • z)) 6= ∅. Among the 9 cases, we shall prove here the last one. The rest of them can be proved in a similar way x • (y ⊕ z) = {x + mty + mtz | m, t ∈ [0, 1]}, (x • z) ⊕ (y • z) = {2rx + rny + rkz | r, n, k ∈ [0, 1]}. If m = n = k = 0 and r = 21 then {x} ⊆ (x • (y ⊕ z)) ∩ ((x • y) ⊕ (x • z)). Also, ((x ⊕ y) • z) ∩ ((x • z) ⊕ (y • z)) 6= ∅, for all x, y, z ∈ Rn . 

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Now, we present some general constructions which can be useful in the theory of representations of several classes of Hv -groups. Let (H, ◦) be a hypergroupoid; by 4H we mean the diagonal of the Cartesian product H × H, i.e., 4H = {[x, x] | x ∈ H}. Let us define a mapping D : H → H × H by D(x) = [x, x] for all x ∈ H, i.e., 4H = D(H). Lemma 4.1. Let (H, ◦) be a hypergroupoid. Define a hyperoperation ? on the diagonal 4H as follows: [x, x] ? [y, y] = D(x ◦ y ∪ y ◦ x) = {[u, u] | u ∈ x ◦ y ∪ y ◦ x} for any pair [x, x], [y, y] ∈ 4H . Then the following assertions hold: (1) For any hypergroupoid (H, ◦) we have that (4H , ?) is a commutative hypergroupoid. (2) If (H, ◦) is a weakly associative hypergroupoid, then the hypergroupoid (4H , ?) is weakly associative as well. (3) If (H, ◦) is a quasihypergroup, the the hypergroupoid (4H , ?) satisfies also the reproduction axiom, i.e., it is a quasihypergroup. (4) If (H, ◦) is associative, then the hypergroupoid (4H , ?) is weakly associative (but not associative in general). Proof. The assertion (1) follows immediately from the above definition of the hyperoperation ?. (2) Suppose that [x, x], [y, y], [z, z] ∈ 4H . Then ([x, x] ? [y, y]) ? [z, z] = D(x ◦ y ∪ y ◦ x) ? [z, z] = (D(x ◦ y) ∪ D(y ◦ x)) ? [z, z] = (D(x ◦ y) ? [z, z]) ∪ (D(y ◦ x) ? [z, z])  S   S  = [u, u] ? [z, z] ∪ [v, v] ? [z, z] u∈x◦y

=

 S

=

 S

v∈y◦x

   S D(v ◦ z ∪ z ◦ v) D(u ◦ z ∪ z ◦ u) ∪ v∈y◦x

u∈x◦y

  S   S   S  D(u ◦ z) ∪ D(z ◦ u) ∪ D(v ◦ z) ∪ D(z ◦ u)

u∈x◦y

=D

 S u∈x◦y

u∈x◦y



u◦z ∪D

 S u∈x◦y

v∈y◦x

v∈y◦x

  S   S  z◦u ∪D v◦z ∪D z◦u v∈y◦x

= D(x ◦ y ◦ z) ∪ D(z ◦ x ◦ y) ∪ D(y ◦ x ◦ z) ∪ D(z ◦ y ◦ x) = D(x ◦ y ◦ z ∪ z ◦ y ◦ x) ∪ D(z ◦ x ◦ y ∪ y ◦ x ◦ z).

v∈y◦x

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On the other hand [x, x] ? ([y, y] ? [z, z]) = ([z, z] ? [y, y]) ? [x, x] = D(z ◦ y ◦ x ∪ x ◦ z ◦ y ∪ y ◦ z ◦ x ∪ x ◦ y ◦ z) = D(x ◦ y ◦ z ∪ z ◦ y ◦ x) ∪ D(x ◦ z ◦ y ∪ y ◦ z ◦ x). Thus ([x, x]?[y, y])?[z, z]∩[x, x]?([y, y]?[z, z]) ⊇ D(x◦y◦z)∪D(z◦y◦x) 6= ∅. (3) Let x ∈ H be an arbitrary element. Then x ◦ H = H = H ◦ x and we have S S [x, x] ? 4H = ([x, x] ? [y, y]) = D(x ◦ y ∪ y ◦ x) y∈H

=

 S



D(x ◦ y) ∪

y∈H

y∈H

 S

  S   S  D(y ◦ x) = D x◦y ∪D y◦x

y∈H

y∈H

y∈H

= D(x ◦ H) ∪ D(H ◦ x) = D(H) = 4H . Since a semihypergroup is also weakly associative, the assertion (4) follows from (2).  Let (R, +, ·) be an Hv -ring. We define the hyperoperations ⊕ and on the diagonal D(R) = 4R by [x, x] ⊕ [y, y] = {[u, u] | u ∈ (x + y) ∪ (y + x)}, [x, x] [y, y] = {[v, v] | v ∈ (x · y) ∪ (y · x)} for all x, y ∈ R. Then we have Proposition 4.4. Let (R, +, ·) be an Hv -ring. Then, (D(R), ⊕, ) is a commutative Hv -ring. Proof. According to Lemma 4.1, we obtain that (D(R), ⊕) is a commutative weakly associative hypergroupoid satisfying the reproduction axiom, thus it is a commutative Hv -group. Similarly, (D(R), ) is a commutative Hv -semigroup. Thus, it remains to prove that [x, x] ([y, y] ⊕ [z, z]) ∩ ([x, x] [y, y]) ⊕ ([x, x] [z, z]) 6= ∅ for arbitrary elements x, y, z ∈ R. Indeed, we have [y, y] ⊕ [z, z] = {[u, u] | u ∈ (y + z) ∪ (z + y)} and S [x, x] ([y, y] ⊕ [z, z]) = [x, x] [u, u] u∈(y+z)∪(z+y)

=

 S

=

 S

u∈y+z



[x, x] [u, u] ∪

 S

[x, x] [u, u]



u∈z+y

  S  {[v, v] | v ∈ x · u ∪ u · x} ∪ {[v, v] | v ∈ x · u ∪ u · x}

u∈y+z

= {[v, v] | v ∈ x · (y + z)} ∪ M (x, y, z),

u∈z+y

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where M (x, y, z) =

S

121

{[v, v] | v ∈ u · x} ∪

u∈y+z

Davvaz-Vougiouklis

S

{[v, v] | v ∈ x · u ∪ u · x}.

u∈z+y

On the other hand, [x, x] [y, y] = {[v, v] | v ∈ x · y ∪ y · x} = {[v, v] | v ∈ x · y} ∪ {[v, v] | v ∈ y · x}, [x, x] [z, z] = {[v, v] | v ∈ x · z} ∪ {[v, v] | v ∈ z · x} and then ([x, x] [y, y]) ⊕ ([x, x] [z, z])     = {[v, v]|v∈x·y} ∪ {[v, v]|v∈x·z} ⊕ {[v, v]|v∈z·x} ∪ {[v, v]|v∈y·x}     = {[v, v]|v∈x·y} ⊕ {[v, v]|v∈x·z} ∪ {[v, v]|v∈x·y} ⊕ {[v, v]|v∈z·x}     ∪ {[v, v]|v∈y·x} ⊕ {[v, v]|v∈x·z} ⊕ {[v, v]|v∈y·x} ∪ {[v, v]|v∈z · x}  S   S  = [v, v] ⊕ [u, u] ∪ [v, v] ⊕ [u, u] v ∈ x·y u ∈ x·z

 ∪

  [v, v] ⊕ [u, u] ∪

S v ∈ y·x u ∈ x·z

S

=

v ∈ x·y u ∈ z·x

 [v, v] ⊕ [u, u]

S v ∈ y·x u ∈ z·x

{[t, t] | t ∈ (v + u) ∪ (u + v)} ∪ K(x, y, z),

v ∈ y·x u ∈ x·z

where K(x, y, z) =  S   [v, v] ⊕ [u, u] ∪ v ∈ x·y u ∈ z·x

S

  [v, v] ⊕ [u, u] ∪

v ∈ y·x u ∈ x·z

S

 [v, v] ⊕ [u, u] .

v ∈ y·x u ∈ z·x

Now, we have ([x, x] [y, y]) ⊕ ([x, x] [z, z])  S   = {[t, t] | t ∈ u + v} ∪ v ∈ x·y u ∈ x·z

S

 {[t, t] | t ∈ u + v} ∪ K(x, y, z).

v ∈ x·y u ∈ x·z

From (x·y+x·z)∩x·(y+x) 6= ∅, it follows that [t0 , t0 ] ∈ {[v, v] | v ∈ x·(y+z)} for some t0 ∈ x · y + x · z, thus {[v, v] | v ∈ x · (y + z)} ∩ {[t, t] | t ∈ x · y + x · z} = 6 ∅, consequently the sets [x, x] ([y, y]⊕[z, z]) and ([x, x] [y, y])⊕([x, x] [z, z]) have a nonempty intersection. 

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From the above proof, it follows that only one (right or left) of the weak distributivity laws for (R, +, ·) ensures the weak distributivity of (D(R), ⊕, ). Definition 4.3. Let R1 and R2 be two Hv -rings. The map f : R1 → R2 is called an Hv -homomorphism or a weak homomorphism if, for all x, y ∈ R1 the following conditions hold: f (x + y) ∩ (f (x) + f (y)) 6= ∅ and f (x · y) ∩ f (x) · f (y) 6= ∅. f is called an inclusion homomorphism if, for all x, y ∈ R, the following relations hold: f (x + y) ⊆ f (x) + f (y) and f (x · y) ⊆ f (x) · f (y). Finally, f is called a strong homomorphism if for all x, y in R1 we have f (x + y) = f (x) + f (y) and f (x · y) = f (x) · f (y). If R1 and R2 are Hv -rings and there exists a strong one to one and onto homomorphism from R1 to R2 , then R1 and R2 are called isomorphic. Corollary 4.1. Let (R, +, ·) be an Hv -ring and rR (x) = [x, x] ∈ D(R) for any x ∈ R. Then, the mapping rR : (R, +, ·) → (D(R), ⊕, ) is an inclusion homomorphism of Hv -rings. Theorem 4.1. For any pair of Hv -rings (R, +, ·), (S, +, ·) and for any inclusion Hv -ring homomorphism f : (R, +, ·) → (S, +, ·) there exists exactly one inclusion homomorphism ψ : (D(R), ⊕, ) → (D(S), ⊕, ) such that the diagram f / (S, +, ·) (R, +, ·) rS

rR

 (D(R), ⊕, )

ψ

 / (D(S), ⊕, )

is commutative. Proof. Consider an arbitrary inclusion homomorphism f : R → S and define ψ : D(R) → D(S) as the restriction of the mapping f × f : R × R → S × S onto D(R) ⊆ R × R, i.e., ψ = (f × f )|D(R) , hence ψ([x, x]) = [f (x), f (x)] for any x ∈ R. Now, we have ψ([x, x] ⊕ [y, y]) = ψ({[u, u] | u ∈ (x + y) ∪ (y + x)} = {[f (u), f (u)] | u ∈ (x + y) ∪ (y + x)} = {[v, v] | v ∈ f (x + y) ∪ f (y + x)} ⊆ {[v, v] | v ∈ (f (x) + f (y)) ∪ (f (y) + f (x))} = [f (x) + f (y)] ⊕ [f (y) + f (x)] = ψ([x, x]) ⊕ ψ([y, y])

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for any elements x, y ∈ R and similarly ψ([x, x] [y, y]) ⊆ ψ([x, x])·ψ([y, y]), which is obtained as above. Now, we show that the above diagram commutes. Let us suppose that f : R → S is an inclusion homomorphism. Then evidently ψ : D(R) → D(S) is an inclusion homomorphism as well. For an arbitrary x in R, we have (rS ◦ f )(x) = rS (f (x)) = [f (x), f (x)] = (f × f )(x, x) = ψ([x, x]) = ψ(rR (x)) = (ψ ◦ rR )(x), and so rS ◦ f = ψ ◦ rR . Now, let g : D(R) → D(S) be an inclusion homomorphism such that rS ◦ f = g ◦ rR . Since rR : R → D(R) and −1 rS : S → D(S) are bijections, there exist the maps rR : D(R) → R and −1 rS : D(S) → S. Then, we obtain −1 −1 −1 ψ = ψ ◦ idD(R) = ψ ◦ rR ◦ rR = rS ◦ f ◦ rR = g ◦ rR ◦ rR = g ◦ idD(R) = g.

 From the above results we obtain the following theorem. Theorem 4.2. Let Hv R be the category of all Hv -rings and their inclusion homomorphisms and AHv R be its full subcategory of all commutative Hv rings. Then there exists the functor φ : Hv R → AHv R defined by φ(R, +, ·) = (D(R), ⊕, ), φ(f ) = ψ for any (R, +, ·) ∈ Ob(Hv R), and any morphism f ∈ M or(Hv R), f : (R, +, ·) → (S, +, ·) is a reflector; more precisely the pair (rR , (4R , ⊕, )) is an AHv R-reflection for any (R, +, ·) ∈ Ob(Hv R). Thus AHv R is a reflective full subcategory of the category Hv R. 4.2

Fundamental relations on Hv -rings

In what follows, we focus our attention on the β ∗ and γ ∗ relations defined on Hv -rings. Notice that two kinds of β ∗ relations can be defined on Hv ∗ rings. We denote them by β+ and β.∗ . They are β ∗ relations with respect to addition and multiplication, respectively. If (R, +, ·) is an Hv -ring, then the relations β+ and β. are defined as follows: xβ+ y ⇔ there exist z1 , ..., zn ∈ R such that {x, y}⊆z1 + ... + zn , xβ. y ⇔ there exist z1 , ..., zn ∈ R such that {x, y}⊆z1 ·...·zn . ∗ β+ and β.∗ are the transitive closures of the relations β+ and β. Note that ∗ the quotient hyperstructures with respect to β+ and β.∗ are Hv -rings. In

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this section, the fundamental relations defined on an Hv -ring are studied. Especially, some connections among different types of fundamental relations are obtained. Definition 4.4. Let (R, +, ·) be an Hv -ring. We define γ ∗ as the smallest equivalence relation such that the quotient R/γ ∗ is a ring. γ ∗ is called the fundamental equivalence relation and R/γ ∗ is called the fundamental ring. An Hv -ring is called an Hv -field if its fundamental ring is a field. Let us denote the set of all finite polynomials of elements of R over N by U. We define the relation γ as follows: xγy if and only if {x, y} ⊆ u, where u ∈ U. The following theorem is similar to Theorem 3.4, where it is proved by Vougiouklis [120]. Theorem 4.3. The fundamental equivalence relation γ ∗ is the transitive closure of the relation γ. Proof. Let γ b be the transitive closure of the relation γ. We denote the equivalence class of a by γ b(a). First, we prove that the quotient set R/b γ is a ring. The sum ⊕ and the product are defined in R/b γ in the usual manner: γ b(a) ⊕ γ b(b) = {b γ (c) | c ∈ γ b(a) + γ b(b)}, γ b(a) γ b(b) = {b γ (d) | d ∈ γ b(a) · γ b(b)}. Take a0 ∈ γ b(a) and b0 ∈ γ b(b). Then, we have a0 γ ba if and only if there exist x1 , ..., xm+1 with x1 =a0 , xm+1 =a and u1 , ..., um ∈ U such that {xi , xi+1 } ⊆ ui (i = 1, ..., m), and b0 γ bb if and only if there exist y1 , ..., yn+1 with y1 = b0 , yn+1 = b and v1 , ..., vn ∈ U such that {yj , yj+1 } ⊆ vj (j = 1, ..., n). Now, we obtain {xi , xi+1 } + y1 ⊆ ui + v1 (i = 1, ..., m − 1), xm+1 + {yj , yj+1 } ⊆ um + vj (j = 1, ..., n). The sums ui +v1 =ti (i=1, ..., m−1) and um +vj =tm+j−1 (j=1, ..., n) are polynomials and so tk ∈ U for all k ∈ {1, ..., m + n − 1}. Now, pick up the elements z1 , ..., zm+n such that zi ∈ xi + y1 (i = 1, ..., m) and zm+j ∈ xm+1 + yj+1 (j = 1, ..., n). Hence, we obtain {zk , zk+1 } ⊆ tk (k = 1, ..., m + n − 1). Therefore, every element z1 ∈ x1 + y1 = a0 + b0

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is γ b equivalent to every element zm+n ∈ xm+1 + yn+1 = a + b. Thus, γ b(a) ⊕ γ b(b) = γ b(c) for all c ∈ γ b(a) + γ b(b). In a similar way, it is proved that γ b(a) γ b(b) = γ b(d) for all d ∈ γ b(a) · γ b(b). The weak associativity and the weak distributivity on R guarantee that the associativity and distributivity are valid in the quotient R/b γ . Therefore, R/b γ is a ring. Now, let σ be an equivalence relation on R such that R/σ is a ring. Denote the equivalence class of a by σ(a). Then, σ(a)⊕σ(b) and σ(a) σ(b) are singletons for all a, b ∈ R, i.e., σ(a) ⊕ σ(b) = σ(c) for all c ∈ σ(a) + σ(b) and σ(a) σ(b) = σ(d) for all d ∈ σ(a) · σ(b). Thus, for every a, b ∈ R and A ⊆ σ(a), B ⊆ σ(b) we can write σ(a) ⊕ σ(b) = σ(a + b) = σ(A + B) and σ(a) σ(b) = σ(a · b) = σ(A · B). By induction, we extend these equalities on finite sums and products. So, for every u ∈ U and for all x ∈ u we have σ(x) = σ(u). Therefore, for every a ∈ R, x ∈ γ(a) implies x ∈ σ(a). Since σ is transitive, we obtain that x∈γ b(a) implies x ∈ σ(a). This means that the relation γ b is the smallest equivalence relation on R such that R/b γ is a ring, i.e., γ b = γ∗.  Definition 4.5. We define the γ1∗ , γ2∗ relations as the transitive closures of the relations γ1 , γ2 respectively, which are defined as follows: xγ1 y if and only if there exist ai ∈ R and Ik , K finite sets of indices such that P  Q  {x, y} ⊆ ai k∈K

i∈Ik

and xγ2 y if and only if there exist bj ∈ R and Js , S finite sets of indices such that  P  Q {x, y} ⊆ s∈S bj . j∈Js

In a multiplicative Hv -ring, the addition is an operation, while in an additive Hv -ring, the multiplication is an operation.

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Proposition 4.5. (1) R/γ1∗ is a multiplicative Hv -ring. (2) R/γ2∗ is an additive Hv -ring. Proof. We prove only (1) and similarly (2) can be proved. The sum of the classes is γ1∗ (x) ⊕ γ1∗ (y) = {γ1∗ (z) | z ∈ γ1∗ (x) + γ1∗ (y)}. PQ In the definition of γ1∗ , expressions of the type v = are used. In the definition of ⊕, the element z belongs to the sums v of the above type, which means that z belongs to a sum of products. In other words, all the elements z are in the same γ1∗ class. So, the sum of γ1∗ -classes is a singleton. Therefore, R/γ1∗ is a multiplicative Hv -ring.  Note that the γ1∗ classes are greater than the β ∗ classes. Actually, the is not the smallest equivalence relation such that (R/γ1∗ , ⊕) is a group. ∗ ∼ In order to see this, consider a multiplicative Hv -ring R. Then R/β+ = R, ∗ but R/γ1 is not isomorphic to R.

γ1∗

∗ . For all Proposition 4.6. For all additive Hv -rings, we have γ1∗ = β+ ∗ ∗ multiplicative Hv -rings, we have γ2 = β. .

Proof. We present the proof for a multiplicative Hv -ring R. In this case, every sum of elements of R is singleton. Therefore, Q  P  Q P bj = ds , where ds = bj . s∈S

j∈Js

s∈S

j∈Js

This means that xγ2∗ y if and only if xβ.∗ y.



Using the above propositions, it follows that (R/γ1∗ )/γ2∗ =(R/γ1∗ )/β.∗ is a ∗ is an additive Hv -ring. multiplication Hv -ring and (R/γ2∗ )/γ1∗ =(R/γ2∗ )/β+ Theorem 4.4. Let (R, +, ·) be an Hv -ring. Then, R/γ ∗ ∼ = (R/β.∗ )/β]∗ , ∗ ∗ where β] is the fundamental relation defined in (R/β. , ]) by setting β.∗ (a)] β.∗ (b) = {β.∗ (c) | c ∈ β.∗ (a) + β.∗ (b)}. Proof. The quotient of the additive Hv -ring (R/β.∗ , ], ⊗) with respect to β]∗ is a ring. Let us denote the equivalence relation associated to the projection ψ : R → (R/β.∗ )/β]∗ by σ. Since ψ is a ring homomorphism, then we obtain γ ∗ (a) ⊆ σ(a) for all a ∈ R. On the other hand, since β.∗ (x) ⊆ γ ∗ (x) for all x ∈ R, we have S S S β.∗ (z) = β.∗ (z) ⊆ γ ∗ (z). β.∗ (z)∈β.∗ (x)]β.∗ (y)

z∈β.∗ (x)+β.∗ (y)

z∈γ ∗ (x)+γ ∗ (y)

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From the fundamental property in (R/γ ∗ , ⊕, ), we know that γ ∗ (x)⊕γ ∗ (y) is a singleton, so γ ∗ (x) ⊕ γ ∗ (y) = γ ∗ (w), where w ∈ x + y. Therefore, S β.∗ (z) ⊆ γ ∗ (w), where w ∈ x + y. β.∗ (z)∈β.∗ (x)]β.∗ (y)

Consequently, for every finite sum of elements in R/β.∗ , we have S P β.∗ (z) ⊆ γ ∗ (w), where w ∈ i∈I xi . P z∈]

i∈I

β.∗ (xi )

Moreover, since γ ∗ is transitive, it follows that S σ(a) = β.∗ (z) ⊆ γ ∗ (a) for all a ∈ R. ∗ (β ∗ (a))} {z|(β.∗ (z)) β] .

∗

Therefore, σ = γ .



Theorem 4.5. If (H, ♦) is an Hv -group, then for every hyperoperation O such that {x, y} ⊆ xOy for all x, y ∈ H, the hyperstructures (H, ♦, O) and (H, O, ♦) are Hv -rings. Proof. Every hyperoperation O that satisfies the condition of hypothesis is weak associative, weak commutative and H/γ ∗ is a singleton. Moreover, every element x ∈ H is a unit element, i.e., y ∈ xOy ∩ yOx for all y ∈ H, and every element x ∈ H is symmetric with respect to the unit x, i.e., x ∈ xOy ∩ yOx. In order to prove that (H, ♦, O) is an Hv -ring we need only to prove the weak distributivity on the left. For every x, y, z in H we have xO(y♦z) ⊇ {x} ∪ (y♦z) and (xOy)♦(xOz) ⊇ {x, y}♦{x, z} = (x♦x) ∪ (x♦z) ∪ (y♦x) ∪ (y♦z). Therefore, y♦z ⊆ [xO(y♦z)] ∩ [(xOy)♦(xOz)] 6= ∅. Thus, the left and similarly the right weak distributivity are valid. Similarly, we need to prove the weak distributivity on the left for (H, O, ♦). For every x, y, x in H we have x♦(yOz) ⊇ x♦{y, z} = (x♦y) ∪ (x♦z) and (x♦y)O(x♦z) ⊇ (x♦y) ∪ (x♦z). So the left distributivity is valid, because (x♦y) ∪ (x♦z) ⊆ [x♦(yOz)] ∩ [(x♦y)O(x♦z)] 6= ∅. Hv -rings (H, O, ♦) and (H, ♦, O) are called associated Hv -rings.



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In the theory of representations of the hypergroups in the sense of Marty, there are three types of associated hyperrings (H, ⊕, ·) with the hypergroup (H, ·). The hyperoperation ⊕ is defined, respectively, for all x, y in H, as follows: type a: x ⊕ y = {x, y}, type b: x ⊕ y = β ∗ (x) ∪ β ∗ (y), type c: x ⊕ y = H. In all the above types, the strong associativity and strong or inclusion distributivity are valid. However, in Hv -structures there exists only one class of associated Hv -rings instead of three types. Theorem 4.6. Let (H, +) be an Hv -group with a scalar zero element 0. Then, for every hyperoperation ⊗ such that {x, y} ⊆ x ⊗ y, for all x, y ∈ H \ {0}, x ⊗ 0 = 0 ⊗ x = 0, for all x ∈ H, the hyperstructure (H, +, ⊗) is an Hv -ring. Proof.

For every nonzero elements x, y, z in H, we have y + z ⊆ [x ⊗ (y + z)] ∩ [(x ⊗ y) + (x ⊗ z)] 6= ∅.

Moreover, if one of the elements x, y, z is zero, then the strong distributivity is valid. The rest of the weak axioms are also valid.  Theorem 4.7. Let (H, ·) be an Hv -group. Take an element 0 6∈ H and denote H 0 = H ∪ {0}. We define the hyperoperation + as follows: 0 + 0 = 0, 0 + x = H = x + 0, x + y = 0 for all x, y ∈ H, and we extend the hyperoperation · in H 0 by putting 0 · 0 = 0 · x = x · 0 = 0 for all x ∈ H. Then, the hyperstructure (H 0 , +, ·) is an Hv -field with H 0 /γ ∗ ∼ = Z2 , where 0 is an absorbing and γ ∗ (0) is a singleton. Proof. From the definition it is clear that 0 is an absorbing element. The hyperoperation + is (strongly) associative because if in any triple (x, y, z) of elements of H 0 there are one or three non-zero elements, then their hypersum is 0; in the other cases, the result is H. The · is weak associative because 0 is an absorbing and (H, ·) is an Hv group. The strong distributivity of + with respect to · is valid, because the only one nonzero case is for x, y ∈ H in which we have x · (0 + y) = (0 + y) · x = x · 0 + x · y = 0 · x + y · x = H.

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Finally, one can check that γ ∗ (0) is a singleton and that there are only two fundamental classes in H 0 . Thus, (H 0 , +, ·) is an Hv -field and H 0 /γ ∗ ∼ =Z2 .  0 Notice that if the Hv -group (H, ·) is strongly associative, then (H , +, ·) is a hyperfield instead of an Hv -field; moreover, the strong distributivity is valid. Since γ ∗ (0) is a singleton, the Hv -fields of this type are very useful. This happens always in an Hv -group H, that we need to represent, for which the cardinality of the hyperproducts of the elements is equal to a power of cardH. On the other hand, the representations are normally of lower dimension and cardH is a small number. The Hv -groups of constant length, such as the P -hypergroups, can be also represented on these Hv -fields. Now, one can prove the following theorem. There is no need to check if the weak axioms are valid since they are obvious. Notice that nondegenerate fundamental rings or fields, which are desired actually, are obtained using this construction. Theorem 4.8. Let (R, +, ·) be a ring and J be an ideal. Then, we can define two Hb -operations
and greater than + and ·, respectively, for all x, y in R as follows: x
y ⊆ x + y + J and x

y ⊆ xy + J.

Then, the hyperstructure (R,
, ) is an Hv -ring for which the fundamental ring R/γ ∗ is a subring of R/J. Notice that the maximum of the above hyperadditions, i.e., x
y = x + y + J, is a P -hyperoperation so that the Hv -ring (R,
, ) can be a P -Hv -ring (see Section 6.4). Remark that for any maximal ideal J, one obtains R/γ ∗ = R/J. This construction leads to an enormous number of Hv -rings. Let us point out that if the products of the ring R are enlarged, then all hyperproducts with any cardinality can be represented and the main theorem of this theory is not trivial. 4.3

Uniting elements

The uniting elements method was introduced by Corsini and Vougiouklis [18] in 1989. With this method one puts in the same class, two or more elements. This leads, through hyperstructures, to structures satisfying additional properties.

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The uniting elements method is described as follows: Let G be an algebraic structure and let d be a property, which is not valid. Suppose that d is described by a set of equations; then, consider the partition in G for which it is put together, in the same partition class, every pair of elements that causes the non-validity of the property d. The quotient by this partition G/d is an Hv -structure. Then, quotient out the Hv -structure G/d by the fundamental relation β ∗ , a stricter structure (G/d)/β ∗ for which the property d is valid, is obtained. An interesting application of the uniting elements is when more than one properties are desired. The reason for this is some of the properties lead straighter to the classes than others. Therefore, it is better to apply the straightforward classes followed by the more complicated ones. The commutativity is one of the easy applicable properties. Moreover it is clear that the reproductivity property is also easily applicable. One can do this because the following is valid. Theorem 4.9. Let (G, ·) be a groupoid, and F = {f1 , ..., fm , fm+1 , ..., fm+n } be a system of equations on G consisting of two subsystems Fm = {f1 , ..., fm } and Fn = {fm+1 , ..., fm+n }. Let σ, σm be the equivalence relations defined by the uniting elements procedure using the systems F and Fm respectively, and let σn be the equivalence relation defined using the induced equations of Fn on the groupoid Gm = (G/σm )/β ∗ . Then, (G/σ)/β ∗ ∼ = (Gm /σn )/β ∗ . Analogous to the above theorem can be proved for rings. Theorem 4.10. Let (R, +, ·) be a ring, and F = {f1 , ..., fm , fm+1 , ..., fm+n } be a system of equations on R consisting of two subsystems Fm = {f1 , ..., fm } and Fn = {fm+1 , ..., fm+n }. Let σ, σm be the equivalence relations defined by the uniting elements procedure using the systems F and Fm respectively, and let σn be the equivalence relation defined using the induced equations of Fn on the ring

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Rm = (R/σm )/γ ∗ . Then (R/σ)/γ ∗ ∼ = (Rm /σn )/γ ∗ i.e., the following diagram is commutative R

ρm

R/σm

φm

Rm

ρ

ρn

R/σ

Rm /σn

φ

φn

(R/σ)/γ*

∼ =

(Rm /σn )/γ*

where all the maps ρ, φ, ρm , φm , ρn and φn are the canonicals. From the above it is clear that the fundamental structure is very important, mainly if it is known from the beginning. This is the problem to construct hyperstructures with desired fundamental structures.

4.4

Multiplicative Hv -rings

Algebraic hyperstructures are a generalization of the classical algebraic structures which, among others, are appropriate in two directions: (a) to represent a lot of application in an algebraic model, (b) to overcome restrictions ordinary structures usually have. Concerning the second direction the restrictions of the ordinary matrix algebra can be overcome by the helixoperations. More precisely, the helix addition and the helix-multiplication can be defined on every type of matrices. In [47], Davvaz et al. studied properties and examples on special classes of matrices. Let A = (aij ) ∈ Mm×n be a matrix and s, t ∈ N be two natural numbers such that 1 ≤ s ≤ m and 1 ≤ t ≤ n. Then, we define the characteristic-like map cst from Mm×n to Ms×t by corresponding to A the matrix Acst = (aij ), where 1 ≤ i ≤ s and 1 ≤ j ≤ t. We call this the cut-projection map of type st. In other words, Acst is a matrix obtained from A by cutting the lines and columns greater than s and t respectively. Let A = (aij ) ∈ Mm×n be a matrix and s, t ∈ N be two natural numbers such that 1 ≤ s ≤ m and 1 ≤ t ≤ n. Then, we define the mod-like map st

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from Mm×n to Ms×t by corresponding to A the matrix Ast = (Aij ) which has as entries the sets Aij = {ai+ks,j+λt | k, λ ∈ N, i + ks ≤ m, j + λt ≤ n}, for 1 ≤ i ≤ s and 1 ≤ j ≤ t. We call this multivalued map helix-projection of type st. Therefore, Ast is a set of s × t-matrices X = (xij ) such that xij ∈ Aij for all i, j. Obviously, Amn = A. Example 4.2. Let us consider the following matrix:   2 1 34 2 3 2 0 1 2   A=  2 4 5 1 −1  . 1 −1 0 0 8 Suppose that s = 3 and t = 2. Then, 

 21 Ac32 =  3 2  24 and A32 = (Aij ), where A11 A12 A21 A22 A31 A32

= {a11 , a13 , a15 , a41 , a43 , a45 } = {2, 3, 2, 1, 0, 8}, = {a12 , a14 , a42 , a44 } = {1, 4, −1, 0}, = {a21 , a23 , a25 } = {3, 0, 2}, = {a22 , a24 } = {2, 1}, = {a31 , a33 , a35 } = {2, 5, −1}, = {a32 , a34 } = {4, 1}.

Therefore, 

 {2, 3, 1, 0, 8} {1, 4, −1, 0} A32 = (Aij ) =  {3, 0, 2} {2, 1}  {2, 5, −1} {4, 1} = {(xij ) | x11 ∈ {0, 1, 2, 3, 8}, x12 ∈ {−1, 0, 1, 4}, x21 ∈ {0, 2, 3}, x22 ∈ {1, 2}, x31 ∈ {−1, 2, 5}, x32 ∈ {1, 4}}. Therefore |A32| = 720. Let A = (aij ) ∈ Mm×n and B = (aij ) ∈ Mu×v be two matrices and s = min(m, u), t = min(n, u). We define an addition, which we call cutaddition, as follows: ⊕c : Mm×n × Mu×v → Ms×t (A, B) 7→ A ⊕c B = Acst + Bcst.

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Let A = (aij ) ∈ Mm×n and B = (aij ) ∈ Mu×v be two matrices and s = min(n, u). Then, we define a multiplication, which we call cutmultiplication, as follows: ⊗c : Mm×n × Mu×v → Mm×v (A, B) 7→ A ⊗c B = Acms · Bcsv. The cut-addition is associative and commutative. Let A = (aij ) ∈ Mm×n and B = (aij ) ∈ Mu×v be two matrices and s = min(m, u), t = min(n, v). We define a hyper-addition, which we call helix-addition or helix-sum, as follows: ⊕ : Mm×n × Mu×v → P(Ms×t ) (A, B) 7→ A ⊕ B = Ast +h Bst, where Ast +h Bst = {(cij ) = (aij + bij ) | aij ∈ Aij , bij ∈ Bij } . Example 4.3. Suppose that   21 A = 0 1 23



 140 and B = . 201

Then,  A22 =

A11 A12 A21 A22



 and B22 =

 B11 B12 , B21 B22

where A11 A12 A21 A22

= {a11 , a31 } = {2}, B11 = {a12 , a32 } = {1, 3}, B12 = {a21 } = {0}, B21 = {a22 } = {1}, B22

= {b11 , b13 } = {1, 0}, = {b12 } = {4}, = {b21 , b23 } = {2, 1}, = {b22 } = {0}.

So  A22 =

   21 23 , , 01 01

and  B22 =

       14 14 04 04 , , , . 20 10 20 10

Therefore, we have  A22 +h B22 =

         35 35 25 25 37 , , , , , 21 11 21 11 21       37 27 27 , , . 11 21 11

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Lemma 4.2. The helix-addition is commutative. Let A = (aij ) ∈ Mm×n and B = (aij ) ∈ Mu×v be two matrices and s = min(n, u). Then, we define a hyper-multiplication, which we call helixhyperoperation, as follows: ⊗ : Mm×n × Mu×v → P(Mm×v ) (A, B) 7→ A ⊗ B = Ams ·h Bsv, where n X  o Ams ·h Bsv = (cij ) = ait btj | aij ∈ Aij , bij ∈ Bij . Example 4.4. We consider the matrices A and B as follows:     1020 −1 1 A= and B = . 3132 0 2 Then,  A22 =

 {1, 2} 0 . 3 {1, 2}

Therefore,  A⊗B =

 {−1, −2} {1, 2} . −3 {5, 7}

Proposition 4.7. (1) The cut-multiplication ⊗c is associative. (2) The helix-multiplication ⊗ is weak associative. Proof.

It is straightforward.



Note that the helix-multiplication is not distributive (not even weak) with respect to the helix-addition. But if all matrices which are used in the distributivity are of the same type Mm×n , Then, we have A ⊗ (B ⊕ C) = A ⊗ (B + C) and (A ⊗ B) ⊕ (A ⊗ C) = (A ⊗ B) + (A ⊗ C). Therefore, the weak distributivity is valid and more precisely the inclusion distributivity is valid. One of the kinds of hyperrings was introduced and studied by Rota [90] in 1982, where the multiplication is a hyperoperation, while the addition is an ordinary operation. Many properties of multiplication hyperrings are investigated in [42, 90]. A multiplicative Hv -ring is a generalization of a multiplicative hyperring and a special case of Hv -rings, where the associative law and the distributive law are replaced by their corresponding weak axioms.

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Definition 4.6. The hyperstructure (R, +, ·) is a multiplicative Hv -ring if (1) (R, +) is an abelian group, (2) (R, ·) is an Hv -semigroup, (3) · is weak distributive with respect to +, i.e., x · (y + z) ∩ (x · y + x · z) 6= ∅ and (x + y) · z ∩ (x · z + y · z) 6= ∅, for all x, y, z ∈ R. The fundamental relation γ ∗ is defined in multiplicative Hv -rings as the smallest equivalence relation, so that the quotient would be ordinary ring. The way to find the fundamental classes is given by Theorem 4.3 to the following. Theorem 4.11. Let (R, +, ·) be a multiplicative Hv -ring and let us denote by U the set of all finite sum of finite product of elements of R. We define the relation γ in R as follows: xγy ⇔ {x, y} ⊆ u, for some u ∈ U. Then, the fundamental relation γ ∗ is the transitive closure of the relation γ. Theorem 4.12. Let R = Mm×n . Then, R together with addition and helix-multiplication becomes a multiplicative Hv -ring. Remark that the helix-operations give a wide class of Hv -rings, therefore the research can be restricted in special classes, finite or infinite, obtained by several types of matrices. Following this remark we can see several examples presented in this paragraph. In the following examples, we denote Eij any type of matrices which have the ij-entry 1 and in all the others entries we have 0. Example 4.5. Consider the 1 × 2 matrices, with entries in Z4 , of the following form: ¯ 11 and B ¯ = λE ¯ 11 + ¯2E12 , Ak¯ = kE λ ¯ λ ¯ ∈ Z4 . Suppose that R = {A¯ , B ¯ | k, ¯ λ ¯ ∈ Z4 }. Clearly, (R, ⊕) is an for k, k λ

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abelian group. Also, we have     ¯ ¯0 Ak¯ ⊗ Aλ¯ = k¯ ¯0 ⊗ λ     ¯ ¯0} ·h λ ¯ ¯0 = {k,   ¯ ¯0} ¯0 = {k¯λ,   ¯ ¯0 , [¯0 ¯0] = k¯λ = {Ak¯λ¯ , A¯0 } ,     ¯ ¯2 Bk¯ ⊗ Bλ¯ = k¯ ¯ 2 ⊗ λ     ¯ ¯2} ·h λ ¯ ¯2 = {k,   ¯ ¯2λ} ¯ {¯2k, ¯ ¯0} = {k¯λ,         ¯ ¯2k¯ , k¯λ ¯ ¯0 , ¯2λ ¯ 2¯k¯ , ¯2λ ¯ 0¯ . = k¯λ Since ¯ 2k¯ = ¯ 0 or ¯ 2, it follows that  ¯ ¯   ¯  k λ ¯ 0 , ¯ 2λ ¯0 = {Ak¯λ¯ , A¯2λ¯ }  Bk¯ ⊗ Bλ¯ = or  ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ k λ 2 , k λ 0 , 2λ 2 , 2λ 0 = {Bk¯λ¯ , Ak¯λ¯ , B¯2λ¯ , A¯2λ¯ } . Also, we have     ¯ ¯2 Ak¯ ⊗ Bλ¯ = k¯ ¯0 ⊗ λ   ¯ ¯0} {¯2k, ¯ ¯0} = {k¯λ,       ¯ ¯2k¯ , k¯λ ¯ ¯0 , ¯0 ¯2k¯ , [¯0 ¯0] . = k¯λ Since ¯ 2k¯ = ¯ 0 or ¯ 2, it  ¯ ¯  kλ Ak¯ ⊗ Bλ¯ = or  ¯ ¯ kλ Finally, we have

follows that  ¯ 0 , [¯ 0 ¯0] = {Ak¯λ¯ , A¯0 }        ¯ ¯0 , ¯0λ ¯ ¯2 , ¯0λ ¯ ¯0 = {B¯ ¯ , A¯ ¯ , B¯0 , A¯0 } . ¯ 2 , k¯λ kλ kλ     ¯ ¯2 ⊗ k¯ ¯2 Bλ¯ ⊗ Ak¯ = λ     ¯ ¯2} ·h k¯ ¯0 = {λ,     ¯ ¯0 , ¯2k¯ ¯0 = k¯λ = {Ak¯λ¯ , A¯2k¯ } .

Therefore, (R, ⊕, ⊗) is a multiplicative Hv -ring. The set {Ak¯ | k¯ ∈ Z4 } is a left Hv -ideal of R.

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Example 4.6. Consider the 2 × 3 matrices of the form A(k,λ) = kE11 + λE22 , where k, λ ∈ Z. Let R = {A(k,λ) | k, λ ∈ Z}. Then, we have A(k,λ) ⊕ A(k0 ,λ0 ) = A(k+k0 ,λ+λ0 ) . Indeed, (R, ⊕) is an abelian group. On the other hand, we have    0  k00 k 0 0 A(k,λ) ⊗ A(k0 ,λ0 ) = ⊗ 0λ0 0 λ0 0    0  {k, 0} 0 k 0 0 = ·h 0 λ 0 λ0 0  0    kk 0 0 0 0 0 = , . 0 λλ0 0 0 λλ0 0  = A(kk0 ,λλ0 ) , A(0,λλ0 ) . Clearly, ⊗ is commutative and associative. Moreover, ⊗ is distributive respect to ⊕. Therefore, (R, ⊕, ⊗) is a commutative multiplicative hyperring. The following sets       k00 000 I= |k∈Z and J = |λ∈Z 000 0λ0 are hyperideals of R. Example 4.7. Consider the 2 × 3 matrices of the form A(x,y,z) = xE11 + yE21 + zE22 , where x, y, z ∈ R. Let R = {A(x,y,x) | x, y, z ∈ R}. Then, (R, ⊕) is an abelian group. On the other hand, A(x,y,z) ⊗ A(x0 ,y0 ,z0 )    0  x00 x 0 0 = ⊗ 0 0 yz0 y z 0    0  {x, 0} 0 x 0 0 = ·h 0 0 {y, 0} z y z 0      0    0 0 0 xx 0 0 xx0 0 0 0 0 0 = , , , yx0 + zy 0 zz 0 0 zy 0 zz 0 0 yx0 + zy 0 zz 0 0 zy 0 zz 0 0  = A(0,xy0 ,zz0 ) , A(0,yx0 +zy0 ,zz0 ) , A(xx0 ,zy0 ,zz0 ) , A(xx0 ,yx0 +zy0 ,zz0 ) .

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Therefore, (R, ⊕, ⊗) is a multiplicative Hv -ring. Note that ⊗ is not (weak) commutative.     100 x00 The unit matrix is Ic = E11 + E22 = . Let is given. 010 yz0   a00 This matrix is invertible if and only if x 6= 0 and z 6= 0. An element bc0   x00 is inverse of if we have yz0       100 a00 x00 ∈ ⊗ 010 bc0 yz0 and       100 x00 a00 ∈ ⊗ . 010 yz0 bc0 From the first one, we obtain (ax = 1, cy = 0, cz = 1) or (ax = 1, bx + cy = 0, cz = 1), and from the second, we obtain (xa = 1, zb = 0, zc = 1) or (xa = 1, ya + zb = 0, zc = 1). 1 the inverse Therefore, we obtain solutions a = x1 , b = −y xz , c = z . Thus,   1  1  x00 0 0 0 0 x x of is the matrix −y 1 . Moreover the element is also yz0 0 z1 0 xz z 0    x00 100 an inverse element of with respect to the unit matrix . yz0 010 The set    000 I= |y∈R y00 is a hyperideal of R.

Example 4.8. (Finite case). Suppose that R = {O, A, B, C}, where         ¯ ¯1 ¯0 ¯0 ¯0 ¯0 ¯0 ¯1 ¯0 ¯0 0¯ 0¯ 0 O= ¯¯¯ , A= ¯¯¯ , B= ¯¯¯ , C= ¯¯¯ , 000 000 100 100 with entries in Z2 . Clearly, (R, ⊕) is an abelian group. For helixmultiplication we obtain the following Cayley’s table: ⊗ O A B C O O O O O A O {O, A} O {O, A} B O {O, B} {O, B} {O, B} C O {O, A, B, C} O {O, A, B, C} Then, (R, ⊕, ⊗) is a multiplicative Hv -ring.

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Hv -fields

More general structures can be defined by using the fundamental structures. An application in this direction is the general hyperfield. Definition 4.7. An Hv -ring (R, +, ·) is called Hv -field if R/γ ∗ is a field. Definition 4.8. The Hv -semigroup (H, ·) is called h/v-group if H/β ∗ is a group. Definition 4.9. An Hv -structure is called very thin if all hyperoperations are operations except one, which has all hyperproducts singletons except one, which is a subset of cardinality more than one. Therefore, in a very thin Hv -structure in H there exists a hyperoperation · and a pair (a, b) ∈ H 2 for which ab = A, with cardA > 1, and all the other products, are singletons. In this section, we denote by [x] the fundamental class of the element x ∈ H. Therefore β ∗ (x) = [x]. Analogous theorems are for Hv -rings, Hv -vector spaces and so on. An element is called single if its fundamental class is singleton, so [x] = {x}. The class of h/v-groups is more general than the Hv -groups since in h/v-groups the reproductivity is not valid. The reproductivity of classes is valid, i.e., if H is partitioned into equivalence classes, then x[y] = [xy] = [x]y, for all x, y ∈ H, because the quotient is reproductive. In a similar way the h/v-rings, h/vfields, h/v-modulus, h/v-vector spaces etc. are defined. Note that h/vstructures were introduced by Vougiouklis for the first time. Remark 4.1. From definition of the Hv -field, we remark that the reproduction axiom in the product, is not assumed, the same is also valid for the definition of the h/v-field. Therefore, an Hv -field is an h/v-field where the reproduction axiom for the sum is also valid. We know that the reproductivity in the classical group theory is equivalent to the axioms of the existence of the unit element and the existence of an inverse element for any given element. From the definition of the h/v-group, since a generalization of the reproductivity is valid, we have to extend the above two axioms on the equivalent classes. Definition 4.10. Let (H, ·) be an Hv -semigroup, and denote [x] the fundamental, or equivalent classe, of the element x ∈ H. We call [e] the unit

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class if we have ([e] · [x]) ∩ [x] 6= ∅ and ([x] · [e]) ∩ [x] 6= ∅, for all x ∈ H, and for each element x ∈ H, we call inverse class of [x], the class [x0 ], if we have ([x] · [x0 ]) ∩ [e] 6= ∅ and ([x0 ] · [x]) ∩ [e] 6= ∅. The “enlarged” hyperstructures were examined in the sense that a new element appears in one result. In enlargement or reduction, most useful are those Hv -structures or h/v-structures with the same fundamental structure. Construction 4.13. (1) Let (H, ·) be an Hv -semigroup and v ∈ / H. We extend the · into H = H ∪ {v} as follows: x · v = v · x = v, for all x ∈ H, and v · v = H. The (H, ·) is an h/v-group, called attach, where (H, ·)/β ∗ ∼ = Z2 and v is a single element. We have core (H, ·) = H. The scalars and units of (H, ·) are scalars and units, respectively, in (H, ·). If (H, ·) is weak commutative (respectively, commutative), then (H, ·) is also weak commutative (respectively, commutative). (2) Let (H, ·) be an Hv -semigroup and {v1 , . . . , vn } ∩ H = ∅, is an ordered set, where if vi < vj , when i < j. Extend · in H n = H ∪ {v1 , . . . , vn } as follows: x · vi = vi · x = vi , vi · vj = vj · vi = vj , for all i < j and vi · vi = H ∪ {v1 , . . . , vi−1 }, for all x ∈ H, i ∈ {1, . . . , n}. Then (H n , ·) is h/v-group, called attach elements, where (H n , ·)/β ∗ ∼ = Z2 and vn is single. (3) Let (H, ·) be an Hv -semigroup, v ∈ / H, and (H, ·) be its attached h/v-group. Take an element 0 ∈ / H and define in H o = H ∪ {v, 0} two hyperoperations: Hypersum +: 0 + 0 = x + v = v + x = 0, 0 + v = v + 0 = x + y = v, 0 + x = x + 0 = v + v = H, for all x, y ∈ H. Hyperproduct ·: remains the same as in H. Moreover, 0 · 0 = v · x = x · 0 = 0, for all x ∈ H. Then, (H o , +, ·) is h/v-field with (H o , +, ·)/γ ∗ ∼ = Z3 . Moreover, + is associative, · is weak associative and weak distributive with respect to +. Also, 0 is zero absorbing and single but not scalar in +. (H o , +, ·) is called the attached h/v-field of the Hv -semigroup (H, ·).

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Let us denote by U the set of all finite products of elements of a hypergroupoid (H, ·). Consider the relation defined as follows: xLy ⇔ there exists u ∈ U such that ux ∩ uy 6= ∅. Then the transitive closure L∗ of L is called left fundamental reproductivity relation. Similarly, the right fundamental reproductivity relation R∗ is defined. Theorem 4.14. If (H, ·) is a commutative semihypergroup, i.e. the strong commutativity and the strong associativity is valid, then the strong expression of the above L relation: ux = uy, has the property: L∗ = L. Proof. Suppose that two elements x and y of H are L∗ equivalent. Therefore, there are u1 , . . . , un+1 elements of U and z1 , . . . , zn elements of H such that u1 x = u1 z1 , u2 z1 = u2 z2 , . . . , un zn−1 = un zn , un+1 zn = un+1 y. From these relations, using the strong commutativity, we obtain un+1 . . . u2 u1 x = un+1 . . . u2 u1 z1 = un+1 . . . u1 u2 z1 = un+1 . . . u2 u1 z2 = · · · = un+1 . . . u2 u1 y. Therefore, setting u = un+1 . . . u2 u1 ∈ U, we have ux = uy.



We present now the small non-degenerate Hv -fields on (Zn , +, ·) which satisfy the following conditions, appropriate in Santilli’s iso-theory: (1) (2) (3) (4)

multiplicative very thin minimal, weak commutative (non-commutative), they have the elements 0 and 1, scalars, when an element has inverse element, then this is unique.

Note that last condition means than we cannot enlarge the result if it is 1 and we cannot put 1 in enlargement. Moreover we study only the upper triangular cases, in the multiplicative table, since the corresponding under, are isomorphic since the commutativity is valid for the underline rings. From the fact that the reproduction axiom in addition is valid, we have always Hv -fields. Example 4.9. All multiplicative Hv -fields defined on (Z4 , +, ·), which have non-degenerate fundamental field, and satisfy the above 4 conditions, are the following isomorphic cases: The only product which is set is 2 ⊗ 3 = {0, 2} or 3 ⊗ 2 = {0, 2}. The fundamental classes are [0] = {0, 2}, [1] = {1, 3} and we have (Z4 , +, ⊗)/γ ∗ ∼ = (Z2 , +, ·).

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Example 4.10. Let us denote by Eij the matrix with 1 in the ij-entry and zero in the rest entries. Take the following 2 × 2 upper triangular Hv matrices on the above Hv -field (Z4 , +, ·) of the case that only 2 ⊗ 3 = {0, 2} is a hyperproduct: I = E11 + E22 , a = E11 + E12 + E22 , b = E11 + 2E12 + E22 , c = E11 + 3E12 + E22 , d = E11 + 3E22 , e = E11 + E12 + 3E22 , f = E11 + 2E12 + 3E22 , g = E11 + 3E12 + 3E22 . Then, we obtain for X = {I, a, b, c, d, e, f, g}, that (X, ⊗) is non-COW Hv group and the fundamental classes are a = {a, c}, d = {d, f }, e = {e, g} and the fundamental group is isomorphic to (Z2 × Z2 , +). In this Hv -group there is only one unit and every element has a unique double inverse. Theorem 4.15. All multiplicative Hv -fields defined on (Z6 , +, ·), which have non-degenerate fundamental field, and satisfy the above 4 conditions, are the following isomorphic cases: We have the only one hyperproduct, • 2 ⊗ 3 = {0, 3} or 2 ⊗ 4 = {2, 5} or 3 ⊗ 4 = {0, 3} or 3 ⊗ 5 = {0, 3} or 4 ⊗ 5 = {2, 5} Fundamental classes: [0] = {0, 3}, [1] = {1, 4}, [2] = {2, 5}, and (Z6 , +, ·)/γ ∗ ∼ = (Z3 , +, ·). • 2 ⊗ 3 = {0, 2} or 2 ⊗ 3 = {0, 4} or 2 ⊗ 4 = {0, 2} or 2 ⊗ 4 = {2, 4} or 2 ⊗ 5 = {0, 4} or 2 ⊗ 5 = {2, 4} or 3 ⊗ 4 = {0, 2} or 3 ⊗ 4 = {0, 4} or 3 ⊗ 5 = {3, 5} or 4 ⊗ 5 = {0, 2} or 4 ⊗ 5 = {2, 4} Fundamental classes: [0] = {0, 2, 4}, [1] = {1, 3, 5}, and (Z6 , +, ⊗)/γ ∗ ∼ = (Z2 , +, ·). Example 4.11. All multiplicative Hv -fields defined on (Z9 , +, ·), which have non-degenerate fundamental field, and satisfy the above 4 conditions, are the following isomorphic cases: We have the only one hyperproduct, 2 ⊗ 3 = {0, 6} or {3, 6}, 2 ⊗ 4 = {2, 8} or {5, 8}, 2 ⊗ 6 = {0, 3} or {3, 6},

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2 ⊗ 7 = {2, 5} 3 ⊗ 5 = {0, 6} 3 ⊗ 8 = {0, 6} 4 ⊗ 8 = {2, 5} 5 ⊗ 8 = {1, 4}

or or or or or

{5, 8}, {3, 6}, {3, 6}, {5, 8}, {4, 7},

2 ⊗ 8 = {1, 7} 3 ⊗ 6 = {0, 3} 4 ⊗ 5 = {2, 5} 5 ⊗ 6 = {0, 3} 6 ⊗ 7 = {0, 6} 7 ⊗ 8 = {2, 5}

143

or or or or or or

{4, 7}, {0, 6}, {2, 8}, {3, 6}, {3, 6}, {2, 8},

3 ⊗ 4 = {0, 3} 3 ⊗ 7 = {0, 3} 4 ⊗ 6 = {0, 6} 5 ⊗ 7 = {2, 8} 6 ⊗ 8 = {0, 3}

or or or or or

{3, 6}, {3, 6}, {3, 6}, {5, 8}, {3, 6},

Fundamental classes are [0] = {0, 3, 6}, [1] = {1, 4, 7}, [2] = {2, 5, 8}, and (Z9 , +, ⊗)/γ ∗ ∼ = (Z3 , +, ·). Example 4.12. All Hv -fields defined on (Z10 , +, ·), which have nondegenerate fundamental field, and satisfy the above 4 conditions, are the following isomorphic cases: • We have the only one hyperproduct, 2 ⊗ 4 = {3, 8}, 2 ⊗ 5 = {2, 5}, 2 ⊗ 6 = {2, 7}, 2 ⊗ 7 = {4, 9}, 2 ⊗ 9 = {3, 8}, 3 ⊗ 4 = {2, 7}, 3 ⊗ 5 = {0, 5}, 3 ⊗ 6 = {3, 8}, 3 ⊗ 8 = {4, 9}, 3 ⊗ 9 = {2, 7}, 4 ⊗ 5 = {0, 5}, 4 ⊗ 6 = {4, 9}, 4 ⊗ 7 = {3, 8}, 4 ⊗ 8 = {2, 7}, 5 ⊗ 6 = {0, 5}, 5 ⊗ 7 = {0, 5}, 5 ⊗ 8 = {0, 5}, 5 ⊗ 9 = {0, 5}, 6 ⊗ 7 = {2, 7}, 6 ⊗ 8 = {3, 8}, 6 ⊗ 9 = {4, 9}, 7 ⊗ 9 = {3, 8}, 8 ⊗ 9 = {2, 7}. Fundamental classes: [0] = {0, 5}, [1] = {1, 6}, [2] = {2, 7}, [3] = {3, 8}, [4] = {4, 9} and (Z10 , +, ⊗)/γ ∗ ∼ = (Z5 , +, ·). • The cases where we have two classes [0] = {0, 2, 4, 6, 8} and [1] = {1, 3, 5, 7, 9}, thus we have fundamental field (Z10 , +, ⊗)/γ ∗ ∼ = (Z2 , +, ·), can be described as follows: Taking in the multiplicative table only the results above the diagonal, we enlarge each of the products by putting one element of the same class of the results. We do not enlarge setting the element 1, and we cannot enlarge only the product 3 ⊗ 7 = 1. The number of those Hv -fields is 103. Example 4.13. In order to see how hard is to realize the reproductivity of classes and the unit class and inverse class, we consider the above Hv -field (Z10 , +, ⊗) where we have 2 ⊗ 4 = {3, 8}. Then, the multiplicative table of

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the hyperproduct is the following: ⊗ 0 1 2 3 4 5 6 7 8 9

0 0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8 9

2 0 2 4 6 8 0 2 4 6 8

3 4 0 0 3 4 6 3, 8 9 2 2 6 5 0 8 4 1 8 4 2 7 6

5 0 5 0 5 0 5 0 5 0 5

6 0 6 2 8 4 0 6 2 8 4

7 0 7 4 1 8 5 2 9 6 3

8 0 8 6 4 2 0 8 6 4 2

9 0 9 8 7 6 5 4 3 2 1

On this table it is easy to see that the reproductivity is not valid but it is very hard to see that the reproductivity of classes is valid. We can see the reproductivity of classes easier if we reformulate the multiplicative table according to the fundamental classes, [0] = {0, 5}, [1] = {1, 6}, [2] = {2, 7}, [3] = {3, 8}, [4] = {4, 9}. Then, we obtain ⊗ 0 5 1 6 2 7 3 8 4 9

0 0 0 0 0 0 0 0 0 0 0

5 0 5 5 0 0 5 5 0 0 5

1 0 5 1 6 2 7 3 8 4 9

6 0 0 6 6 2 2 8 8 4 4

2 0 0 2 2 4 4 6 6 8 8

7 0 5 7 2 4 9 1 6 8 3

3 0 5 3 8 6 1 9 4 2 7

8 4 0 0 0 0 8 4 8 4 6 3, 8 6 8 4 2 4 2 2 6 2 6

9 0 5 9 4 8 3 7 2 6 1

From this it is easy to see the unit class and the inverse class of each class. 4.6

Hv -rings endowed with P -hyperoperations

In this section, we study a wide class of Hv -rings obtained from an arbitrary ring by using P -hyperoperations. The notion of P -hyperoperations and their generalizations are introduced in [112, 114]. We use the results obtained by S. Spartalis [103, 105].

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Let (R, +, ·) be a ring and P1 , P2 be nonempty subsets of R. We shall use of the following right P -hyperoperations: xP1∗ y = x + y + P1 , xP2∗ y = xyP2 for all x, y ∈ R. We denote the center of the semigroup (R, ·) by Z(R). Theorem 4.16. If 0 ∈ P1 and P2 ∩ Z(R) 6= ∅, then (R, P1∗ , P2∗ ) is an Hv -ring called a P -Hv -ring or an Hv -ring with P -hyperoperations. Proof.

The proof is straightforward.



Let J be an Hv -ideal of (R, P1∗ , P2∗ ). Since 0 ∈ RP2∗ J ∩ JP2∗ R ⊆ J and P1 = 0P1∗ 0 ⊆ J, we have JP1∗ x = J + x = xP1∗ J for all x ∈ R. Moreover, the addition ⊕ and the multiplication between classes are defined in a usual manner: (JP1∗ x) ⊕ (JP1∗ y) = {JP1∗ z| z ∈ (JP1∗ x)P1∗ (JP1∗ y)} = {J + x + y}, (JP1∗ x) (JP1∗ y) = {J + w| w ∈ (JP1∗ x)P2∗ (JP1∗ y)} = {J + w| w ∈ xyP2 }. Theorem 4.17. If (R, P1∗ , P2∗ ) is a P -Hv -ring and J is an Hv -ideal, then (R/J, ⊕, ) is a multiplicative Hv -ring. Proof. Obviously, (R/J, ⊕) is an abelian group. Moreover, (R/J, ) is an Hv -semigroup, because is well defined and for all x, y, z ∈ R, we have (JP1∗ x) [(JP1∗ y) (JP1∗ z)]= {J + v | v ∈ xyzP2 P2 }, [(JP1∗ x) (JP1∗ y)] (JP1∗ z)= {J + u | u ∈ xyP2 zP2 }. But, since zP2 P2 ∩ P2 zP2 6= ∅, it follows that the multiplication is weak associative. Finally, (JP1∗ x) [(JP1∗ y) ⊕ (JP1∗ z)] = {J + v | v ∈ x(y + z)P2 } ⊆ {J + u | u ∈ xyP2 + xzP2 } = [(JP1∗ x) (JP1∗ y)] ⊕ [(JP1∗ x) (JP1∗ z)]. In the same way, the right distributivity is proved and so (R/J, ⊕, ) is a multiplicative Hv -ring.  Theorem 4.18. Let (R, P1∗ , P2∗ ) be a P -Hv -ring over the ring (R, +, ·) and J be an ideal of the ring R, containing P1 . Then (R/J, ⊕, ) is a multiplicative Hv -ring, which is a P -Hv -ring.

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Proof. It is easy to see that J is an Hv -ideal of (R, P1∗ , P2∗ ). So, according to the previous theorem, (R/J, ⊕, ) is a multiplicative Hv -ring. On the other hand, consider the quotient ring (R/J, +, ·) and take L1 = {J}, L2 = {J + a | a ∈ P2 }. Then, for all x ∈ R, we have (J + x)L2 L2 = {J + v | v ∈ xP2 P2 }, L2 (J + x)L2 = {J + w | w ∈ P2 xP2 }. Since for all x ∈ R, xP2 P2 ∩ P2 xP2 6= ∅, it follows that (J + x)L2 L2 ∩ L2 (J + x)L2 6= ∅. Consequently, since J is the zero element of R/J, from Theorem 4.16 it follows that (R/J, L∗1 , L∗2 ) is a P -Hv -ring over the ring (R/J, +, ·). Therefore, the hyperoperation is the P -hyperoperation L∗2 and ⊕ is the degenerate P -hyperoperation L∗1 .  Theorem 4.19. Let (R, P1∗ , P2∗ ) be a P -Hv -ring and J be an Hv -ideal of R. If H is an Hv -subring of R containing P1 , then HP1∗ J/J ∼ = H/H ∩ J. Proof. It is easy to see that H, J are subgroups of (R, +) and so HP1∗ J = H + J and H ∩ J are two groups containing P1 . Since (HP1∗ J)P2∗ (HP1∗ J) = (H + J)(H + J)P2 ⊆ HHP2 + J we have that (HP1∗ J, P1∗ , P2∗ ) is an Hv -subring of (R, P1∗ , P2∗ ). Moreover, J, J ∩ H are Hv -ideals of the Hv -rings HP1∗ J and H respectively. On the other hand, the quotients HP1∗ J/J = {J + x | x ∈ H} and H/H ∩ J = {(H ∩ J) + y | y ∈ H} are multiplicative Hv -rings. We consider the bijection map f : HP1∗ J/J → H/H ∩ J such that J + x 7−→ (H ∩ J) + x. The map f is a homomorphism, since for all x, y ∈ R, we have f (J+x ⊕ J+y) = f (J + x + y) = (H ∩ J) + x + y = f (J + x) ⊕ f (J + y), f (J+x J+y) = f (J + s | s ∈ xyP2 }) = {(H ∩ J) + s | s ∈ xyP2 } = f (J + x) f (J + y). Consequently, f is an isomorphism.



Theorem 4.20. Let J and K be two Hv -ideals of the P − Hv -ring (R, P1∗ , P2∗ ). If J ⊆ K, then (R/J)/(K/J) ∼ = R/K.

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Proof. The quotients R/J, K/J and R/K are multiplicative Hv -rings and K/J is an Hv -ideal of (R/J, ⊕, ). Indeed, K/J ⊆ R/J and for all x ∈ K, y ∈ R, we have (J + x) (J + y) = {J + z | z ∈ xP2∗ y} ⊆ K/J. Therefore, K/J R/J ⊆ K/J. Similarly, R/J K/J ⊆ K/J. On the other hand, ((R/J)/(K/J), , ∗) is a multiplicative Hv -ring, where  and ∗ are the usual addition and multiplication of classes. Now, we consider the map f : (R/J)/(K/J) → R/K such that (K/J) ⊕ (J + x) 7−→ K + x. Since, for all x, y ∈ R, (K/J) ⊕ (J + x) = (K/J) ⊕ (J + y) ⇔ {J + z | z ∈ K + x} = {J + w | w ∈ K + y} ⇔y−x∈K ⇔K +x=K +y it follows that f is well defined and one to one. Obviously, it is onto, so it remains to prove that f is a homomorphism. Indeed, for all x, y ∈ R we have f [(K/J) ⊕ (J + x)  (K/J) ⊕ (J + y)] = f ({(K/J) ⊕ (J + z) | J + z ∈ (K/J) ⊕ (J + x) ⊕ (K/J) ⊕ (J + y)}) = f (K/J ⊕ (J + x + y)) = K + x + y = (K + x) ⊕ (K + y) = f ((K/J) ⊕ (J + x)) ⊕ f ((K/J) ⊕ (J + y)) and f [(K/J) ⊕ (J + x) ∗ (K/J) ⊕ (J + y)] = f [{(K/J) ⊕ (J + w) | J + w ∈ (J + x) (J + y)}] = f ({(K/J) ⊕ (J + w) | w ∈ xyP2 }] = {K + w | w ∈ xP2∗ y} = (K + x) (K + y) = f ((K/J) ⊕ (J + x)) f ((K/J) ⊕ (J + y)). Hence f is an isomorphism.



Theorem 4.21. Let R and A be two rings, f ∈ Hom(R, A) and (R, P1∗ , P2∗ ) be an Hv -ring with P -hyperoperations. Then, the following assertions hold: (1) If (A, L∗1 , L∗2 ) is an Hv -ring such that f (P2 ) ∩ L2 6= ∅, then f : (R, P1∗ , P2∗ ) → (A, L∗1 , L∗2 ) is an Hv -homomorphism. (2) If f (P2 ) ∩ Z(A) 6= ∅, then f : (R, P1∗ , P2∗ ) → (A, f (P1 )∗ , f (P2 )∗ ) is a strong homomorphism. A particular case: f : (R, P1∗ , P2∗ ) → (Imf, f (P1 )∗ , f (P2 )∗ ).

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(1) For all x, y ∈ R we have

f (xP1∗ y)

= f (x) + f (y) + f (P1 ) and f (x)L∗1 f (y) = f (x) + f (y) + L1 .

From the hypothesis it follows that 0 ∈ P1 and so 0 = f (0) ∈ f (P1 ) ∩ L1 and f (xP1∗ y) ∩ f (x)L∗1 f (y) 6= ∅. Moreover, the condition f (xP2∗ y) ∩ f (x)L∗2 f (y) 6= ∅ holds obviously. Hence f is an Hv -homomorphism. (2) The structure (A, f (P1 )∗ , f (P2 )∗ ) is an Hv -ring, because 0 ∈ f (P1 ) and f (P2 ) ∩ Z(A) 6= ∅. The Hv -homomorphism f is strong, since for all x, y ∈ R, we have f (xP1∗ y) = f (x) + f (y) + f (P1 ) = f (x)f (P1 )∗ f (y), f (xP2∗ y) = f (x)f (y)f (P2 ) = f (x)f (P2 )∗ f (y). In the particular case when f is the Hv -homomorphism from (R, P1∗ , P2∗ ) to (Imf, f (P1 )∗ , f (P2 )∗ ), from P2 ∩ Z(R) 6= ∅, we can deduce easily that f (P2 ) ∩ Z(Imf ) 6= ∅. Hence, (2) is valid.  Proposition 4.8. Let (R, P1∗ , P2∗ ) be an Hv -ring with P -hyperoperations. If α ∈ Z(R), then the translation of the semigroup (R, ·) by α: fα : x → αx is a multiplicatively strong homomorphism from (R, L∗1 , (αP2 )∗ ) to (R, P1∗ , P2∗ ), where 0 ∈ L1 ⊆ R. Proof. First, we can observe that the structure (R, L∗1 , (αP2 )∗ ) is an Hv ring because 0 ∈ L1 and αP2 ∩ Z(R) 6= ∅. Moreover, for all x, y, z ∈ R, we have fα (xL∗1 y) = α(x+y+L1 ) = αx+αy+αL1 and fα (x)P1∗ fα (y) = αx+αy+P1 and since 0 ∈ αL1 ∩ P1 , it follows that fα (xL∗1 y) ∩ fα (x)P1∗ fα (y) 6= ∅. Moreover, fα (x(αP2 )∗ y) = α(xyαP2 ) = (αx)(αy)P2 = fα (x)P2∗ fα (y). Hence, fα is a multiplicatively strong homomorphism.



Proposition 4.9. Let (R, P1∗ , P2∗ ) be an Hv -ring with P -hyperoperations and α ∈ Z(R). If the element α is simplificable and reproductive in (R, ·), then for each subset L1 of R such that αL1 = P1 , we have (R, L∗1 , (αP2 )∗ ) ∼ = (R, P1∗ , P2∗ ).

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Proof.

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Let us consider the translation of the semigroup (R, ·) by α fα : (R, L∗1 , (αP2 )∗ ) → (R, P1∗ , P2∗ ), fα (x) = αx,

which is a multiplicatively strong homomorphism of Hv -rings. Moreover, fα is additively strong because, for all x, y ∈ R, fα (xL∗1 y) = αx + αy + αL1 = αx + αy + P1 = fα (x)P1∗ fα (y). Finally, according to the hypothesis, the map fα is one to one and onto, hence fα is an isomorphism of Hv -rings.  Suppose that the conditions of Proposition 4.9 hold and α2 P1 = P1 . Therefore, every translation of the sets P1 , P2 by α ∈ Z(R) gives an isomorphism of Hv -rings, i.e., (R, (αP1 )∗ , (αP2 )∗ ) ∼ = (R, P1∗ , P2∗ ). In case the ∗ ∗ Hv -ring (R, P1 , P2 ) is derived from a ring with unit 1, we obtain the following isomorphism: (R, (−P1 )∗ , (−P2 )∗ ) ∼ = (R, P1∗ , P2∗ ), since −1 satisfies the hypothesis of Proposition 4.9 and (−1)2 P1 = P1 . Now, we calculate the number of Hv -rings with P -hyperoperations, which can be constructed starting from a finite ring (R, +, ·). By Proposition 4.8, it follows that this number can be substantially reduced, because some of these Hv -rings are isomorphic. Proposition 4.10. Let (R, +, ·) be a ring with cardR = n, n > 1 and cardZ(R)=m. The number of Hv -rings (R, P1∗ , P2∗ ) is at most 2n−1 (2n −2n−m ). Proof. The number of the subsets P1 of R, which satisfy the condition 0 ∈ P1 , is 2n−1 . The number of the subsets P2 of R, which satisfy the condition P2 ∩ Z(R) 6= ∅, is 2n−1 + 2n−2 + ... + 2n−m = 2n − 2n−m . Hence, the number of Hv -rings is at most 2n−1 (2n − 2n−m ).  If · is commutative, the above number is 22n−1 − 2n−1 . Proposition 4.11. If (R, +, ·) is a commutative ring without non-zero divisors and cardR = n, n > 1, then the number of hyperrings (R, P1∗ , P2∗ ) which are not rings is at most 5 · 2n−1 − n − 4. Proof. First of all it is easy to check that the structure (R, P1∗ ) is a hypergroup. Moreover, because of the commutativity of the multiplication, we have xP2∗ (yP2∗ z) = xyzP2 P2 = (xP2∗ y)P2∗ z,

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for all x, y, z ∈ R. The necessary and sufficient condition for the validity of the inclusion distributivity is RP2 P1 ⊆ P1 . We suppose that there exist P1 , P2 ⊆ R, P1 6=R, satisfying the previous condition. Then, for any p1 ∈ P1 , p2 ∈ P2 , p1 6= 0 6= p2 , the condition Rp2 p1 ⊆ P1 is valid and so, there exist a, b ∈ R, a 6= b such that ap2 p1 = bp2 p1 . Therefore, (a − b)p2 p1 = 0, i.e., p1 or p2 is a zero divisor, which is a contradiction. Hence, the only cases, in which the condition of distributivity is satisfied, are P1 = R, P1 = {0} and P2 = {0}. Therefore, the number of hyperrings (R, P1∗ , P2∗ ) is at most (2n − 1 + 2n − 1 + 2n−1 ) − 2 = 5 · 2n−1 − 4, because the hyperrings (R, {0}∗ , {0}∗ ) and (R, R∗ , {0}∗ ) are calculated twice. Finally, the number of hyperrings which are not rings is at most 5 · 2n−1 − n − 4.  We remark that the following facts are also valid. • There are 2n − 1 hyperrings of the form (R, R∗ , P2∗ ), where P2 ⊆ R, because xR∗ y = R holds for all x, y ∈ R. • There are 2n − n − 1 multiplicative hyperrings (R, {0}∗ , P2∗ ) in the sense of Rota, where P2 ⊆ R. Indeed, the hyperoperation {0}∗ of each hyperring of the above form is the addition + of the ring (R, +, ·). Moreover, we have (−x)P2∗ y = −(xP2∗ y) = xP2∗ (−y). • If · ≡ ◦, where x ◦ y = 0, for all x, y ∈ R, then the number of Hv -rings is at most 2n−1 , because (R, P1∗ , ◦) ∼ = (R, P1∗ , {0}∗ ). Example 4.14. In the case of the ring (Zp , +, ·), where p is a prime number, there are at most 22p−1 − 2p−1 Hv -rings, from which 5 · 2p−1 − p − 4 are hyperrings, that are not rings. In the particular case p = 3 we have 28 Hv -rings. Observe that 13 Hv -rings are hyperrings which are not rings. We have the following isomorphisms: (Z3 , {0}∗ , {1}∗ ) ∼ = (Z3 , {0}∗ , {2}∗ ), ∗ ∗ ∼ (Z3 , {0} , {0, 1} ) = (Z3 , {0}∗ , {0, 2}∗ ), (Z3 , {0, 1}∗ , {0}∗ ) ∼ = (Z3 , {0, 2}∗ , {0}∗ ), ∗ ∗ ∼ (Z3 , {0, 1} , {1} ) = (Z3 , {0, 2}∗ , {2}∗ ), (Z3 , {0, 1}∗ , {2}∗ ) ∼ = (Z3 , {0, 2}∗ , {1}∗ ), (Z3 , {0, 1}∗ , {0, 1}∗ ) ∼ = (Z3 , {0, 2}∗ , {0, 2}∗ ), ∗ ∗ ∼ (Z3 , {0, 1} , {0, 2} ) = (Z3 , {0, 2}∗ , {0, 1}∗ ), (Z3 , {0, 1}∗ , {1, 2}∗ ) ∼ = (Z3 , {0, 2}∗ , {1, 2}∗ ), ∗ ∗ ∼ (Z3 , {0, 1} , Z3 ) = (Z3 , {0, 2}∗ , Z∗3 ), (Z3 , Z∗3 , {1}∗ ) ∼ = (Z3 , Z∗3 , {2}∗ ), ∗ ∗ ∼ (Z3 , Z3 , {0, 1} ) = (Z3 , Z∗3 , {0, 2}∗ ).

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So, the number of Hv -rings is reduced to 17 and observe that 9 of them are hyperrings which are not rings. Example 4.15. We consider the finite field (F, +, ·) and suppose that cardF = pn , where p is a prime number, p > 2, n ≥ 1. Consequently, if we consider the translation of F fa−1 b : x 7−→ a−1 bx, for all a, b ∈ R \ {0}, then we have fa−1 b (a) = b. Now, we obtain the following isomorphisms between Hv -rings with P -hyperoperations: (F, {0}∗ , {a}∗ ) ∼ = (F, {0}∗ , {b}∗ ), ∗ ∗ ∼ (F, F , {a} ) = (F, F ∗ , {b}∗ ), ∼ (F, {0}∗ , {0, b}∗ ), (F, {0}∗ , {0, a}∗ ) = (F, F ∗ , {0, a}∗ ) ∼ = (F, F ∗ , {0, b}∗ ), So, the number of Hv -rings (R, P1∗ , P2∗ ) is at most 22p

n

−1

−2p

n

−1

−4(pn −2).

Theorem 4.22. Let (R, P1∗ , P2∗ ) be an Hv -ring with the P -hyperoperations P1∗ , P2∗ over the ring (R, +, ·). Consider the subgroup < P1 > of (R, +) ∗ generated by P1 . Then for all a in R, we have β+ (a) =< P1 > +a and ∗ R/β+ is a multiplicative Hv -ring with the inclusion distributivity. ∗ We denote the fundamental class of a ∈ R by β+ (a) and any ∗ P ∗ hypersum with respect to the P1∗ by . Let a ∈ R and x ∈ β+ (a). Then, there exist z1 , ..., zn+1 and there are yj ∈ R and the finite sets of indices Ii , i = 1, ..., n such that ∗ P {zi , zi+1 } ⊆ yj for i = 1, ..., n.

Proof.

j∈Ii

Set ui =

∗ P

yj and si = cardIi . Then, {zi , zi+1 } ⊆ ui + (si − 1)P1 for

j∈Ii

i = 1, ..., n. Therefore, for i = 1, ..., n − 1 we have zi+1 ∈ (ui + (si − 1)P1 ) ∩ (ui+1 + (si+1 − 1)P1 ) and so ui ∈ zi+1 − (si − 1)P1 . Consequently, ui ∈ ui+1 + (si+1 − 1)P1 − (si − 1)P1 . We obtain u1 ∈ un + (s2 + s3 + ... + sn − n + 1)P1 − (s1 + s2 + ... + sn−1 − n + 1)P1 . But zn+1 ∈ un + (sn − 1)P1 , so un ∈ zn+1 − (sn − 1)P1 . Moreover, z1 ∈ u1 + (s1 − 1)P1 . Thus, we have z1 ∈ zn+1 + (s1 + s2 + ... + sn − n)P1 − (s1 + s2 + ... + sn − n)P1 .

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Finally, x ∈ (s1 + s2 + ... + sn − n)(P1 − P1 ) + a. This means that x ∈< ∗ P1 > +a, so β+ ⊆< P1 > +a. Now, let a ∈ R and take x ∈< P1 > +a. Then, there exists s ∈ N such that x ∈ s(P1 − P1 ) + a. So {x, a} ⊆ s(P1 − P1 ) + a = aP1∗ (−P1 )...P1∗ (−P1 ) ∗ ∗ which means that xβ+ a. Therefore, we proved that β+ (a) =< P1 > +a. ∗ The sum ] and the product ⊗ of the elements of R/β+ are defined in the ∗ usual manner, and (R/β+ , ]) is a group. Moreover, the weak associativity of ⊗ is valid. Finally, for all x, y, z ∈ R, we have ∗ ∗ ∗ ∗ ∗ (z))= β+ (x) ⊗ β+ (y + z) β+ (x) ⊗ (β+ (y) ] β+ ∗ = {β+ (u) | u ∈ (< P1 > +x)(< P1 > +y + z)P2 }.

On the other hand, ∗ ∗ ∗ ∗ (z)) (x) ⊗ β+ (y)) ] (β+ (x) ⊗ β+ (β+ ∗ = {β+ (v) | v ∈ (< P1 > +x)(< P1 > +y)P2 } ∗ (w) | w ∈ (< P1 > +x)(< P1 > +z)P2 } ]{β+ ∗ = {β+ (v + w) | v + w ∈ (< P1 > +x)(< P1 > +y)P2 +(< P1 > +x)(< P1 > +z)P2 }.

Consequently, the inclusion distributivity is valid.



Let (R, P1∗ , P2∗ ) be an Hv -ring with the P -hyperoperations P1∗ , P2∗ over the ring (R, +, ·) such that RP2 ⊆ P2 . Denote the set of all finite polynomials of elements of R by A. Then, for every ai ∈ A, i ∈ N, there exist ri ∈ R, Ii finite set of indices, P2j ∈ P(P2 ), j ∈ Ii and si ∈ N such that P ai = ri + P2j + si Pi , j∈Ii

where si P1 = P1 + ... + P1 . {z } | si

Theorem 4.23. Let (R, P1∗ , P2∗ ) be an Hv -ring with the P -hyperoperations P1∗ , P2∗ over the ring (R, +, ·). If RP2 ⊆ P2 and < P1 , P2 > is the subgroup of (R, +) generated by P1 ∪ P2 , then for all x ∈ R, γ ∗ (x) ⊆< P1 , P2 > +x. Proof. Suppose that x ∈ R and y ∈ γ ∗ (x). Then, there exist z1 , ..., zm+1 ∈ R with z1 = y, zm+1 = x and a1 , ..., am ∈ A such that {zi , zi+1 } ⊆ ai , (i = 1, ..., m). Then, for i = m, we have x = zm+1 ∈ P rm + P2j +sm P1 . So rm ∈ x+tm (−P2 )+sm (−P1 ), where tm = cardIm . j∈Im

Moreover, for all i = 1, ..., m, we have P P zi+1 ∈ (ri + P2j + si P1 ) ∩ (ri+1 + P2j + si+1 P1 ) j∈Ii

j∈Ii+1

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and hence ri ∈ ri+1 + ti+1 P2 + ti (−P2 ) + si+1 P1 + si (−P1 ), where ti+1 = cardIi+1 , ti = cardIi . We obtain y = z1 ∈ x + t(P2 − P2 ) + s(P1 − P1 ), where t = t1 + ... + tm , ti = cardIi , i ∈ {1, ..., m}, s = s1 , ..., sm . This means that y ∈ x+ < P2 > + < P1 >. Hence, γ ∗ (x) ⊆< P1 , P2 > +x.  Theorem 4.24. Let (R, P1∗ , P2∗ ) be an Hv -ring with the P -hyperoperations P1∗ , P2∗ over the unitary ring (R, +, ·). If P2 is a left ideal and < P1 > is the subgroup of (R, +) generated by P1 , then R/γ ∗ = R/(< P1 > +P2 ). Proof. Suppose that x ∈ R. From the previous theorem, we have that γ ∗ (x) ⊆< P1 , P2 > +x. Since P2 is a subgroup of (R, +), it follows that γ ∗ (x) ⊆< P1 > +P2 + x. Conversely, for all z ∈< P1 > +P2 + x there exist p2 ∈ P2 and n ∈ N such that z ∈ x + p2 + n(P1 − P1 ). Moreover, p2 ∈ P2 = 1P2∗ 1 and so {z, x} ⊆ xP1∗ (1P2∗ 1)P1∗ (−P1 )P1∗ ...P1∗ (−P1 ) where P1∗ (−P1 ) appears n times. Hence γ ∗ (x) =< P1 > +P2 + x.



Corollary 4.2. If (R, +, ·) is a ring and P2 is a left ideal, then for all multiplicative P − Hv -rings over R, we have R/γ ∗ = R/P2 . Proof. Suppose that (R, P1∗ , P2∗ ) is a multiplicative P -Hv -ring over the ring (R, +, ·). Then, cardP1 = 1 and since the necessary and sufficient condition for the weak distributivity is 0 ∈ P1 , we have P1 = {0}. From the previous theorem, it follows that R/γ ∗ = R/P2 .  4.7

∂-hyperoperations and Hv -rings

In this section, we consider a hyperoperation, denoted by ∂f . This hyperoperation studied by Vougiouklis in [129, 132]. The motivation for this hyperoperation is the property which the “derivative” has on the product of functions. Since there is no confusion, we can simply write ∂. To more precise, we introduce the following definition. Definition 4.11. Let H be a set with n operations (or hperoperations) ⊗1 , ⊗2 , ..., ⊗n and one map (or multivalued map) f : H → H, then n hyperoperations ∂1 , ∂2 , ..., ∂n on H are defined, called ∂-hyperoperations by putting x∂i y = {f (x) ⊗i y, x ⊗i f (y)}, for all x, y ∈ H, i ∈ {1, 2, ..., n}

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or in case where ⊗i is hyperoperation or f is multivalued map we have x∂i y = (f (x) ⊗i y) ∪ (x ⊗i f (y)), for all x, y ∈ H, i ∈ {1, 2, ..., n}. One can see that if ⊗i is associative then ∂i is weak associative. Remark that one can use several maps f , instead of only one, in an analogous way. We can define ∂-hyperoperations on the union of maps: Definition 4.12. Let (G, ·) groupoid and fi : G → G, i ∈ I, set of maps on G. Take the map f∪ : G → P ∗ (G) such that f∪ (x) = {fi (x)|i ∈ I} and we call it the union of the fi (x). We call the union ∂-hyperoperation (∂), on G if we consider the map f∪ (x). An important case for a map f, is to take the union of this with the identity id. Thus, we consider the map f ≡ f∪ (id), so f (x) = {x, f (x)}, for all x ∈ G, which is called b-∂-hyperoperation, we denote it by (∂), so we have x∂y = {xy, f (x) · y, x · f (y)}, for all x, y ∈ G. Remark that ∂ contains the operation (·), so it is b-operation. Moreover, if f : G → P ∗ (G) is multivalued then the b-∂-hyperoperations is defined by using the f (x) = {x} ∪ f (x), for all x ∈ G. Motivation for the definition of ∂-hyperoperation is the derivative where only multiplication of functions can be used. Therefore, for functions s(x), t(x), we have s∂t = {s0 t, st0 }, (0 ) is the derivative. Example 4.16. Application on derivative: consider all polynomials of first degree gi (x) = ai x + bi . We have g1 ∂g2 = {a1 a2 x + a1 b2 , a1 a2 x + b1 a2 }, so it is a hyperoperation in the set of first degree polynomials. Moreover, all polynomials x + c, where c be a constant, are units. Example 4.17. If R+ is the set of positive reals, and a ∈ R+ , then we take the map f : x 7→ xa . The theta-operation is x∂y = {xa y, xy a }, for all x, y ∈ R+ . The only unit is 1, and any element x ∈ R+ has two inverses, x−a and x−1/a . Lemma 4.3. Let (G, ·) semigroup. Then, (1) For every f : G → G, the hyperoperation ∂ is weak associative. (2) For every f : G → G, the b-∂-hyperoperation ∂ is weak associative.

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(3) If f is homomorphism and projection, i.e., f 2 = f , then (∂) is associative. Proof.

(1) For all x, y, z ∈ G, we have

(x∂y)∂z = {f (f (x) · y) · z, f (x) · y · f (z), f (x · f (y)) · z, x · f (y) · f (z)}. x∂(y∂z) = {f (x) · f (y) · z, x · f (f (y)) · z), f (x) · y · f (z), x · f (y · f (z))}. Therefore, (x∂y)∂z∩x∂(y∂z) = {f (x)·y·f (z)} = 6 ∅, so ∂ is weak associative. (2) Since ∂ is greater than ∂, we obtain that ∂ is weak associative. Moreover, for every x, y, z ∈ G, we have (x∂y)∂z = {f (x · y) · z, x · y · z, x · y · f (z), f (f (x) · y) · z, f (x) · y · z, f (x) · y · f (z), f (x · f (y)) · z, x · f (y) · z, x · f (y) · f (z)}, x∂(y∂z) = {f (x) · y · z, x · y · z, x · f (y · z), f (x) · f (y) · z, x · f (y) · z, x · f (f (y) · z), f (x) · y · f (z), x · y · f (z), x · f (y · f (z))}. Thus, we obtain (x∂y)∂z∩x∂(y∂z) = {x·y·z, f (x)·y·z, x·f (y)·z, x·y·f (z), f (x)·y·f (z)} = 6 ∅. (3) If f is a homomorphism and projection, then we obtain (x∂y)∂z = {f (x) · f (y) · z, f (x) · y · f (z), x · f (y) · f (z)} = x∂(y∂z). Therefore, ∂ is an associative hyperoperation.



Note that projection without homomorphism does not give the associativity. Furtheremore, commutativity does not improve the result. Properties 4.25. Reproductivity. For the reproductivity we must have S x∂G = {f (x) · g, x · f (g)} = G g∈G

and G∂x =

S

{f (g) · x, g · f (x)} = G.

g∈G

So, if (·) is reproductive, then (∂) is reproductive, since S S {f (x) · g} = G and {g(x) · f } = G. g∈G

g∈G

Commutativity. If · is commutative, then ∂ is commutative. If f is into the centre of G, then ∂ is commutative. If · is weak commutative, then ∂ is weak commutative.

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Unit elements. u is a right unit element if x∂u = {f (x) · u, x · f (u)} 3 x. So, f (u) = e, where e is a unit in (G, ·). The elements of the kernel of f , are the units of (G, ∂). Inverse elements. Let (G, ·) is a monoid with unit e and u be a unit in (G, ∂). Then, f (u) = e. For given x, the element x0 is an inverse with respect to u, if x∂x0 = {f (x) · x0 , x · f (x0 )} 3 u and x0 ∂x = {f (x0 ) · x, x0 · f (x)} 3 u. So, x0 = (f (x))−1 u and x0 = u(f (x))−1 , are the right and left inverses, respectively. We have two-sided inverses if and only if f (x)u = uf (x). Similar properties for multivalued maps, are obtained. Proposition 4.12. If (G, ·) is a group, then (G, ∂) is an Hv -group, for all f : G → G. Proof.

It is straightforward.



Proposition 4.13. Let (G, ·) be a group and f (x) = a, constant map on G. Then, (G, ∂)/β ∗ is singleton. If f (x) = e, then we obtain x∂y = {x, y}, the smallest incidence hyperoperation. For every x ∈ G, we have

Proof.

−1

a

∂(a−1 · x) = {f (a−1 · a−1 · x, a−1 · f (a−1 · x)} = {x, e}.

Thus, xβe, for all x ∈ G. So, β ∗ (x) = β ∗ (e) and (G, ∂)/β ∗ is singleton.  Every f : G → G defines a partition of G, by setting two elements x, y in the same class, if and only if f (x) = f (y). We call this partition f -partition (this is the associated map), and we denote the class of x, by f [x]. So, in Proposition 4.13, for constant maps, we have that f [x] = G = β ∗ (x), for all x ∈ G, where β ∗ (x) is referred to (G, ∂). Proposition 4.14. Let (G, ·) be a group and f be a homomorphism. Then, f [x] ⊆ β ∗ (x), for all x ∈ G. Proof.

It is straightforward.



Example 4.18. Let (G, ·) is a commutative semigroup and P ⊆ G. Consider the multivalued map f such that f (x) = P · x, for all x ∈ G. Then, we have x∂y = x · y · P, for all x, y ∈ G. So, the ∂-operation coincides with the well known class of P hyperoperations.

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Example 4.19. Let (G, ·) be a group and f : x 7→ x−1 , for all x ∈ G, be the inverse map. Then, the ∂-hyperoperation is defined by x∂y = {x−1 · y, x · y −1 }, forall x, y ∈ G. The hyperstructure (G, ∂) has a unique unit, the unit e, of (G, ·). All elements in (G, ∂) are self inverses and unique. Now, we define ∂-hyperoperation on rings and analogously to other more complicate structures, where more than one θ-operations can be defined. Moreover, one can replace structures by hyperstructures or by Hv structures as well. Definition 4.13. Let (R, +, ·) be ring and f : R → R, g : R → R be two maps. We define two hyperoperations ∂+ and (∂· ), called both ∂hyperoperations, on R as follows: x∂+ y = {f (x)+y, x+f (y)} and x∂· y = {g(x)·y, x·g(y)}, for all x, y ∈ G. A hyperstructure (R, +, ·), where +, · are two hyperoperations which satisfy all Hv -ring axioms, except the weak distributivity, will be called an Hv -semi-near-ring. Proposition 4.15. Let (R, +, ·) ring and f : R → R, g : R → R maps. The hyperstructure (R, ∂+ , ∂· ), called theta, is an Hv -semi-near-ring. Moreover, + is commutative. Proof. One can see that all properties of Hv -rings, expect the distributivity, are valid.  More properties are valid if we replace ∂ by the corresponding b-∂hyperoperations ∂. Proposition 4.16. Let (R, +, ·) ring and f : R → R, g : R → R maps. The (R, ∂+ , ∂· ), is an Hv -ring. Proof. The only axiom we check is the one of distributivity. Therefore, for all x, y, z ∈ R, we have x∂· (y∂+ z) = {g(x)(y + z), g(x)(f (y) + z), g(x)(y + f (z)), xg(y + z), xg(f (y) + z), xg(y + f (z))}

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and (x∂· y)∂ + (x∂· z) = {g(x)(y + z)f (g(x)y) + g(x)z, g(x)y + f (g(x)z), g(x)y + xg(z), f (g(x)y) + xg(z), g(x)y + f (xg(z)), xg(y) + g(x)z, f (xg(y)) + g(x)z, xg(y) + f (g(x)z), x(g(y) + g(z)), f (xg(y)) + xg(z), xg(y) + f (xg(z))}. Therefore, we obtain x∂· (y∂+ z) ∩ (x∂· y)∂ + (x∂· z) = {g(x)(y + z)} = 6 ∅.  Properties 4.26. (Special classes). The theta hyperstructure (R, ∂+ , ∂· ) takes a new form and have some properties in several forms as the following ones. (1) If f (x) ≡ g(x), ∀x ∈ R, i.e., the two maps coincide, (R, ∂+ , ∂· ) is an Hv -ring. (2) If g(x) = x, ∀x ∈ R, i.e., only the f in addition is used, then we have x(y∂+ z) = {xf (y) + xz, xy + xf (z)}, (xy)∂+ (xz) = {f (xy) + xz, xy + f (xz)} Therefore, x(y∂+ z) ∩ (xy)∂+ (xz) = ∅. (3) If f (x) = x, for all x ∈ R, then (R, +, ∂· ) becomes a multiplicative Hv -ring. Example 4.20. Consider the ring (Z6 , +, ·), the map f : 0 7→ 2 and f (x) = x, for all x ∈ Z6 \ {0}. Then the operations ∂+ and ∂· are given in the following tables: ∂+

0

0

2

1

2

3

4

5

{1, 3} {2, 4} {3, 5} {4, 0} {5, 1}

1 {1, 3}

2

3

4

5

0

2 {2, 4}

3

4

5

0

1

3 {3, 5}

4

5

0

1

2

4 {4, 0}

5

0

1

2

3

5 {5, 1}

0

1

2

3

4

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and ∂·

0

0

0

1

2

3

4

5

{0, 2} {0, 4} 0 {0, 2} {0, 4}

1 {0, 2}

1

2

3

4

5

2 {0, 4}

2

4

0

2

4

3

3

0

3

0

3

4 {0, 2}

4

2

0

4

2

5 {0, 4}

5

4

3

2

1

0

We obtain (Z6 , ∂+ , ∂· )/γ ∗ = {{0, 2, 4}, {1, 3, 5} ∼ = Z2 . Generalization of Example 4.20, on rings and groups, can be the following. Theorem 4.27. Consider the group of integers (Z, +) and let n 6= 0 be a natural number. Take the map f such that f (0) = n and f (x) = x, for all x ∈ Z \ {0}. Then, (Z, ∂)/β ∗ ∼ = (Zn , +). Proof.

The proof is clear from the following: 0∂0 = {n}, 0∂(0∂0) = {n, 2n}, 0∂{n, 2n} = {n, 2n, 3n}, 0∂(−n) = {0, −n}, 0∂x = {x, x + n}, for all x ∈ Z \ {0}

and x∂{y, y + n} = {x + y, x + y + n}, for all x, y ∈ Z \ {0}.

4.8



(H, R)-Hv -rings

In this section, we present the following structures. The main reference is [104]. Let (H, ∗, ◦) be an Hv -ring, (R, +, ·) be a ring with the zero element denoted by 0 and {Ai }i∈R be a family of nonempty sets indexed in R such that A0 = H and for all i, j ∈ R, i 6= j, Ai ∩ Aj = ∅. Moreover, for all Ai , i ∈ R∗ there exists a set of indices Ii and a unique family {Bk }k∈Ii

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S

such that Bk ⊆ Ai and

Bk 6= ∅. Set K =

k∈Ii

S

Ai and consider the

i∈R

hyperoperations ⊕, defined in K as follows: ∀(x, y) ∈ H 2 , x ⊕ y = x ∗ y, x y = x ◦ y,  H if ij = 0 S ∀(x, y) ∈ Ai × Aj = 6 H 2 , x ⊕ y = Ai+j , x y = B if ij 6= 0. k  k∈Iij

It is clear that (K, ⊕) is an Hv -group. Moreover, (K, ) is an Hv -semigroup since the hyperoperation ◦ is weak associative and for all (x, y, z) ∈ Ai × Aj × Ar 6= H 3 , we have (1) if i = 0 = j (similarly, if i = 0, j 6= 0 6= r), then x (y z) ⊆ (x y) z = H; (2) if j = 0 = r (similarly, if r = 0, i 6= 0 6= j), then H = x (y z) ⊇ (x y) z; (3) if i = 0 = r, then (x ◦ H = x (y z)) ∩ ((x y) z = H ◦ z) 6= ∅; (4) if j = 0, i 6= 0 6= r, then x (y z) = H = (x y) z; (5) if i 6= 0 6= j 6= 0 6= r, then x (y z) ∩ (x y) z 6= ∅. Finally, the weak distributive law is verified and so (K, ⊕, ) is an Hv -ring. Definition 4.14. The previous Hv -ring (K, ⊕, ) is called (H, R)−Hv -ring S with the support K = Ai . i∈R

Theorem 4.28. If γ ∗ is the fundamental equivalence relation in K, then K/γ ∗ ∼ = R. Proof. Let a ∈ K. Then there exists r ∈ R such that a ∈ Ar . In order to determine γ ∗ (a), we consider x ∈ γ ∗ (a). Then, there exist z1 , ..., zn+1 ∈ K such that z1 = x, zn+1 =a and ui ∈ U, i ∈ {1, ..., n} such that {zi , zi+1 } ⊆ ui , (i = 1, ..., n). Moreover, it is clear that for all ui ∈ U, i = 1, ..., n there exists an appropriate ri ∈ R, such that ui ⊆ Ari . Consequently, Ari = Ari+1 , i = 1, ..., n − 1, because zi+1 ∈ Ari ∩ Ari+1 . Hence, {x, a} ⊆ Arn = Ar and so γ ∗ (a) ⊆ Ar . For the converse, let y ∈ Ar . If r ∈ R∗ , then we consider u ∈ H, w ∈ Ar , and we have {y, a} ⊆ u w = Ar . Hence yγ ∗ a, i.e., y ∈ γ ∗ (a). If r = 0, then we have {y, a} ⊆ u w = H, i.e., y ∈ γ ∗ (a). Therefore, Ar ⊆ γ ∗ (a) and consequently, γ ∗ (a) = Ar . Finally, the map f : K/γ ∗ → R such that f (Ai ) = i is an isomorphism and so K/γ ∗ ∼  = R.

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We denote the kernel of the canonical map φK : K → K/γ ∗ such that φK (x) = γ ∗ (x) by ωk . According to the previous theorem, for all i ∈ R, x ∈ Ai , we have γ ∗ (x) = Ai and hence, we can write K/γ ∗ = {φK (Ai ) | i ∈ R}. Consequently, ωK = H. Now, we consider the (H1 , R1 )-Hv -ring (K1 , ⊕, ) with the support S K1 = Ai and the (H2 , R2 )-Hv -ring (K2 ,
, ) with support K2 = i∈R1 S Gj . We prove the following theorems. j∈R2

Theorem 4.29. If f : K1 → K2 is an inclusion homomorphism, then (1) f (γ ∗ (x)) ⊆ γ ∗ (f (x)) for all x ∈ K1 . (2) We define the induced homomorphism f ∗ : K1 /γ ∗ → K2 /γ ∗ of f by f ∗ (φK1 (x)) = φK2 (f (x)). (3) f (H1 ) ⊆ H2 . Proof. have

(1) Let x ∈ γ ∗ (x) = Ai , i ∈ R. Then, for y ∈ Aj , z ∈ Ai−j , we f (x) ∈ f (γ ∗ (x)) = f (y ⊕ z) ⊆ f (y)
f (z) = γ ∗ (f (x)).

Therefore, f (γ ∗ (x)) ⊆ γ ∗ (f (x)). (2) The map f ∗ is well defined. In fact, if φK1 (x) = φK1 (y), then xγ ∗ y and so f (x)γ ∗ f (y), i.e., φK2 (f (x)) = φK2 (f (y)) and hence f ∗ (φK1 (x)) = f ∗ (φK1 (y)). Moreover, f ∗ is a homomorphism, because for all x, y ∈ K1 , z ∈ x ⊕ y, w ∈ x y, we obtain f ∗ (φK1 (x) + φK2 (y)) = f ∗ (φK1 (z)) = φK2 (f (z)) = φK2 (f (x)
f (y)) = φK2 (f (x)) + φK2 (f (y)) = f ∗ (φK1 (x)) + f ∗ (φK2 (y)) and f ∗ (φK1 (x) · φK2 (y)) = f ∗ (φK1 (w)) = φK2 (f (w)) = φK2 (f (x) f (y)) = φK2 (f (x)) · φK2 (f (y)) = f ∗ (φK1 (x)) · f ∗ (φK2 (y)). ∗ (3) From (1) and (2) it follows that f (x) = φ−1 K2 (f (φK1 (x))). Therefore, ¯ ¯ if x ∈ H and 0K1 /γ ∗ , 0K2 /γ ∗ are the zero elements of the rings K1 /γ ∗ , K2 /γ ∗ , respectively, then −1 ¯ ∗ ¯ f (x) ∈ φ−1 K2 (f (0K1 /γ ∗ )) = φK2 (0K2 /γ ∗ ) = H2 ,

which implies that f (H1 ) ⊆ H2 . Theorem 4.30. If K1 ∼ = K2 , then H1 ∼ = H2 and R1 ∼ = R2 .



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Proof. From the previous theorem it follows that for all x ∈ K1 , f (γ ∗ (x)) = γ ∗ (f (x)) and f ∗ is a homomorphism. Therefore, K1 /γ ∗ ∼ = K2 /γ ∗ . Consequently, from Theorem 4.28, we obtain R1 ∼ = R2 . Next we consider the map g : H1 → H2 defined by g(x) = f (x). The map g is well defined, one to one and onto. We show that it is a strong homomorphism. In fact, for all (x, y) ∈ H 2 , we obtain g(x ∗1 y) = f (x ⊕ y) = f (x)
f (y) = g(x) ∗2 g(y), g(x ◦1 y) = f (x y) = f (x) f (y) = g(x) ◦2 g(y).



Theorem 4.31. If g : H1 → H2 , f : R1 → R2 are homomorphisms and for all i ∈ R1∗ , cardAi ≤ cardGf (i) , then there is an additively strong and one to one homomorphism from K1 to K2 . Proof. Let i ∈ R1∗ and let Fi = Fi (Ai , Gf (i) ) be the set of all the one to one maps from Ai to Gf (i) . If {Bk }k∈Ii , {Bs }s∈If (i) are the families of subsets of Ai and Gf (i) , respectively, then we denote  S n   S  o Fi∗ = hf (i) ∈ Fi | hf (i) Bk ∩ Bs 6= ∅ k∈Ii

s∈If (i)

and we consider the map  t : K1 → K2 : x 7→ t(x) =

g(x) if x ∈ H1 hf (i) (x) where hf (i) ∈ Fi∗ if x ∈ Ai 6= H1 .

The map t is well defined and one to one. We will show that it is an additively strong homomorphism. If (x, y) ∈ H12 , then t(x ⊕ y) = g(x ∗1 y) = g(x) ∗2 g(y) = t(x)
t(y) t(x y) = g(x ◦1 y) = g(x) ◦2 g(y) = t(x) t(y). If (x, y) ∈ Ai × Aj 6= H12 , then t(x ⊕ y) = t(Ai+j ) = Gf (i+j) and t(x)
t(y) = Gf (i)+f (j) = Gf (i+j) . In order to check the multiplications, we notice that: If ij = 0, then t(x y) = g(H1 ) = H2 and t(x) t(y) = Gf (ij) = H2 . S If ij 6= 0, then t(x y) = hf (ij) ( Br ), where Br ⊆ Aij , Sr∈Iij t(x) t(y) = Bm , where Bm ⊆ Gf (ij) .  S   S m∈If(ij) Since hf (ij) Br ∩ Bm 6= ∅, it follows that r∈Iij

m∈If (ij)

t(x y) ∩ t(x)

t(y) 6= ∅.



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The Hv -ring of fractions

It is well-known that if S is a multiplicatively closed subset of a commutative ring R, then there is a natural way to define the ring of fractions of R with respect to S. This ring is denoted by S −1 R. A natural question that arises, is the following one: how the Hv -ring of fractions can be defined? In this section, our aim is to answer the above question and obtain some properties of the Hv -ring of fractions. We use the results obtained by Darafsheh and Davvaz [19]. Throughout this section, R is a commutative (general) hyperring with a unit denoted by 1. Recall that a hyperstructure (R, +, ·) is a hyperring if (R, +) is a hypergroup, (·) is an associative hyperoperation and the distributivity is valid, that is x · (y + z) = x · y + x · z, (x + y) · z = x · z + y · z for all x, y, z ∈ R. Under the above conditions, we define the Hv -ring of fractions of R. Definition 4.15. A non-empty subset S of R is called a strong multiplicatively closed subset (s.m.c.s.) if the following axioms hold: (1) 1 ∈ S; (2) a · S = S · a = S, for all a ∈ S. Now, as we have indicated earlier, suppose that R is a commutative hyperring with scalar unit. Furthermore we assume that S is a s.m.c.s. of R. Let M be the set of all the ordered pairs (r, s) where r ∈ R, s ∈ S. For A ⊆ R and B ⊆ S, we denote the set {(a, b)|a ∈ A, b ∈ B} by (A, B). We define the following relation ∼. (A, B) ∼ (C, D) if and only if there exists a subset X of S such that X · (A · D) = X · (B · C). Lemma 4.4. ∼ is an equivalence relation. Proof. Obviously ∼ is reflexive and symmetric. To verify that ∼ is transitive, we assume (A1 , B1 ) ∼ (A2 , B2 ) and (A2 , B2 ) ∼ (A3 , B3 ), where (Ai , Bi ) ∈ P(M), 1 ≤ i ≤ 3. By definition of ∼ there exist the subsets X1 and X2 of S such that X1 · (A1 · B2 ) = X1 · (A2 · B1 ),

(4.1)

X2 · (A2 · B3 ) = X2 · (A3 · B2 ).

(4.2)

Multiplying both sides of (4.2) by X1 .B1 we get X1 · X2 · A2 · B3 · B1 = X1 · X2 ·A3 ·B2 ·B1 which implies that X2 ·(X1 ·A2 ·B1 )·B3 = X1 ·X2 ·A3 ·B2 ·B1 .

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Using (4.1), we obtain X2 · (X1 · A1 · B2 ) · B3 = X1 · X2 · A3 · B2 · B1 which implies that (X1 · X2 · B2 ) · (A1 · B3 ) = (X1 · X2 · B2 ) · (A3 · B1 ). If we take X = X1 · X2 · B2 , then X · (A1 · B3 ) = X · (A3 · B1 ) which implies that (A1 , B1 ) ∼ (A3 , B3 ).  The equivalence class containing (A, B) is denoted by k A, B k. We consider the restriction of the relation ∼ on M. We obtain the following two corollaries. Corollary 4.3. For (r, s), (r1 , s1 ) ∈ M, we have (r, s) ∼ (r1 , s1 ) if and only if there exists A ⊆ S such that A · (r · s1 ) = A · (r1 · s). Corollary 4.4. ∼ is an equivalence relation on M. In M, the equivalence class containing (r, s) is denoted by [r, s] and we denote the set of all the equivalence classes by S −1 R. We define: S  A, B  = {[a1 , b1 ]|a1 ∈ A1 , b1 ∈ B1 }. (A1 ,B1 )∈kA,Bk

Now, we define the following hyperoperations on S −1 R, S [r1 , s1 ] ] [r2 , s2 ] = {[r, s]|r ∈ A, s ∈ B} (A,B)∈kr1 ·s2 +r2 ·s1 ,s1 ·s2 k

=  r1 · s2 + r2 · s1 , s1 · s2 , S [r1 , s1 ] ⊗ [r2 , s2 ] = {[r, s]|r ∈ A, s ∈ B} (A,B)∈kr1 ·r2 ,s1 ·s2 k

=  r1 · r2 , s1 · s2  . Lemma 4.5. ] and ⊗ defined above are well-defined hyperoperations. Proof. Suppose that [r1 , s1 ] = [a1 , t1 ] and [r2 , s2 ] = [a2 , t2 ]. Then there exist subsets A and B of S such that A · r1 · t1 = A · a1 · s1 ,

(4.3)

B · r2 · t2 = B · a2 · s2 .

(4.4)

Multiplying (4.3) by B · s2 · t2 and (4.4) by A · t1 · s1 we obtain A · B · s1 · s2 · t2 · a1 = A · B · s2 · t2 · t1 · r1 and A · B · s1 · s2 · t1 · a2 = A · B · s1 · t1 · t2 · r2 . Adding the above equalities, we obtain A · B · (s1 · s2 · (t2 · a1 + t1 · a2 )) = A · B · (t1 · t2 · (s2 · r1 + s1 · r2 )).

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Therefore, k r1 · s2 + r2 · s1 , s1 · s2 k=k a1 · t2 + a2 · t1 , t1 · t2 k which implies that  r1 · s2 + r2 · s1 , s1 · s2 = a1 · t2 + a2 · t1 , t1 · t2 , hence ] is well defined. Now, multiplying (4.3) with (4.4) we obtain A · B · (r1 · r2 ) · (t1 · t2 ) = A · B · (a1 · a2 ) · (s1 · s2 ) and so k r1 · r2 , s1 · s2 k=k a1 · a2 , t1 · t2 k which implies that  r1 · r2 , s1 · s2 = a1 · a2 , t1 · t2  . Therefore, ⊗ is well defined.



Corollary 4.5. For all r ∈ R, s ∈ S, we have  r, s = r · s, s · s . Theorem 4.32. (S −1 R, ], ⊗) is an Hv -ring, that we shall call the Hv -ring of fractions. Proof.

If [r1 , s1 ], [r2 , s2 ], [r3 , s3 ] ∈ S −1 R, then we have:

{[r, s] | r ∈ r1 · (s2 · s3 ) + (r2 · s3 + r3 · s2 ) · s1 , s ∈ s1 · (s2 · s3 )} ⊆ [r1 , s1 ] ] ([r2 , s2 ] ] [r3 , s3 ]), {[r, s] | r ∈ (r1 · s2 + r2 · s1 )s3 + r3 (s1 · s2 ), s ∈ (s1 · s2 ) · s3 } ⊆ ([r1 , s1 ] ] [r2 , s2 ]) ] [r3 , s3 ]. Since R is associative and distributive, we obtain that (S −1 R, ]) is weak associative. The weak distributivity of (S −1 R, ⊗) can be proved in a similar way. Now, we prove the reproduction axioms for (S −1 R, ]). For every [r, s], [r1 , s1 ] ∈ S −1 R, we have s ∈ S, s1 ∈ S and then by the definition of S there exists s2 ∈ S such that s ∈ s1 · s2 . On the other hand, since reproduction axioms hold for the additive law in R, we obtain r1 · s2 + (s1 + 1)R = R. Therefore, there exists r2 ∈ R such that r ∈ r1 · s2 + s1 · r2 + r2 which implies that r ∈ r1 · s2 + (r2 + r2 · s3 ) · s1 where 1 ∈ s3 · s1 . Therefore, there exists a ∈ r2 + r2 · s3 such that r ∈ r1 · s2 + a · s1 . Hence [r, s] ∈ [r1 , s1 ] ] [a, s2 ] = r1 · s2 + a · s1 , s1 · s2  which implies that S −1 R ⊆ [r1 , s1 ] ] S −1 R, therefore S −1 R = [r1 , s1 ] ] S −1 R.

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Finally, we prove the weak distributivity of ⊗ with respect to ]. We have {[r, s]|r ∈ s1 · r1 · (r2 · s3 + s2 · r3 ), s ∈ s1 · s1 · (s2 · s3 )} ⊆ [r1 , s1 ] ⊗ ([r2 , s2 ] ] [r3 , s3 ]), by Corollary 4.5, and hence {[r, s] | r ∈ (r1 · r2 ) · (s1 · s3 ) + (r1 · r3 ) · (s1 · s2 ), s ∈ (s1 · s2 ) · (s1 · s3 )} ⊆ ([r1 , s1 ] ⊗ [r2 , s2 ]) ] ([r1 , s1 ] ⊗ [r3 , s3 ]). Therefore, [r1 , s1 ] ⊗ ([r2 , s2 ] ] [r3 , s3 ]) ∩ ([r1 , s1 ] ⊗ [r2 , s2 ]) ] ([r1 , s1 ] ⊗ [r3 , s3 ]) 6= ∅ and in the similar way we obtain (([r1 , s1 ] ] [r2 , s2 ]) ⊗ [r3 , s3 ]) ∩ (([r1 , s1 ] ⊗ [r3 , s3 ]) ] ([r2 , s2 ] ⊗ [r3 , s3 ])) 6= ∅ thus, (S −1 R, ], ⊗) is an Hv -ring.



Theorem 4.33. Let R1 and R2 be two commutative hyperrings with scalar unit and S be a s.m.c.s. of R1 and let g : R1 → R2 be a strong homomorphism of Hv -rings such that g(1) = 1. Then, g induces an Hv homomorphism g : S −1 R1 → g(S)−1 R2 by setting g([r, s]) = [g(r), g(s)]. Proof. It is clear that g(S) is a s.m.c.s. of R2 . First, we prove that g is well defined. If [r, s] = [r1 , s1 ] then there exists A ⊆ S such that A · r · s1 = A · r1 · s which implies that g(A · r · s1 ) = g(A · r1 · s) or g(A) · g(r) · g(s1 ) = g(A) · g(r1 ) · g(s). Since g(A) ⊆ g(S), it follows that [g(r), g(s)] = [g(r1 ), g(s1 )] or g([r, s]) = g([r1 , s1 ]). Thus, g is well defined. Moreover, g is an Hv -homomorphism because we have {[a, b] | a ∈ g(r1 · s2 + r2 · s1 ), b ∈ g(s1 · s2 )} ⊆ g([r1 , s1 ] ] [r2 , s2 ]) and {[a, b]|a ∈ g(r1 )·g(s2 )+g(r2 )·g(s1 ), b ∈ g(s1 )·g(s2 )} ⊆ g([r1 , s1 ])]g([r2 , s2 ]). Therefore, g([r1 , s1 ] ] [r2 , s2 ]) ∩ (g([r1 , s1 ]) ] g([r2 , s2 ])) 6= ∅ and similarly we obtain g([r1 , s1 ] ⊗ [r2 , s2 ]) ∩ (g([r1 , s1 ]) ⊗ g([r2 , s2 ])) 6= ∅, which proves that g is an Hv -homomorphism.



Definition 4.16. Let R be an Hv -ring. A non-empty subset I of R is called an Hv -ideal if the following conditions hold:

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(1) (I, +) is an Hv -subgroup of (R, +); (2) I · R ⊆ R and R · I ⊆ I. An Hv -ideal I is called an Hv -isolated ideal if it satisfies the following axiom, • For all X ⊆ I, Y ⊆ S if (M, N ) ∈k X, Y k, then M ⊆ I. Lemma 4.6. If I is an Hv -isolated ideal of R, then the set S −1 I = {[a, s] | a ∈ I, s ∈ S} is an Hv -ideal of S −1 R. Proof. First, we prove that (S −1 I, ]) is an Hv -subgroup of (S −1 R, ]). For every [a1 , s1 ], [a2 , s2 ] ∈ S −1 I, we have S [a1 , s1 ] ] [a2 , s2 ] = {[a, s]|a ∈ A, s ∈ B}. (A,B)∈ka1 ·s2 +a2 ·s1 ,s1 ·s2 k

From a1 , a2 ∈ I we obtain a1 · s2 + a2 · s1 ⊆ I and since I is an Hv -isolated ideal of R, it follows that A ⊆ I. Therefore, [a1 , s1 ] ] [a2 , s2 ] ⊆ S −1 I. Now, we prove the equality S −1 I = [a1 , s1 ]]S −1 I, for all [a1 , s1 ] ∈ S −1 I. Suppose that [a, s] ∈ S −1 I, a ∈ I. Since s, s1 ∈ S, there exists s2 ∈ S such that s ∈ s1 ·s2 . Moreover, since I is an Hv -ideal, we have a1 ·s2 +(s1 +1)I = I. Hence there exists a2 ∈ I, such that a ∈ a1 · s2 + s1 · a2 + a2 and so a ∈ a1 ·s2 +(a2 +a2 ·s3 )·s1 , whence 1 ∈ s3 ·s1 . So there exists b ∈ a2 +a2 ·s3 such that a ∈ a1 · s2 + b · s1 , therefore [a, s] ∈ [a1 , s1 ] ] [b, s2 ] implying S −1 I ⊆ [a1 , s1 ] ] S −1 I. It remains to prove the second condition of the definition of an Hv -ideal. In order to do this, suppose that [a, t] ∈ S −1 I and [r, s] ∈ S −1 R. Then S [a, t] ⊗ [r, s] = {[x, y]|x ∈ A, y ∈ B}. (A,B)∈ka·r,t·sk

Since a ∈ I and I is an Hv -isolated ideal of R, we have a · r ⊆ I and so A ⊆ I. Consequently [a, t] ⊗ [r, s] ⊆ S −1 I. Therefore, S −1 I is an Hv -ideal of S −1 R.  Lemma 4.7. If I, J are two Hv -isolated ideals of R, then (1) S −1 (I ∩ J) = S −1 I ∩ S −1 J, (2) S −1 (I · J) = S −1 I ⊗ S −1 J, (3) S −1 (I + J) ⊆ S −1 I ] S −1 J. Proof.

The proof is straightforward and is omitted.



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The natural mapping ψ : R → S −1 R, where ψ(r) = [r, 1], is an inclusion homomorphism. Theorem 4.34. Let I be an Hv -isolated ideal of R. Then, S ∩ I 6= ∅ if and only if S −1 I = S −1 R. Proof. If t ∈ S ∩ I, then [t, t] = [1, 1] ∈ S −1 I. Therefore, for every [r, s] ∈ S −1 R, we have [1, 1] ⊗ [r, s] ⊆ S −1 I. From [r, s] ∈ [1, 1] ⊗ [r, s] we obtain [r, s] ∈ S −1 I and this prove that S −1 R ⊆ S −1 I. Conversely, assume that S −1 I = S −1 R. If we consider the natural inclusion homomorphism ψ : R → S −1 R, then ψ(1) = [1, 1]. On the other hand, ψ(1) ∈ S −1 R, consequently ψ(1) ∈ S −1 I and so ψ(1) = [a, s] for some a ∈ I, s ∈ S. Now, we have [1, 1] = [a, s], therefore, there exists A ⊆ S such that A · s = A · a. Since A · s ⊆ S and A · a ⊆ I, we get I ∩ S 6= ∅.  Theorem 4.35. Let I be an Hv -isolated ideal of R. Then the following assertions hold: (1) I ⊆ ψ −1 (S −1 I), (2) If I = ψ −1 (J) for some Hv -ideal J of S −1 R, then S −1 I = J. Proof. The proof of (1) is obvious. In order to prove (2), let I = ψ −1 (J) where J is an Hv -ideal of S −1 R. Then [r, s] ∈ S −1 I implies r ∈ I and so ψ(r) = [r, 1] ∈ J. Therefore, [1, s] ⊗ [r, 1] ⊆ J. Since [r, s] ∈ [1, s] ⊗ [r, 1], we obtain [r, s] ∈ J which implies that S −1 I ⊆ J. Now, let [r, s] ∈ J. Then ψ(r) = [r, 1] ∈ [r, 1] ⊗ [s, s] = [r, s] ⊗ [s, 1] ⊆ J. Therefore r ∈ ψ −1 (J) = I, hence [r, s] ∈ S −1 I, and this proves that J ⊆ S −1 I.  Definition 4.17. Let A be a commutative Hv -ring. An Hv -ideal P is called an Hv -prime ideal of A, if a · b ⊆ P implies a ∈ P or b ∈ P . Theorem 4.36. If P is an Hv -isolated prime ideal of R such that S ∩P =∅, then S −1 P is an Hv -prime ideal of S −1 R and ψ −1 (S −1 P ) = P . Proof. By Lemma 4.6, S −1 P is an Hv -ideal of S −1 R. Now, we check that S −1 P is prime. If [r, s] ⊗ [r1 , s1 ] ⊆ S −1 P , then {[b, s2 ] | b ∈ r · r1 , s2 ∈ s · s1 } ⊆ r · r1 , s · s1 ⊆ S −1 P . It follows that for every b ∈ r · r1 , s2 ∈ s · s1 there exists a ∈ P and t ∈ S such that [b, s2 ] = [a, t]. Therefore, there exists a subset A of S such that A · b · t = A · a · s2 . Since A · a · s2 ⊆ P , we have A · b · t ⊆ P . Now, for every x ∈ A · t we obtain x · b ⊆ P . Since A · t ⊆ S and S ∩ P = ∅, it follows that x 6∈ P and so b ∈ P . Consequently

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r · r1 ⊆ P which implies that r ∈ P or r1 ∈ P . Therefore, [r, s] ∈ S −1 P or [r1 , s1 ] ∈ S −1 P . On the other hand, by Theorem 4.35, we have P ⊆ ψ −1 (S −1 P ). Conversely, assume that r ∈ ψ −1 (S −1 P ). Then, ψ(r) ∈ S −1 P and since ψ(r) = [r, 1], there exists a ∈ P, t ∈ S such that [r, 1] = [a, t]. Therefore, there is a subset A of S such that A · r · t = A · a. Since A · a ⊆ P , we have A · r · t ⊆ P . Now, for every x ∈ A · t, we obtain x · r ⊆ P . Since A · t ⊆ S, it follows that x 6∈ P and so, r ∈ P . Therefore, ψ −1 (S −1 P ) = P .  Lemma 4.8. For every a ∈ S, γ ∗ (a) is invertible in R/γ ∗ . Proof. Since 1 ∈ R, it follows that γ ∗ (1) ∈ R/γ ∗ . Now, for every γ ∗ (x) ∈ R/γ ∗ , we have γ ∗ (x) γ ∗ (1) = γ ∗ (1) γ ∗ (x) = γ ∗ (x), i.e., γ ∗ (1) is the identity of the ring R/γ ∗ . On the other hand, by the definition of S, for every a ∈ S there exists b ∈ S such that 1 ∈ a · b = b · a. Therefore, γ ∗ (1) = γ ∗ (a · b) = γ ∗ (b · a) and so γ ∗ (1) = γ ∗ (a) γ ∗ (b) = γ ∗ (b) γ ∗ (a) which implies that γ ∗ (b) is the inverse of γ ∗ (a).  Theorem 4.37. If all subsets A of S are finite polynomials of elements of R over N, then there exists an Hv -homomorphism f : S −1 R → R/γ ∗ such that f ψ = ϕ, i.e., the following diagram is commutative. ψ

R ϕ

! { R/γ ∗

/ S −1 R f

Proof. We define f : S −1 R → R/γ ∗ by setting f ([r, s]) = γ ∗ (r) γ ∗ (s)−1 . First, we prove that f is well defined. If [r, s] = [r1 , s1 ], then there exists A ⊆ S such that A · r · s1 =A · r1 · s and so ϕ(A · r · s1 )=ϕ(A · r1 · s) which implies that γ ∗ (A) γ ∗ (r) γ ∗ (s1 )=γ ∗ (A) γ ∗ (r1 ) γ ∗ (s). By hypothesis γ ∗ (A) = γ ∗ (a) for every a ∈ A, so we obtain γ ∗ (a) γ ∗ (r) γ ∗ (s1 ) = γ ∗ (a) γ ∗ (r1 ) γ ∗ (s). Multiplying the above relation by γ ∗ (a)−1 γ ∗ (s)−1 γ ∗ (s1 )−1 , we have γ ∗ (r) γ ∗ (s)−1 = γ ∗ (r1 ) γ ∗ (s1 )−1 . Therefore, f ([r, s]) = f ([r1 , s1 ]). Thus, f is well defined. Moreover, f is an Hv -homomorphism, because we have γ ∗ (r1 · s2 + r2 · s1 ) γ ∗ (s1 · s2 )−1 ∈ f ([r1 , s1 )] ] [r2 , s2 ]), (γ ∗ (r1 ) γ ∗ (s1 )−1 ) ⊕ (γ ∗ (r2 ) γ ∗ (s2 )−1 ) = f ([r1 , s1 ]) ⊕ f ([r2 , s2 ]), γ ∗ (r1 · r2 ) γ ∗ (s1 · s2 )−1 ∈ f ([r1 , s1 ]) ⊗ [r2 , s2 ]), (γ ∗ (r1 ) γ ∗ (s1 )−1 ) (γ ∗ (r2 ) γ ∗ (s2 )−1 ) = f ([r1 , s1 ]) f ([r2 , s2 ]).

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Finally, it is clear that f ψ = ϕ.



Let γs∗ be the fundamental equivalence relation on S −1 R and let Us denotes the set of finite polynomials of elements of S −1 R over N. Theorem 4.38. There exists a homomorphism h : R/γ ∗ → S −1 R/γs∗ . Proof. We define h(γ ∗ (r)) = γs∗ ([r, 1]). First, we prove that h is well defined. Suppose that γ ∗ (r1 ) = γ ∗ (r2 ), so r1 γ ∗ r2 . Hence there exist x1 , ..., xm+1 ∈ R; u1 , ..., um ∈ U with x1 = r1 , xm+1 = r2 such that {xi , xi+1 } ⊆ ui , i = 1, ..., m which implies that {[xi , 1], [xi+1 , 1]} ⊆ ui , 1 ∈ Us . ∗ Therefore, [r1 , 1]γs [r2 , 1] and so γs∗ ([r1 , 1]) = γs∗ ([r2 , 1]). Thus, h is well defined. h is a homomorphism, because h(γ ∗ (a) ⊕ γ ∗ (b)) = h(γ ∗ (c)) = γs∗ ([c, 1]) for all c ∈ γ ∗ (a) + γ ∗ (b) and h(γ ∗ (a)) ⊕ h(γ ∗ (b)) = γs∗ ([a, 1]) ⊕ γs∗ ([b, 1]) = γs∗ ([d, s]) for all [d, s] ∈ γs∗ ([a, 1]) ] γs∗ ([b, 1]). Thus, setting d = c ∈ a + b, s = 1, we obtain h(γ ∗ (a) ⊕ γ ∗ (b)) = h(γ ∗ (a)) ⊕ h(γ ∗ (b)). Similarly, we obtain h(γ ∗ (a) γ ∗ (b)) = h(γ ∗ (a)) h(γ ∗ (b)). Therefore, h is a homomorphism of rings.  Corollary 4.6. The following diagram is commutative, i.e., ϕs ψ = hϕ where ϕ and ϕs are the canonical projections. ψ

R ϕ

/ S −1 R ϕs

 R/γ ∗

h

 / S −1 R/γs∗

Corollary 4.7. If ϕ : R → R/γ ∗ is the canonical projection, then the map θ : S −1 R → ϕ(S)−1 (R/γ ∗ ) defined by θ([r, s]) = [γ ∗ (r), γ ∗ (s)] is an Hv -homomorphism. Corollary 4.8. The following diagram is commutative, i.e., θψ = ψ1 ϕ. R

ψ

ϕ

 R/γ ∗

ψ1

/ S −1 R 

θ

/ ϕ(S)−1 (R/γ ∗ )

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Hv -group rings

In an Hv -group, several convolutions can be defined. In this paragraph, we present a convolution and obtain an Hv -group ring. The results are mainly from [106, 120]. Examples and applications in known classes of hyperstructures are also investigated. Definition 4.18. Let (H, ·) be a hypergroupoid. The following set is called a set of fundamental maps on H with respect to · : Θ = {θ : H × H →onto H | θ(x, y) ∈ x · y}. Any subset Θµ ⊆ Θ define a hyperoperation ◦µ on H as follows: x ◦µ y = {z | z = θ(x, y) for some θ ∈ Θµ }. Obviously, ◦µ ≤ · and Θµ ⊆ Θ◦µ , where Θ◦µ denotes of the fundamental maps on H with respect to ◦µ . A set Θα ⊆ Θ is called associative (respectively weak associative) if and only if for every subset Θµ ⊆ Θα the hyperoperation ◦µ is associative (respectively weak associative). A hypergroupoid (H, ·) will be called Θ-weak associative if there exists an element θ◦ ∈ Θ which defines an associative operation ◦ in H. • For every θ ∈ Θ we have 1 ≤ |θ−1 (g)| ≤ n2 − n + 1 for all g ∈ H where θ−1 (g) is the inverse image of g. However, for every θ ∈ Θ P −1 we have |θ (g)| = n2 . g∈H

• If (H, ·) is Θ-weak associative, then every greater hypergroupoid is Θ-weak associative. • All Hb -groups are Θ-weak associative. • Any Hb -semigroup, which has a b-structure with the property H 2 = H is Θ-weak associative. • If (H, ·) is an Hv -group containing a scalar, then all the maps f : H × H → H with f (x, y) ∈ xy for all x, y ∈ H, are onto, i.e., f ∈ Θ. Indeed, if s is a scalar, then f (s, H) = H. Example 4.21. An example of a type of b-semigroups is defined as follows: Fix an element s ∈ H. We define the product ◦ by setting:  s if x 6= y x◦y = x if x = y. Then H ◦ H = H and ◦ is associative, since  x if x = y = z (x ◦ y) ◦ z = x ◦ (y ◦ z) = s otherwise.

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Definition 4.19. Let (H, ·) be Θ-weak associative with |G| = n and θ◦ ∈ Θ be associative. Let F be a field and F [H] be the set of formal linear combinations of elements of H with coefficients from F . In F [H] the ordinary addition + can be defined by setting (f1 + f2 )(g) = f1 (g) + f2 (g) for all g ∈ H and f1 , f2 ∈ F [H]. Furthermore, consider the hyperproduct ? , called convolution, defined for every f1 , f2 of F [H] as follows: n o P f1 ? f2 = fθ | fθ (g) = f1 (x)f2 (y); θ ∈ Θ . θ(x,y)=g

Theorem 4.39. The structure (F [H], +, ?) is a multiplicative Hv -ring where the inclusion distributivity is valid. Obviously, (F [H], +) is a group. Let f1 , f2 , f3 ∈ F [H]. Then n o P (f1 ? f2 ) ? f3 = fθ | fθ (u) = f1 (x)f2 (y); θ ∈ Θ ? f3

Proof.

θ(x,y)=u

n = fϕ | fϕ (g) =

P

o f1 (x)f2 (y)f3 (z); θ, ϕ ∈ Θ 3 fθ◦

P

ϕ(u,z)=g θ(x,y)=u

P

where fθ◦ (g) =

P

f1 (x)f2 (y)f3 (z). On the other hand,

θ◦ (u,z)=g θ◦ (x,y)=u

n f1 ? (f2 ? f3 ) = fϕ | fϕ (g) =

P

o f1 (x)f2 (y)f3 (z); θ, ϕ ∈ Θ 3 fθ◦ ,

P

ϕ(x,v)=g θ(y,z)=v

where fθ◦ (g) =

P

P

f1 (x)f2 (y)f3 (z). Since θ◦ is associative.

θ◦ (x,v)=g θ◦ (y,z)=v

Thus, (f1 ? f2 ) ? f3 ∩ f1 ? (f2 ? f3 ) 6= ∅. Now, for the left distributivity, we have n o P f1 ? (f2 + f3 ) = fθ | fθ (g) = f1 (x)[(f2 + f3 )(y)]; θ ∈ Θ θ(x,y)=g

n = fθ | fθ (g) =

P

[f1 (x)f2 (y) + f1 (x)f3 (y)]; θ ∈ Θ

θ(x,y)=g

and n f1 ? f2 + f1 ? f3 = fρ | fρ (g) =

P

f1 (x)f2 (y); ρ ∈ Θ

o

ρ(x,y)=g

n + fϕ | fϕ (g) =

P

f1 (x)f3 (y); ϕ ∈ Θ

o

ϕ(x,y)=g

n = fρϕ = fρ + fϕ | fρϕ (g) =

P ρ(x,y)=g

+

P ϕ(r,s)=g

o f1 (r)f3 (s); ρ, ϕ ∈ Θ .

f1 (x)f2 (y)

o

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Thus, f1 ?(f2 +f3 ) ⊆ f1 ?f2 +f1 ?f3 , which is the left inclusion distributively. Similarly, the right inclusion distributivity is valid.  Definition 4.20. The above Hv -ring is called a hypergroupoid Hv -algebra or an Hv -group ring. Given a hypergroupoid (H, ·) one can define an Hv -group ring by “enlarging” the hyperoperation · as follows: Take any ? on H such that (H, ?) is Θ-weak associative hypergroupoid then take the union ♦ = · ∪ ?, i.e., x♦y=(xy) ∪ (x ? y), for all x, y ∈ H. An Hv -group ring is defined on (H, ♦). The most important condition in order to define the Hv -group ring is the Θ-weak associative condition. Therefore, in what follows we focus on our attention on classes which satisfy the Θ-weak associative condition. Now, we prove the following theorem. Theorem 4.40. In every Hv -group ring (F [H], +, ?) we have (−f1 ) ? f2 = −(f1 ? f2 ) = f1 ? (−f2 ), for all f1 , f2 ∈ F [H], and there exists an absorbing element. Proof.

For every f1 , f2 ∈ F [H], we have n o P −f1 ? f2 = fθ | fθ (h) = (−f1 )(x)f2 (y); θ ∈ Θ θ(x,y)=h

=

n

P

− fθ | fθ (h) =

o f1 (x)f2 (y); θ ∈ Θ

θ(x,y)=h

= −(f1 ? f2 ). Take the element f0 ∈ F [H] such that f0 (h) = 0, for all h ∈ H. Then, for every f ∈ F [H] we have n o P f0 ? f = fθ | fθ (h) = f0 (x)f (y); θ ∈ Θ θ(x,y)=h

n o = fθ | fθ (h) = 0; h ∈ H, θ ∈ Θ = f0 . Therefore, f0 is the absorbing element.



Example 4.22. Consider the Hb -group (Z3 , ⊕) which has the b-group (Z3 , +) and the non-singleton products are: 1⊕1 = {0, 1}, 2⊕2 = {0, 1, 2}. The 0 is scalar. Therefore, every map θ : Z23 → Z3 with θ(x, y) ∈ x ⊕ y, is

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an element of Θ. Thus, |Θ| = 2 · 3 = 6 and we see Θ = {θ1 , θ2 , θ3 , θ4 , θ5 , θ6 }. Then, for all θ ∈ Θ, we have θ(0, 0)=0, θ(0, 1)=θ(1, 0)=1, θ(0, 2)=θ(2, 0) = 2, θ(1, 2)=θ(2, 1)=0, and θ1 (1, 1)=0, θ1 (2, 2)=0; θ2 (1, 1)=0, θ2 (2, 2)=1; θ3 (1, 1)=0, θ3 (2, 2)=2; θ4 (1, 1)=1, θ4 (2, 2)=0; θ5 (1, 1)=1, θ5 (2, 2)=1; θ6 (1, 1)=1, θ6 (2, 2)=2. Every hyperproduct of elements of F [Z3 ] has at most 6 elements. Let r, s ∈ F [Z3 ], then n o P r ? s = tθ | tθ (g) = r(x)s(y); θ ∈ Θ . θ(x,y)=g

For every θ ∈ Θ, we have to calculate 9 products of the form r(x)s(y) in order to obtain tθ . tθ1 (0) = r(0)s(0) + r(1)s(2) + r(2)s(1) + r(1)s(1) + r(2)s(2), tθ1 (1) = r(0)s(1) + r(1)s(0), tθ1 (2) = r(0)s(2) + r(2)s(0), tθ2 (0) = r(0)s(0) + r(1)s(2) + r(2)s(1) + r(1)s(1), tθ2 (1) = r(0)s(1) + r(1)s(0) + r(2)s(2), tθ2 (2) = r(0)s(2) + r(2)s(0), tθ3 (0) = r(0)s(0) + r(1)s(2) + r(2)s(1) + r(1)s(1), tθ3 (1) = r(0)s(1) + r(1)s(0), tθ3 (2) = r(0)s(2) + r(2)s(0) + r(2)s(2) and similar for the tθ4 , tθ5 tθ6 . Example 4.23. Consider the Hb -group (Zmn , ⊕) defined in Example 3.2. Then, Θ has only two elements Θ = {θ1 , θ2 }. For all θ ∈ Θ, we have θ(x, y) = x + y if (x, y) 6= (0, m) and θ1 (0, m) = 0, θ2 (0, m) = m. The map θ2 leads to the known convolution on (Zmn , +). For every element g 6= 0 P and m, the sum tθ1 (g) = r(x)s(y) has mn elements. Moreover, θ1 (x,y)=g

tθ1 (0) =

P

r(x)s(y)

θ1 (x,y)=0

is a sum of mn + 1 terms of the form r(x)s(y) and P tθ1 (m) = r(x)s(y) θ1 (x,y)=m

is a sum of mn − 1 terms of r(x)s(y).

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Let (H, ◦) be an Hv -group, (G, +) be a group with the zero element 0, {Ai }i∈G be a family of non empty sets with A0 = H and Ai ∩ Aj = ∅, S for all i, j ∈ G, i 6= j. Set K = Ai and consider the hyperoperation i∈G

defined in K as follows:  x y =

x ◦ y if (x, y) ∈ H 2 Ai+j if (x, y) ∈ Ai × Aj 6= H 2 .

Then, (K, ) becomes an Hv -group. which is called an (H, G)-Hv -group. It is easy to see that K/β ∗ ∼ = G. Theorem 4.41. If cardAi = n for all i ∈ G and (H, ◦) is Θ-weak associative, then (K, ) is Θ-weak associative. Moreover, cardΘK ≥ (n!)m−1 cardΘH , where m = cardG. Proof. We consider a family of one to one maps {pi }i∈G such that pi : H→Ai , i 6= 0 and p0 is the identity map. Notice that all these maps are also onto. Take θ ∈ ΘH which defines an associative operation · and consider the mapping θ0 : K × K → K which defines the operation ♦ in K, as follows: −1 x♦y = pi+j (p−1 i (x) · pj (y)) for all x ∈ Ai , y ∈ Aj .

This mapping is, obviously, onto, so it remain to prove that ♦ is associative. Suppose that (x, y, z) ∈ Ai × Aj × Ar , we have −1 x♦(y♦z)= x♦pj+r (p−1 j (y) · pr (z)) −1 −1 = pi+(j+r) [p−1 i (x) · (pj (y) · pr (z))] = (x♦y)♦z.

Therefore, (K, ⊕) is Θ-weak associative. Now, we remark that the number of the families pi , i ∈ G, i 6= 0 of bijective maps is (n!)m−1 . Therefore, cardΘK ≥ (n!)m−1 cardΘH .



Theorem 4.42. Let (K, ·) be Θ-weak associative such that cardAi = 1, for all i 6= 0. Then, for all θ ∈ ΘK which define an associative operation ♦ in K, there exists an element x ∈ H such that y♦x = x = x♦y, for all y ∈ H, z♦w = x, for all (z, w) ∈ Ar × As 6= H 2 for which r + s = 0.

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Proof. Take (z, w) ∈ Ar × As 6= H 2 for which r + s = 0 and set z♦w = x ∈ H. Then, for all y ∈ H, y♦x = y♦(z♦w) = (y♦z)♦w = z♦w = x. Similarly, x♦y = x. This element x is unique, because if there exists another element x0 such that (u, v) ∈ Ap × Aq 6= H 2 with p + q = 0 and u♦v = x0 , then x = x♦x0 = x0 .  Note that in the above theorem, the operation induced by the restriction of θ to H × H is weak associative. This means that (H, ◦) is Θ-weak associative. From the above theorem we obtain the following construction. Theorem 4.43. Let (H, ◦) be such that there exists an associative operation ♦ on H and a special element x ∈ H such that x♦y = x = y♦x for all y ∈ H. Then there exists a Θ-weak associative (H, G)-Hv -group (K, ) with cardAi = 1, for all i ∈ G, i 6= 0. Proof. Consider the extension of ♦ to K for which z♦w = x for all (z, w) ∈ Ai × Aj 6= H with i + j = 0. It is easy to check that this operation is associative on K.  In the case of the (H, G)-Hv -groups with cardAi = 1, for all i ∈ G \ {0}, the cardinality of the set of onto maps K × K → K : (x, y) 7→ z ∈ x y is less or equal to nm+n

2

−1

, where n = cardH and m = cardG.

Definition 4.21. Let {Si }i∈I be a pairwise disjoint family of Hv semigroups, where |I| > 1. We define a hyperoperation ⊗, called an SS hyperoperation on the set S = Si as follows: i∈I

xi ⊗ yi = xi yi for all (xi , yi ) ∈ Si2 , xi ⊗ xj = Si ∪ Sj for all (xi , xj ) ∈ Si × Sj , i 6= j. Then, the hyperstructure (S, ⊗), called an S-construction, is an Hv -group. Let {Si }i∈I be a family of pairwise disjoint sets, where cardI > 1. On Si we consider the total hyperoperation ab = Si , for all a, b ∈ Si or the least incidence hyperoperation ab = {a, b}, for all a, b ∈ Si . In each case, we obtain the S-construction (S, ⊗). In what follows, we consider the finite case. Let cardI = n and

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cardSi = si , i ∈ I and suppose that for each i ∈ I, Si is a group or a groupoid with the associated set of fundamental maps Θi 6= ∅. Let Θi be the set of fundamental maps on S with respect to ⊗. We obtain i ⊕ < yα >= {< zα > | zα ∈ xα + yα , α ∈ Γ}, < xα > < yα >= {< zα > | zα ∈ xα · yα , α ∈ Γ}. Q Q It follows that Rα is an Hv -ring. We call Rα the external direct α∈Γ

α∈Γ

product of Rα (α ∈ Γ). Let µ and λ be fuzzy subsets of a non-empty set X. The Cartesian product µ × λ is usually defined by: (µ × λ)(x, y) = min{µ(x), λ(y)}, for all x, y ∈ X. Let {Xα | α ∈ Γ} be a collection of non-empty sets and let µα be a fuzzy subset of Xα for all α ∈ Γ. Define the Cartesian product of the µα by Q ( µα )(x) = inf {µα (xα )} where x =< xα > and < xα > denotes an α∈Γ α∈Γ Q element of the Cartesian product Xα . α∈Γ

Proposition 4.19. Let {Rα | α ∈ Γ} be a collection of Hv -rings and let Q Q µα be a fuzzy Hv -ideal of Rα . Then µα is a fuzzy Hv -ideal of Rα . α∈Γ

Proof.

Let x =< xα >, y =< yα >

α∈Γ

∈

Q

Rα . Then, for every z =

α∈Γ

< zα >∈ x + y =< xα > ⊕ < yα > we have Q ( µα )(z) = inf {µα (zα )} α∈Γ

α∈Γ

≥ inf {min{µα (xα ), µα (yα )}} α∈Γ n o = min inf {µα (xα )}, inf {µα (yα )} α∈Γ n α∈Γ o Q Q = min ( µα )(x), ( µα )(y) . α∈Γ

α∈Γ

Therefore, the first condition of the definition of an Hv -ideal is satisfied. Now, we prove the second condition as follows. For every x =< xα > Q Q and a =< aα > in Rα there exists y =< yα > in Rα such that α∈Γ

α∈Γ

min{µα (xα ), µα (aα )} ≤ µα (yα ). Therefore, we have < xα >∈< aα > ⊕ < yα > and Q ( µα )(y) = inf {µα (yα )} ≥ inf {min{µα (xα ), µα (aα )}} α∈Γ α∈Γ α∈Γ n o = min inf {µα (xα )}, inf {µα (aα )} α∈Γ n α∈Γ o Q Q = min ( µα )(x), ( µα )(a) . α∈Γ

α∈Γ

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The proof of third condition is similar to that of second condition. To verify the fourth condition, for every z =< zα >∈ x y =< xα > < yα > we have Q Q ( µα )(z) = inf {µα (zα )} ≥ inf {µα (yα )} = ( µα )(y). α∈Γ

α∈Γ

α∈Γ

α∈Γ

n Q o Q µα )(z) . Hence, ( µα )(y) ≤ inf ( α∈Γ

z∈x·y



α∈Γ

The following corollary is exactly obtained from the above proposition. Corollary 4.13. Let µ be a fuzzy subset of an Hv -ring R. Then, µ × µ is a fuzzy left (right) Hv -ideal of R × R if and only if µ is a fuzzy left (right) Hv -ideal of R. Definition 4.32. Let µ be a fuzzy Hv -ideal of R1 × R2 and let (x1 , x2 ), (a1 , a2 ) ∈ R1 ×R2 . Then, there exists (y1 , y2 ) ∈ R1 ×R2 such that (x1 , x2 ) ∈ (a1 , a2 ) ⊕ (y1 , y2 ) and min{µ(x1 , x2 ), µ(a1 , a2 )} ≤ µ(y1 , y2 ). Now, if for every r1 , s1 ∈ R1 there exists t1 ∈ R1 such that (r1 , x2 ) ∈ (s1 , a2 ) ⊕ (t1 , y2 ) and min{µ(r1 , x2 ), µ(s1 , a2 )} ≤ µ(t1 , y2 ), and for every r2 , s2 ∈ R2 there exists t2 ∈ R2 such that (x1 , r2 ) ∈ (a1 , s2 ) ⊕ (y1 , t2 ) and min{µ(x1 , r2 ), µ(a1 , s2 )} ≤ µ(y1 , t2 ), then we say that µ satisfies in the regular left fuzzy reproduction axiom. Similarly, we can define the regular right fuzzy reproduction axiom. µ is called the regular fuzzy Hv -ideal of R if µ satisfies the regular left and right fuzzy reproduction axioms. Theorem 4.55. Let R1 , R2 be Hv -rings with scalar units and µ be a regular fuzzy Hv -ideal of R1 × R2 . Then, µi , i = 1, 2 is a fuzzy Hv -ideal of Ri , i = 1, 2, respectively, where µ1 (x) = sup {µ(x, a)} and µ2 (y) = sup {µ(b, y)}. a∈R2

b∈R1

Proof. We show that µ1 is a fuzzy Hv -ideal of R1 . Suppose that x, y ∈ R1 . Then, for every α ∈ x + y we have µ1 (α) = sup {µ(α, a)}. For every a∈R2

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a ∈ R2 there exist ra , sa ∈ R2 such that a ∈ a + ra and a ∈ sa + a. Now, we have µ1 (α) ≥ sup {min{µ(x, a), µ(y, ra )}} a∈R2

≥ sup {min{µ(x, a), inf {µ(y, b)}}} b∈R2 a∈R2n o = min sup {µ(x, a)}, inf {µ(y, b)} b∈R2 na∈R2 o = min µ1 (x), inf {µ(y, b)} b∈R2

and also we have µ1 (α) ≥ sup {min{µ(x, sa ), µ(y, a)}} a∈R2 n o ≥ sup min{ inf {µ(x, c)}, µ(y, a)} c∈R2 a∈R2n o = min inf {µ(x, c)}, sup {µ(y, a) a∈R2 o nc∈R2 = min inf {µ(x, c)}, µ1 (y) . c∈R2

Therefore,  o o n n µ1 (α) ≥ max min µ1 (x), inf {µ(y, b)} , min inf {µ(x, c)}, µ1 (y) c∈R2 b∈R2  n o n o = min max µ1 (x), inf {µ(x, c) , max inf {µ(y, b)}, µ1 (y) c∈R2

b∈R2

≥ min{µ1 (x), µ1 (y)}, Now, if x, a ∈ R1 , then for every r, s ∈ R2 there exists (y, yr,s ) ∈ R1 × R2 such that (x, r) ∈ (a, s) ⊕ (y, yr,s ) and min{µ(x, r), µ(a, s)} ≤ µ(y, yr,s ). Thus, n o min{µ1 (x), µ1 (a)} = min

sup {µ(x, r)}, sup {µ(a, s)}

r∈R2

s∈R2

= sup {min{µ(x, r), µ(a, s))}} r∈R2 s∈R2

≤ sup {µ(y, yr,s )} r∈R2 s∈R2

≤ sup {µ(y, z)} = µ1 (y). z∈R2

The proof of the third condition is similar to the second condition. Now, we verify the fourth condition of the definition. Suppose that 1 be the unit scalar of R2 . For every α ∈ x · y, we have (α, a) ∈ (x, 1) (y, a). Since µ is a fuzzy Hv -ideal of R1 × R2 , we obtain µ(α, a) ≥ µ(y, a) which implies that supa∈R2 {µ(α, a)} ≥ sup {µ(y, a)}. Therefore µ1 (α) ≥ µ1 (y) for every a∈R2

α ∈ x · y, and so inf {µ1 (α)} ≥ µ1 (y). Hence, µ1 is a fuzzy Hv -ideal of R1 . α∈x·y

Similarly, we can prove that µ2 is a fuzzy Hv -ideal of R2 .



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Corollary 4.14. Let R1 , R2 be two Hv -rings with scalar units and µ, λ be fuzzy subsets of R1 , R2 , respectively. If sup {µ(x)} = sup {λ(y)} = 1 and x∈R1

y∈R2

µ × λ is a strong fuzzy Hv -ideal of R1 × R2 , then µ, λ are fuzzy Hv -ideals of R1 , R2 , respectively. Proof.

Suppose that

µ1 (x) = sup {(µ × λ)(x, a)} and µ2 (y) = sup {(µ × λ)(b, y)}, a∈R2

b∈R1

then it is enough to show that µ(x) = µ1 (x) and λ(y) = λ2 (y). Let x ∈ R1 then µ(x) = min{µ(x), 1} n

= min µ(x), sup {λ(y)}

o

y∈R2

= sup {min{µ(x), λ(y)}} y∈R2

= sup {(µ × λ)(x, y)} y∈R2

= µ1 (x).



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Chapter 5

Hv -modules

5.1

Hv -modules and fundamental relations

Definition 5.1. Let M be a non-empty set. Then, M is called an Hv module over an Hv -ring R if (M, +) is a weak commutative Hv -group and there exists a map · : R × M → P ∗ (M ) denoted by (r, m) 7→ rm such that for every r1 , r2 ∈ R and every m1 , m2 ∈ M, we have (1) r1 (m1 + m2 ) ∩ (r1 m1 + r1 m2 ) 6= ∅, (2) (r1 + r2 )m1 ∩ (r1 m1 + r2 m1 ) 6= ∅, (3) (r1 r2 )m1 ∩ r1 (r2 m1 ) 6= ∅. Definition 5.2. Let M1 and M2 be two Hv -modules over an Hv -ring R. A mapping f : M1 → M2 is called a strong Hv -homomorphism if for every x, y ∈ M1 and every r ∈ R, we have f (x + y) = f (x) + f (y) and f (rx) = rf (x). The Hv -modules M1 and M2 are called isomorphic if the Hv homomorphism f is one to one and onto. It is denoted by M1 ∼ = M2 . Similar to Hv -groups and Hv -rings, the fundamental equivalence relation ε∗ on Hv -modules is introduced by Vougiouklis [120]. The fundamental equivalence relation ε∗ on an Hv -module M can be defined as follows. Definition 5.3. Consider the Hv -module M over an Hv -ring R. If ϑ denotes the set of all expressions consisting of finite hyperoperations of either on R and M or of the external hyperoperations applying on finite sets of elements of R and M . A relation ε can be defined on M whose transitive closure is the fundamental relation ε∗ . The relation ε is defined as follows: 193

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for every x, y ∈ M , x ε y if and only if {x, y} ⊆ u for some u ∈ ϑ; i.e: xεy ⇔ x, y ∈

n P i=1

0

0

0

mi , mi = mi or mi =

ni k ij P Q ( rijk )mi , j=1 k=1

where mi ∈ M, rijk ∈ R. Suppose that γ ∗ (r) is the equivalence class containing r ∈ R and ε∗ (x) is the equivalence class containing x ∈ M . On M/ε∗ the ⊕ and the external product using the γ ∗ classes in R, are defined as follows: For every x, y ∈ M , and for every r ∈ R, ε∗ (x) ⊕ ε∗ (y) = ε∗ (c), for all c ∈ ε∗ (x) + ε∗ (y), γ ∗ (r) ε∗ (x) = ε∗ (d), for all d ∈ γ ∗ (r) · ε∗ (x). The kernel of canonical map φ : M → M/ε∗M is called heart of M and it is denoted by ωM , i.e., ωM = {x ∈ M | φ(x) = 0}, where 0 is the unit element of the group (M/ε∗ , ⊕). One can prove that the unit element of the group (M/ε∗ , ⊕) is equal to ωM . By the definition of ωM , we have ωωM = Ker(φ : ωM → ωM /ε∗ωM = 0) = ωM . The kernel of a strong Hv -homomorphism f : A → B is defined as follows Ker(f ) = {a ∈ A | f (a) ∈ ωB }. Lemma 5.1. Let M1 and M2 be two Hv -modules over an Hv -ring R and let ε∗M1 , ε∗M2 and ε∗M1 ×M2 be the fundamental relations on M1 , M2 and M1 × M2 respectively. Then, (x1 , x2 )ε∗M1 ×M2 (y1 , y2 ) ⇔ x1 ε∗M1 y1 and x2 ε∗M2 y2 , for all (x1 , x2 ), (y1 , y2 ) ∈ M1 × M2 . Proof.

It is straightforward.



Theorem 5.1. Let M1 and M2 be two Hv -modules over an Hv -ring R and let ε∗M1 , ε∗M2 and ε∗M1 ×M2 be the fundamental relations on M1 , M2 and M1 × M2 respectively. Then, (M1 × M2 )/ε∗M1 ×M2 ∼ = M1 /ε∗M1 × M2 /ε∗M2 . Proof.

It is straightforward.



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Definition 5.4. Let M be an Hv -module and X, Y are non-empty subsets w of M . We say X is weak equal to Y and write X = Y if and only if for every x ∈ X there exists y ∈ Y such that ε∗M (x) = ε∗M (y) and for every y ∈ Y there exists x ∈ X such that ε∗M (x) = ε∗M (y). f1

f2

fn

Definition 5.5. Let M0 → M1 → M2 → · · · → Mn−1 → Mn be a sequence of Hv -modules and strong Hv -homomorphisms. We say this sequence is w exact if for every 2 ≤ i ≤ n, Im(fi−1 ) = Ker(fi ). Definition 5.6. A function f : M1 → M2 is called weak-monic if for every 0 0 0 m1 , m1 ∈ M1 , f (m1 ) = f (m1 ) implies ε∗M1 (m1 ) = ε∗M1 (m1 ) and f is called weak-epic if for every m2 ∈ M2 there exists m1 ∈ M1 such that ε∗M2 (m2 ) = ε∗M1 (f (m1 )). Finally f is called weak-isomorphism if f is weak-monic and weak-epic. We present the following example for the above definitions. Example 5.1. Let R be an Hv -ring. Consider the following Hv -modules on R. (1) M = {a, b} together with the following hyperoperations: ∗M a b

a a b

b b a

and ·M : R × M → P ∗ (M ) (r,m)7→{a}

(2) M1 = {0, 1, 2} together with the following hyperoperations: ∗M1 0 1 2

0 0 1 2

1 1 0,2 1

(3) M2 = {¯ 0,¯ 1,¯ 2} together with ¯ ¯1 ∗M2 0 ¯ ¯ ¯1 0 0 ¯ ¯ ¯2 1 1 ¯ ¯ ¯0 2 2

2 2 1 0

and ·M1 : R × M1 → P ∗ (M1 ) (r,m1 )7→{0}

the following hyperoperations: ¯2 ¯2 and ·M2 : R × M2 → P ∗ (M2 ) ¯0 (r,m2 )7→M2 ¯1

Since {0, 2} ⊆ 1 ∗M1 1, r · m1 = 0 for every r ∈ R and every m1 ∈ M1 and 0 ∗M1 0 = 0, we obtain M1 /ε∗M1 = {ε∗M1 (0) = ε∗M1 (2) = {0, 2}, ε∗M1 (1) = {1}}. Also, since ε∗M1 (0)+ε∗M1 (1) = ε∗M1 (1), it follows that ωM1 = ε∗M1 (0) = {0, 2}. Since r ·M2 m2 = M2 for every r ∈ R and every m2 ∈ M2 , we obtain M2 /ε∗M2 = {{¯ 0,¯ 1,¯ 2}} and ωM2 = ε∗M2 (¯0) = ε∗M2 (¯1) = ε∗M2 (¯2) = M2 .

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Since (M1 × M2 )/ε∗M1 ×M2 ∼ = M1 /ε∗M1 × M2 /ε∗M2 , it follows that ε∗M1 ×M2

M1 × M2 / ¯ ¯ ¯ = {{(0,0), (0,1), (0,2), (2,¯0), (2,¯1), (2,¯2)}, {(1,¯0), (1,¯1), (1,¯2)}}.

Note that ωM1 ×M2 = ωM1 ×ωM2 . The subsets X = {(2,¯1), (2,¯2), (1,¯1), (1,¯2)} and Y = {(0,¯ 2), ((1,¯ 0)} of M1 × M2 are weakly equal. Now, consider f ∈ M [M1 × M2 ], where f (a) = (2,¯2), f (b) = (1,¯0) and g ∈ M1 [M1 × M2 ], where g(0) = (1,¯ 1), g(1) = (2,¯2), g(2) = (1,¯1). Then, f is weak-epic and g is weak-monic. 5.2

Hv -module of fractions

In Section 4.9, we introduced the concept of Hv -ring of fractions S −1 R of a commutative hyperring. The construction of S −1 R can be carried through with a hypermodule M over a hyperring R in place of the hyperring R. In this section, we introduce the set of fractions S −1 M and define addition and multiplication by elements of S −1 R. Then we show that S −1 M is an Hv -module over the Hv -ring S −1 R and is called Hv -module of fractions. The main reference for this part is [21]. Let X be the set of all ordered pairs (m, s) where m ∈ M , s ∈ S. For A ⊆ M and B ⊆ S, we denote the set {(a, b) | a ∈ A, b ∈ B} by (A, B). The relation ∼ is defined as follows: (A, B) ∼ (C, D) ⇔ there exists a subset T of S such that T ·(B·C) = T ·(D·A). Lemma 5.2. ∼ is an equivalence relation. Proof.

The proof is similar to the proof of Lemma 4.4.



The equivalence class containing (A, B) is denoted by k A, B k. If we consider the relation ∼ on X, we obtain the following two corollaries. Corollary 5.1. For (m1 , s1 ), (m2 , s2 ) ∈ X, we have (m1 , s1 ) ∼ (m2 , s2 ) if and only if there exists T ⊆ S such that T · (s1 · m2 ) = T · (s2 · m1 ). Corollary 5.2. ∼ is an equivalence relation on X. In X, the equivalence class containing (m, s) is denoted by [m, s] and we let S −1 M to be the set of all the equivalence classes. We define S  A, B = {[c, d] | c ∈ C, d ∈ D}. (C,D)∈kA,Bk

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Corollary 5.3. For all m ∈ M , s ∈ S, we have  m, s = sm, ss . Now we define addition and multiplication by elements of S −1 R, as follows: S [m1 , s1 ] ⊕ [m2 , s2 ] = {[a, b] | a ∈ A, b ∈ B} (A,B)∈ks1 ·m2 +s2 ·m1 ,s1 ·s2 k

[r, s] [m1 , s1 ] =

= s1 m2 + s2 m1 , s1 s2 , S {[a, b] | a ∈ A, b ∈ B} = rm1 , ss1 , (A,B)∈krm1 ,ss1 k

for all [m1 , s1 ], [m2 , s2 ] ∈ S −1 M and [r, s] ∈ S −1 R. Theorem 5.2. The above definitions are independent of the choices of representatives [m1 , s1 ], [m2 , s2 ] and [r, s] and that S −1 M satisfies the axioms of an Hv -module over S −1 R. If we define r [m1 , s1 ] = rm1 , s1 , then S −1 M is an Hv -module over R. Theorem 5.3. Let M1 and M2 be two hypermodules over a hyperring R and let f : M1 → M2 be a strong R-Hv -homomorphism. Then the map S −1 (f ) : S −1 M1 → S −1 M2 defined by S −1 (f )[m, s] = [f (m), s], is a strong S −1 R-Hv -homomorphism. Proof. Suppose that [m1 , s1 ], [m2 , s2 ] ∈ S −1 M and [r, s] ∈ S −1 R. First, we show that S −1 (f ) is well defined. If [m1 , s1 ] = [m2 , s2 ], then there exists T ⊆ S such that T · (s1 · m2 ) = T · (s2 , m1 ) which implies f (T · (s1 · m2 )) = f (T · (s2 · m1 )) and so T · (s1 · f (m2 )) = T · (s2 · f (m1 )) or [f (m1 ), s1 ] = [f (m2 ), s2 ]. Therefore, S −1 (f ) is well defined. Moreover, S −1 (f ) is an S −1 R-Hv -homomorphism because, we have S −1 (f )([m1 , s1 ] ⊕ [m2 , s2 ]) S = S −1 (f )( {[a, b] | a ∈ A, b ∈ B}) (A,B)∈ks1 m2 +s2 m1 ,s1 s2 k S = S −1 (f )({[a, b] | a ∈ A, b ∈ B}) (A,B)∈ks1 m2 +s2 m1 ,s1 s2 k S = {[f (a), b] | a ∈ A, b ∈ B} (A,B)∈ks1 m2 +s2 m1 ,s1 s2 k

and S −1 (f )([m1 , s1 ]) ⊕ S −1 (f )([m2 , s2 ]) = [f (m1 ), s1 ] ⊕ [f (m2 ), s2 ] S = {[a, b] | a ∈ A, b ∈ B} (A,B)∈ks1 f (m2 )+s2 f (m1 ),s1 s2 k S = {[a, b] | a ∈ A, b ∈ B}. (A,B)∈kf (s1 m2 +s2 m1 ),s1 s2 k

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Therefore, we have {[f (a), b] | a ∈ s1 m2 + s2 m1 , b ∈ s1 s2 } ⊆ S −1 (f )([m1 , s1 ] ⊕ [m2 , s2 ]), {[a, b] | a ∈ f (s1 m2 +s2 m1 ), b ∈ s1 s2 } ⊆ S −1 (f )([m1 , s1 ]) ⊕ S −1 (f )([m2 , s2 ]). Thus, S −1 (f )([m1 , s1 ] ⊕ [m2 , s2 ]) ∩ S −1 (f )([m1 , s1 ]) ⊕ S −1 (f )([m2 , s2 ]) 6= ∅. Similarly, we obtain {[f (a), b] | a ∈ rm1 , b ∈ ss1 } ⊆ S −1 (f )([r, s] [m1 , s1 ]), {[a, b] | a ∈ rf (m1 ), b ∈ ss1 } ⊆ [r, s] S −1 (f )([m1 , s1 ]). So, S −1 (f )([r, s] [m1 , s1 ]) ∩ [r, s] S −1 (f )([m1 , s1 ]) 6= ∅, which proves that S −1 (f ) is an S −1 R-Hv -homomorphism.



Lemma 5.3. The natural mapping Ψ : M → S −1 M , where Ψ(m) = [m, 1], is an inclusion R-Hv -homomorphism. Proof.

For every m1 , m2 ∈ M , we have Ψ(m1 + m2 ) = {[α, 1] | α ∈ m1 + m2 } S ⊆ {[a, b] | a ∈ A, b ∈ B} (A,B)∈km1 +m2 ,1k

= [m1 , 1] ⊕ [m2 , 1] = Ψ(m1 ) ⊕ Ψ(m2 ), and for every r ∈ R and m ∈ M we have Ψ(rm) = {[α, 1] | α ∈ rm} ⊆ rm, 1 = r Ψ(m). Therefore, Ψ is an inclusion R-Hv -homomorphism. ε∗s

 −1

Let be the fundamental equivalence relation on S M and Us is the set of all expressions consisting of finite hyperoperations of either on S −1 R and S −1 M or of external hyperoperation. In this case S −1 M/ε∗s is an S −1 R/γs∗ -module. Theorem 5.4. S −1 M/ε∗s is an R/γ ∗ -module. Proof.

We can define γ ∗ (r) ∗ ε∗s ([m, s]) = γs∗ ([r, 1])  ε∗ ([m, s]).

Then, it is clear that S −1 M/ε∗s is a module over the ring R/γ ∗ .



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Theorem 5.5. There is an R/γ ∗ -homomorphism h : M/ε∗ → S −1 M/ε∗s . Proof. We define h(ε∗ (m)) = ε∗s ([m, 1]). First we prove that h is well defined. Suppose that ε∗ (m1 ) = ε∗ (m2 ). So, m1 ε∗ m2 . m1 ε∗ m2 if and only if there exist x1 , . . . , xm+1 ; u1 , . . . , um ∈ U with x1 = m1 , xm+1 = m2 such that {xi , xi+1 } ⊆ ui , i = 1, . . . , m which implies {[xi , 1], [xi+1 , 1]} ⊆  ui , 1 ∈ Us . Therefore, [m1 , 1]ε∗s [m2 , 1] and so ε∗s ([m1 , 1]) = ε∗s ([m2 , 1]). Thus, h is well defined. Moreover, h is a homomorphism because h(ε∗ (a) ◦ ε∗ (b)) = h(ε∗ (c)) = ε∗s ([c, 1]), for all c ∈ ε∗ (a) + ε∗ (b), and h(ε∗ (a)) ◦ h(ε∗ (b)) = ε∗s ([a, 1]) ◦ ε∗s ([b, 1]) = ε∗s ([d, s]), for all [d, s] ∈ ε∗s ([a, 1]) ⊕ ε∗s ([b, 1]). Thus, setting d = c ∈ a + b, s = 1. Then, it is proved that h(ε∗ (a) ◦ ε∗ (b)) = h(ε∗ (a) ◦ h(ε∗ (b)). Also, we have h(γ ∗ (r)  ε∗ (m) = h(ε∗ (r · m)) = ε∗ ([a, 1]), for all a ∈ r · m, γ ∗ (r) ∗ h(ε∗ (m)) = γ ∗ (r) ∗ ε∗s ([m, 1]) = γs∗ ([r, 1])  ε∗s ([m, s]) = ε∗s ([b, s]), for all [b, s] ∈ [r, 1] [m, 1]. Hence, we obtain h(γ ∗ (r)  ε∗ (m) = γ ∗ (r) ∗ h(ε∗ (m)). Therefore, h is a homomorphism of modules.

5.3



Direct system and direct limit of Hv -modules

The construction of the direct system and direct limit is similar to the usual module theory (see [58, 79, 91]). In [58], Ghadiri and Davvaz considered the category of Hv -modules and prove that the direct limit always exists in this category. Direct limits are defined by a universal property, and so are unique. Also, already Leoreanu [78, 79] and Romeo [88] studied the notions of direct limits of hyperstructures. A partially ordered set I is said to be a direct set, if for each i, j ∈ I there exists k ∈ I such that i ≤ k and j ≤ k. Let I be a direct set and ϑ the category of Hv -modules. Let (Mi )i∈I be a family of Hv -modules indexed by I. For each pair i, j ∈ I such that i ≤ j, let φij : Mi → Mj be a homomorphism and suppose that the following axioms are satisfied:

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(1) φii is the identity for all i ∈ I, (2) φik = φjk φij whenever i ≤ j ≤ k. Then, the Hv -modules Mi and strong Hv -homomorphisms φij are said to be a direct system M = (Mi , φij ) over the direct set I. Let M = (Mi , φij ) be a direct system in ϑ. The direct limit of this system, denoted by lim Mi , is an → Hv -module and a family of strong Hv -homomorphisms αi : Mi → lim Mi , →

with αi = αj φij whenever i ≤ j satisfying the following universal mapping property: lim Mi −→ Y b

/X ?G

β fi αi

Mi

αj

fj

φ0j

 Mj

for every Hv -module X and every family of strong Hv -homomorphism fi : Mi → X with fi = fj φij , whenever i ≤ j, there is a unique strong homomorphism β : lim Mi −→ X making the above diagram commute. −→ S Let X be the disjoint union Mi . We define an equivalence relation on X by ai ρaj , ai ∈ Mi , aj ∈ Mj ⇔ there exists an index k ≥ i, j with φik ai = φk jaj . The equivalence class of ai is denoted by ρ(ai ). Suppose that X/ρ is the set of all equivalent classes. It is clear that a1 ρφiJ aj for j ≥ i. Now, for r ∈ R and ρ(ai ), ρ(aj ) ∈ X/ρ, we define ρ(ai )⊕ρ(aj ) = {ρ(x) | ak +a0k , where ak = φik ai , a0k = φjk aj , for some k ≥ i, j}, r ◦ ρ(ai ) = {ρ(x) | x ∈ rai }. Proposition 5.1. (X/ρ, ⊕, ◦) is an Hv -module over R. Proof.

It is straightforward.



Theorem 5.6. Let (Mi , φij ) be a direct system of Hv -modules indexed by I. Then, the Hv -module X/ρ is lim Mi . →

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Proof. We define αi : Mi → X/ρ by ai 7→ ρ(ai ) and consider the following diagram. X/ρ X a αi αj

Mi φ0j

 Mj

Then, αj (φij ai ) = ρ(φij ai ) = αi (ai ). Thus, αj φij = αi , and so the diagram is commutative. Now, suppose that M is an Hv -module and {fi | fi : Mi → M } be a family of strong Hv -homomorphisms with fi = fj φij . Define ζ : X/ρ → M by ρ(ai ) 7→ fi ai . We show that ζ is a strong Hv -homomorphism and so the universal mapping property holds. First, we show that zeta is well defined. Suppose that ρ(ai ) = ρ(bj ). Then, there exists k ≥ i, j such that φik ai = φjk bj . Hence, fk φik ai = fk φjk bj which implies that fi ai = fj bj . Therefore, zeta is well defined. Now, let ρ(ai ), ρ(bj ) ∈ X/ρ and r ∈ R. Then, we obtain ζ(ρ(ai ) ⊕ ρ(bj )) = {ζ(ρ(x)) | x ∈ ak + a0k , where ak = φik ai , a0k = φjk bj f or some k ≥ i, j} = {fk x | x ∈ ak + a0k , where ak = φik ai , a0k = φjk bj f or some k ≥ i, j} = fk (ak + a0k ), where ak = φik ai , a0k = φjk bj = fk ak + fk a0k = fk φik ai + fk φjk bj = fi ai + fj bj = ζ(ρ(ai )) + ζ(ρ(bi )) and ζ(r ◦ ρ(ai )) = ζ(ρ(rai )) = fi (rai ) = rζ(ρ(ai )). Therefore, ζ is a strong Hv -homomorphism and ζαi = fi .



Lemma 5.4. Let M1 , M2 be two Hv -modules over an Hv -ring R and f : M1 → M2 be a strong Hv -homomorphism. Let ε∗1 , ε∗2 and γ ∗ be the fundamental relations on M1 , M2 and R. Then, the map F : M1 /ε∗1 → M2 /ε∗2 defined by F (ε∗1 )) = ε∗2 (f (a)) is an R/γ ∗ -homomorphism of modules. Proof. By hypotheses M1 /ε∗1 and M2 /ε∗2 are modules. First, we show that F is well defined. If ε∗1 (a) = ε∗1 (b), then there exist x1 , ..., xm+1 ∈ M1

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and u1 , ..., um ∈ UM1 with x1 = a, xm+1 = b such that {xi , xi+1 } ⊆ ui , i = 1, ..., m. Since f is a strong Hv -homomorphism, it follows that f (ui ) ∈ UM2 . Therefore, f (a)ε∗2 f (b) which implies that ε∗2 (f (a)) = ε∗2 (f (b)), and so F (ε∗1 (a)) = F (ε∗1 (b)). Thus, F is well defined. Now, we have F (ε∗1 (a) ⊕ ε∗1 (b)) = F (ε∗1 (a + b)) = ε∗2 (f (a + b)) = ε∗2 (f (a) + f (b)) = ε∗2 (f (a)) ⊕ ε∗2 (f (b) = F (ε∗1 (a)) ⊕ F (ε∗1 (b)) and F (γ ∗ ε∗1 (a)) = F (ε∗1 (ra)) = ε∗2 (f (ra)) = ε∗2 (rf (a)) = γ ∗ F (ε∗1 (a)).



Proposition 5.2. Let (Mi , φij ) be a direct system of Hv -modules over an Hv -ring R indexed by a directed set I. Then, (Mi /ε∗Mi , φi∗ j ) is a direct system of modules over the ring R/γ ∗ , where ∗ ∗ φi∗ j : Mi /εMi → Mj /εMj i ∗ ∗ εMi (ai ) 7→ εMj (φj ai ). ∗ Proof. By Lemma 5.4, (Mi /ε∗Mi , φ∗i j ) is a family of R/γ -modules and i∗ R/γ-homomorphisms. It is clear that φi is the identity for all i ∈ I. Now, for i ≤ j ≤ k, we have j i ∗ ∗ i ∗ (φjk φij )∗ (ε∗Mi (ai )) = φi∗ k (εMi (ai )) = εMk (φk ai )) = εMk (φk φj )ai ) j∗ i∗ ∗ ∗ i = ε∗Mk (φjk (φij ai )) = φj∗ k (εMj (φj ai )) = φk φj (εMi (ai )). j∗ i∗ Therefore, (φjk φij )∗ = φi∗ k = φk φ j .



Proposition 5.3. Let ρ(ai ), ρ(bj ) ∈ X/ρ. We define ρ(ai ) θ ρ(bj ) ⇔ there exists k ≥ i, j such that φik ai εMk φjk bj . Then, θ = εX/ρ . Proof. Suppose that ρ(ai ) εX/ρ ρ(bj ). Then, there exist r1 , ..., rn ∈ R S and t1 , ..., tm ∈ Mi such that {ρ(ai ), ρ(bj )} ⊆ r1 ρ(t1 ) + ... + rm ρ(tm ). Suppose that t1 ∈ Mi1 , ..., tm ∈ Mim and k ≥ i1 , ..., im , i, j. Hence, r1 ρ(t1 ) + ... + rm ρ(tm ) = r1 ρ(φik1 t1 ) + ... + rm ρ(φikm tm ) = {ρ(x) | x ∈ φik1 r1 t1 + ... + φikm rm tm }, and so {ρ(φik ai ), ρ(φjk bj )} ⊆ {ρ(x) | x ∈ φik1 r1 t1 + ... + φikm rm tm }. Now, for n ≥ k, we obtain {φin ai , φjn bj } ⊆ φin1 r1 t1 + ... + φinm rm tm which implies that φin ai εMn φjn bj . Hence, ρ(ai ) θ ρ(bj ). Conversely, if ρ(ai ) θ ρ(bj ), then there exists k ≥ i, j such that

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φik ai εMk φjk bj . So, there exist t1 , ..., tm ∈ Mk ⊆ such that

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S

Mi and r1 , ..., rm ∈ R

{φik ai , φjk bj } ⊆ r1 t1 + ... + rm tm . Thus, {ρ(φik ai ), ρ(φjk bj )} ⊆ r1 ρ(t1 ) + ... + rm ρ(tm ) which implies that ρ(ai ) εX/ρ ρ(bj ). Therefore, θ = εX/ρ .  Theorem 5.7. Let (Mi , φij ) be a direct system of Hv -modules over an Hv ring R indexed by a directed set I, and let ε∗ be the fundamental relation of lim Mi . Then, −→

lim(Mi /ε∗Mi ) ∼ = (lim Mi )/ε∗ . −→

Proof.

5.4

−→

It is straightforward.



M[-] and -[M] Functors

Vaziri, Ghadiri and Davvaz introduced the Hv -module M [A] and determined it’s heart. They studied the connection between equivalence relations ε∗M [A] and ε∗A . Moreover, they introduced M [−] and −[M ] functors. The main reference for this section is [109, 110]. Let f : A → B be a strong Hv -homomorphism of Hv -modules over an Hv -ring R. Then, F : A/ε∗A → B/ε∗B where F (ε∗A (a)) = ε∗B (f (a)) is an R/ γ ∗ -homomorphism of R/γ ∗ -modules. Let R be a weak-commutative Hv ring and H be the set of all Hv -modules and all strong R-homomorphisms. One can show that H is a category. Also, set H* the category of R/γ ∗ modules and R/γ ∗ -homomorphisms. Then we have the following theorem. Theorem 5.8. Let T : H → H*, defined by T (M1 ) = M1 /ε∗1 and when f : M1 → M2 is a strong Hv -homomorphism T (f ) : M1 /ε∗1 → M2 /ε∗2 , ε∗1 (a) 7→ ε∗2 (f (a)) where ε∗1 and ε∗2 are fundamental relations on M1 and M2 , respectively. Then, T is a functor, and is called natural functor. Proof.

It is easy to see that T is well defined. We have the following:

(1) If M1 ∈ obj H, then M1 /ε∗1 ∈ obj H*;

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(2) If f : M1 → M2 is a strong Hv -homomorphism, then by Lemma 5.4, F : M1 /ε∗1 → M2 /ε∗2 is an R/γ ∗ -homomorphism of modules. f

g

(3) Suppose that ε∗3 is the fundamental relation on M3 . If M1 → M2 → M3 is an exact sequence of Hv -modules and strong Hv -homomorphisms in H, then T (gf )(ε∗1 (a)) = ε∗3 (gf (a)) = T (g)ε∗2 (f (a)) = T (g)T (f )(ε∗1 (a)), so T (gf ) = T (g)T (f ); (4) For every M1 ∈ obj H, we have T (1M1 )(ε∗1 (a)) = ε∗1 (1M1 (a)) = ε∗1 (a), so T (1M1 ) = 1T (M1 ) . Therefore, T is a functor.



Now, we want to introduce M [−] and −[M ] functors and investigate some related concepts. Suppose that M and N are two Hv -modules and M [N ] is the set of all functions on M with values in N . First we equip M [N ] to appropriate hyperoperations to be an Hv -module. Then, we introduce the functors M [−] and −[M ] and investigate some related concepts. Throughout this paper, the hyperoperations in M, N and M [N ] will be shown with same symbols. Theorem 5.9. The M [N ] with the following hyperoperations is an Hv module. f + g = {h ∈ M [N ] | h(x) ∈ f (x) + g(x), ∀x ∈ M }, r · f = {k ∈ M [N ] | k(x) ∈ r · f (x), ∀x ∈ M }. Proof. The hyperoperations + and · in M [N ] are well-define, for + and · in N are well-define. Let f, g, h ∈ M [N ]. We have S (f +g)+h = {l ∈ M [N ] | l(x) ∈ f (x)+g(x), ∀x ∈ M }+h = l∈f +g l+h = {L ∈ M [N ] | L(x) ∈ l(x)+h(x), ∀x ∈ M, l(x) ∈ f (x)+g(x)} and S f +(g+h) = f +{k ∈ M [N ] | k(x) ∈ g(x)+h(x), ∀x ∈ M } = k∈g+h f +k = {K ∈ M [N ] | K(x) ∈ f (x)+k(x), ∀x ∈ M, k(x) ∈ g(x)+h(x)}. Since N is an Hv -group, for all x ∈ M there exists nx ∈ [(f (x) + g(x)) + h(x)] ∩ [f (x) + (g(x) + h(x))]. We define u ∈ M [N ] by u(x) = nx , according

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to the choice axiom. Then, u ∈ [(f + g) + h] ∩ [f + (g + h)] and associativity is satisfied. For the reproduction axiom let f, g ∈ M [A]. Then, for all x ∈ M , f (x), g(x) ∈ N and so, there exists yx ∈ N such that f (x) ∈ g(x) + yx . We 0 define h ∈ M [N ] by h(x) = yx , then f ∈ g + h. Similarly, there exists h ∈ 0 M [N ] such that f ∈ h +g. Since N is an Hv -module, the conditions of Hv modules satisfy in M [N ]. We check only one of the Hv -module conditions. Let r1 , r2 ∈ R and f ∈ M [N ]. Since N is an Hv -module, it follows that for every x ∈ M there exists nx ∈ [(r1 + r2 )f (x)] ∩ [r1 f (x) + r2 f (x)]. We define h ∈ M [N ] by h(x) = nx . Obviously, h ∈ [(r1 + r2 )f ] ∩ [(r1 f + r2 f )] 6= ∅.  Lemma 5.5. Let f : A → B be a strong Hv -homomorphism and M be an Hv -module. Then, −

−

(1) The map f : M [A] → M [B] defined by f (φ) = f ◦ φ is a strong Hv homomorphism. −

−

(2) The map f : B[M ] → A[M ] defined by f (φ) = φ ◦ f is a strong Hv homomorphism. (1) Let φ1 , φ2 ∈ M [A]. Then,

Proof. −

f(φ1 +φ2 ) = {f ◦ h | h ∈ M [A], h(m) ∈ φ1 (m)+φ2 (m), ∀m ∈ M }, −

−

0

0

f(φ1 )+ f(φ2 ) = f ◦ φ1 +f ◦ φ2 = {h ∈ M [B] | h (m) ∈ f ◦ φ1 (m)+f ◦ φ2 (m)}. −

Suppose that f ◦h ∈f (φ1 +φ2 ), where h ∈ M [A] and h(m) ∈ φ1 (m)+φ2 (m) for every m ∈ M . Then, f (h(m)) ∈ f (φ1 (m) + φ2 (m)) = f (φ1 (m)) + −

−

−

f (φ2 (m)). Therefore, f (φ1 + φ2 ) ⊆f (φ1 )+ f (φ2 ). −

0

−

Conversely, suppose that h ∈f (φ1 )+ f (φ2 ). We need to find an 0 h ∈ M [A] such that h = f oh and h(m) ∈ φ1 (m) + φ2 (m). By hypothesis for m ∈ M, we have 0

h (m) = bm ∈ f ◦ φ1 (m) + f ◦ φ2 (m) = f (φ1 (m) + φ2 (m)) ⊆ Im(f ). So, bm ∈ f (φ1 (m) + φ2 (m)). Now, according to the choice axiom we can select a ∈ f −1 (bm ) such that a ∈ φ1 (m) + φ2 (m) and define h(m) = a. −

−

Similarly, one can show that f (rφ) = r f (φ). (2) Let φ1 , φ2 ∈ B[M ]. Then, −

f(φ1 +φ2 ) = {h ◦ f | h ∈ B[M ], h(b) ∈ φ1 (b)+φ2 (b)}, −

−

0

0

f(φ1 )+ f(φ2 ) = φ1 ◦ f +φ2 ◦ f = {h ∈ A[M ] | h (a) ∈ φ1 ◦ f (a)+φ2 ◦ f (a)}.

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Suppose that h ◦ f ∈f (φ1 + φ2 ), where h ∈ B[M ] and h(b) ∈ φ1 (b) + φ2 (b) for every b ∈ B. Since Im(f ) ⊆ B, we have h(f (a)) ∈ φ1 (f (a)) + φ2 (f (a)) −

−

−

for every a ∈ A. Therefore, f (φ1 + φ2 ) ⊆f (φ1 )+ f (φ2 ). 0

−

−

Conversely, suppose that h ∈f (φ1 )+ f (φ2 ). We need to find an h ∈ 0 B[M ] such that h = h ◦ f and h(b) ∈ φ1 (b) + φ2 (b). For every b ∈ Im(f ) ⊆ 0 B we define h(b) = h (a), where f (a) = b and for every b ∈ B\Im(f ) according to the choice axiom we select an mb in φ1 (b) + φ2 (b) ⊆ M and define h(b) = mb . Then, h satisfies the requirement conditions. −

−

Similarly, one can show that f (rφ) = r f (φ).



Lemma 5.6. Let M be an Hv -module and f : A → B be a morphism in the category H. Then, (1) M [−]

:H→H define by M [−](A)

M [A] and M [−](f )

=

f : M [A] → M [B], where f (φ) = f ◦ φ, is a covariant functor. (2) −[M ] :H→H define by −[M ](A) = A[M ] and −[M ](f )

=

−

=

−

−

−

f : B[M ] → A[M ], where f (φ) = φ ◦ f , is a contravariant functor. Proof. (1) By Theorem 5.9 if A is an Hv -module, then M [−](A) = M [A] is an Hv -module. By Lemma 5.5 if f : A → B is a strong Hv −

homomorphism, then M [−](f ) =f is a strong Hv -homomorphism. Now, f

g

let A → B → C be a strong Hv -homomorphism in H. Then, M [−](g◦f )(φ) = g◦f ◦φ = g(f ◦φ) = M [−](g)(f ◦φ) = M [−](g)◦M [−](f )(φ) and for every A ∈ objH we have M [−](1A )(φ) = 1A ◦ φ = φ. Then, M [−](1A ) = 1M [−](A) and so, M [−] is a covariant functor. (2) By Theorem 5.9 if A is an Hv -module, then −[M ](A) = A[M ] is an Hv -module. By Lemma 5.5 if f : A → B is a strong Hv -homomorphism, −

f

g

then −[M ](f ) =f is a strong Hv -homomorphism. Now, let A → B → C be a strong Hv -homomorphism in H. Then, −[M ](g◦f )(φ) = φ◦g◦f = (φ◦g)f = −[M ](f )(φ◦g) = −[M ](f )◦−[M ](g)(φ), and for every A ∈ objH we have −[M ](1A )(φ) = φ ◦ 1A = φ. Then, −[M ](1A ) = 1−[M ](A) and so, −[M ] is a contravariant functor. 

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Lemma 5.7. Let A

f

k

h

 A1

/B

g

 / B1

be a commutative diagram of Hv -modules and strong Hv -homomorphisms. Then, the following diagrams are commutative. A/ε∗A

F

/ B/ε∗ B

H

 A1 /ε∗A1

K

G

 / B1 /ε∗ B1

−

M [A]

f

/ M [B]

−

−

h

k

 M [A1 ]

−

 / M [B1 ]

g

Proof. We have T (A) = A/ε∗A and T (f : A → B) = F : A/ε∗ → B/ε∗B , where F (ε∗A (a)) = ε∗B (f (a)). Therefore, K ◦ F = T (k) ◦ T (f ) = T (k ◦ f ) = T (g ◦ h) = T (g) ◦ T (h) = G ◦ H. −

We have M [−](A) = M [A], M [−](f : A → B) =f : M [A] → M [B] −

where f (φ) = f ◦ φ. Therefore, −

−

k ◦ f = M [−](k) ◦ M [−](f ) = M [−](k ◦ f ) = M [−](g ◦ h) − − = M [−](g) ◦ M [−](h) = g ◦ h .



We know that the combination of two covariant functors is a covariant functor. Therefore, the map S = T ◦ M [−]:H→H* is a covariant functor, where − S(A) = M [A]/ε∗M [A] and S(f : A → B) =F : M [A]/ε∗M [A] → M [B]/ε∗M [B] , −

where F (ε∗M [A] (φ)) = ε∗M [B] (f ◦ φ).

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Lemma 5.8. For every A ∈ objH, τA : T (A) → S(A) defined by τA (ε∗A (a)) = ε∗M [A] (φa ) is a R/γ ∗ -homomorphism, where φa : M → A defined by φa (m) = a for every m ∈ M . Then, the family τ = (τA : T (A) → S(A))A∈objH is a natural transformation from T to S. Proof.

We have τA (ε∗A (a) ⊕ ε∗A (b)) = τA (ε∗A (a + b)) = ε∗M [A] (φt ),

where t ∈ a + b. On the other hand, we obtain τA (ε∗A (a))⊕τA (ε∗A (b)) = ε∗M [A] (φa )⊕ε∗M [A] (φb ) = ε∗M [A] (φa +φb ) = ε∗M [A] ({φ ∈ M [A] | φ(m) ∈ φa (m)+φb (m), ∀m ∈ M }) = ε∗M [A] ({φ ∈ M [A] | φ(m) ∈ a+b, ∀m ∈ M }) = ε∗M [A] (φt ), where t ∈ a + b. Therefore, τA (ε∗A (a) ⊕ ε∗A (b)) = τA (ε∗A (a)) ⊕ τA (ε∗A (b)). Similarly, we have τA (γ ∗ (r) ε∗A (a)) = τA (ε∗A (d)), for some d ∈ γ ∗ (r) · ε∗A (a) = ε∗M [A] (φd ), for some d ∈ r · a and γ ∗ (r) τA (ε∗A (a)) = γ ∗ (r) ε∗M [A] (φa ) = ε∗M [A] (h) for some h ∈ r · φa = ε∗M [A] (h), where for every m ∈ M , h(m) ∈ r · φa (m) = r · a. Therefore, τA (γ ∗ (r) ε∗A (a)) = γ ∗ (r) τA (ε∗A (a)). Now, let f : A → B be a morphism in H and consider the following diagram. T (A)

τA

T (f )

 T (B)

/ S(A)

S(f )

τB

 / S(B)

We have S(f ) ◦ τA (ε∗A (a)) = S(f )(ε∗M [A] (φa )) = ε∗M [B] (f ◦ φa ), τB ◦ T (f )(ε∗A (a)) = τB (ε∗B (f (a))) = ε∗M [B] (φf (a) ). Obviously, f ◦ φa = φf (a) and so, S(f ) ◦ τA = τB ◦ T (f ) and τ : T → S is a natural transformation. 

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Lemma 5.9. Let H1 and H2 be two Hv -modules. Then, H1 × H2 is a product object in H category. Proof.

The proof is straightforward.



Notice that Lemma 5.9 can be generalized to the cartesian product of n arbitrary Hv -modules. Theorem 5.10. Let M be an Hv -module. Then, M [H1 × H2 ] ∼ = M [H1 ] × M [H2 ]. Proof. It is easy to see that the map φ : M [H1 ] × M [H2 ] → M [H1 × H2 ] defined by φ(f1 , f2 ) = f : M → H1 × H2 , where f (m) = (f1 (m), f2 (m)) is well-defined. Now, we have φ((f1 , g1 )+(f2 , g2 )) = φ({(f, g) | f ∈ f1 +f2 , g ∈ g1 +g2 }) = {h | h(m) = (f (m), g(m)), f (m) ∈ f1 (m)+f2 (m), g(m) ∈ g1 (m)+g2 (m)}. On the other hand, we have φ((f1 , g1 )) = h ∈ M [H1 × H2 ] such that h(m) = (f1 (m), g1 (m)), φ((f2 , g2 )) = k ∈ M [H1 × H2 ] such that k(m) = (f2 (m), g2 (m)). And h+k = {l | l(m) ∈ h(m)+k(m) = (f1 (m), g1 (m))+(f2 (m), g2 (m))} = {l | l(m) = (f (m), g(m)), f (m) ∈ f1 (m)+f2 (m), g(m) ∈ g1 (m)+g2 (m)}. Therefore, φ((f1 , g1 ) + (f2 , g2 )) = φ((f1 , g1 )) + φ((f2 , g2 )). Similarly, one can show that φ(r(f, g)) = rφ((f, g)). Now, let f ∈ M [H1 × H2 ] where f (m) = (h1m , h2m ). We define f1 ∈ M [H1 ] by f (m) = h1m and f2 ∈ M [H2 ] by f (m) = h2m . Obviously, φ((f1 , f2 )) = f . Finally, suppose that φ((f1 , f2 )) = φ((g1 , g2 )). Then, for every m ∈ M , we obtain (f1 (m), f2 (m)) = (g1 (m), g2 (m)) and so, (f1 , f2 ) = (g1 , g2 ).  Notice that in finite mode in Theorem 5.10 we have |M [H1 ] × M [H2 ]| = |M [H1 ]| × |M [H2 ]| = |H1 ||M | × |H2 ||M | |M | = |H1 × H2 | = |M [H1 × H2 ]|. So, it is enough to show that φ is one to one or onto. Corollary 5.4. Let M , H1 , H2 , . . . , Hn be Hv -modules. Then, M [H1 × H2 × H3 × · · · × Hn ] ∼ = M [H1 ] × M [H2 ] × M [H3 ] × · · · × M [Hn ].

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5.5

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A Walk Through Weak Hyperstructures: Hv -Structures

Five short lemma and snake lemma in Hv -modules

In this section, we investigate the five short lemma and snake lemma in Hv -modules. The main reference is [109, 110]. Lemma 5.10. If ε∗M [A] (f ) = ε∗M [A] (g), then ε∗A (f (m)) = ε∗A (g(m)), for every m ∈ M ; i.e. if for some m ∈ M , ε∗A (f (m)) 6= ε∗A (g(m)) then ε∗M [A] (f ) 6= ε∗M [A] (g). Proof. Suppose that f ε∗M [A] g. Then, there exist f0 = f, f1 , · · · , fn = g in M [A] such that fi εM [A] fi+1 for i = 0, 1, . . . , n − 1. So, {fi , fi+1 } ⊆ n ni 0 ij lQ ijk P P 0 0 gij , for i = 0, 1, . . . , n − 1, where gij = gij or gij = ( rijkl )gij for j=1

k=1 l=1

gij ∈ M [A] and rijkl ∈ R. Now, since ni X 0 0 0 0 gij = {h ∈ M [N ] | h(m) ∈ gi1 (m) + gi2 (m) + · · · + gini (m), ∀m ∈ M }, j=1

we have {fi (m), fi+1 (m)} ⊆

ni P j=1

0

gij (m) for every m ∈ M and so, there exist

a0 = f0 (m) = f (m), a1 = f1 (m), . . . , an = fn (m) = g(m) ∈ A such that ai εA ai+1 , for i = 0, 1, . . . , n − 1. Therefore, for every m ∈ M, we have f (m)  ε∗A g(m). In the following example we show that the converse of Lemma 5.10 is not true in general. Example 5.2. Consider f, g ∈ M [M1 × M2 ] as in Example 5.1 and define f (a) = (2,¯ 2), f (b) = (1,¯0) and g(a) = (0,¯1), g(b) = (1,¯2). By Example 5.1 we have ε∗M1 ×M2 (f (a)) = ε∗M1 ×M2 (g(a)) and ε∗M1 ×M2 (f (b)) = ε∗M1 ×M2 (g(b)). Since for every r ∈ R and every m1 ∈ M1 , rm1 = {0} and 0 0 on the other hand, for every two elements m2 and m2 of M2 , m2 ∗M2 m2 is singleton, it follows ε∗M [M1 ×M2 ] (f ) 6= ε∗M [M1 ×M2 ] (g). In the following lemma, we determine the heart of M [A]. Lemma 5.11. Let M and A be two Hv -modules. Then, ωM [A] = M [ωA ]. Proof.

Suppose that f ∈ ωM [A] . Then, for every g ∈ M [A] we have   ε∗M [A] (g) = ε∗M [A] (f ) ⊕ ε∗M [A] (g) = ε∗M [A] (f + g) .

Now, by Lemma 5.10 for every m ∈ M we obtain ε∗A ((f + g)(m)) = ε∗A (g(m)).

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But for every m ∈ M , we have (f + g)(m) = {l(m) | l ∈ f + g} = f (m) + g(m). So, ε∗A ((f + g)(m)) = ε∗A (f (m) + g(m)) = ε∗A (f (m)) ⊕ ε∗A (g(m)) = ε∗A (g(m)). Therefore, for every m ∈ M , we obtain ε∗A (f (m)) ∈ ωA and so, f ∈ M [ωA ]. Conversely, suppose that f ∈ M [ωA ]. Then, for every g ∈ M [A] and all m ∈ M , we have ε∗A (f (m) + g(m)) = ε∗A (f (m)) ⊕ ε∗A (g(m)) = ε∗A (g(m)). So, for every g ∈ M [A] and all m ∈ M , we have f (m) + g(m) ∈ ε∗A (g(m)) and we obtain ε∗M [A] (f )⊕ε∗M [A] (g) = ε∗M [A] (f+g) = ε∗M [A] ({l | l(m) ∈ f (m)+g(m)}) = ε∗M [A] (g) and consequently f ∈ ωM [A] .



In the following, we want to investigate the exactness of −[M ] and M [−] functors. f g Let A → B → C be an exact sequence. Then, for every a ∈ A, we have w f (a) ∈ Im(f ) = Ker(g) and so, ε∗B (f (a)) = ε∗B (b) for some b ∈ Ker(g). Now, we obtain ε∗C (g(f (a))) = G(ε∗B (f (a))) = G(ε∗B (b)) = ε∗C (g(b)) = ωC . Therefore, for every a ∈ A we have g(f (a)) ∈ ωC . f

g

Now, by considering −[M ] functor on the exact sequence A → B → C, we obtain −

g

−

f

C[M ] → B[M ] → A[M ]. We want to check the exactness of this sequence. We have −

−

Im( g ) = { g (φ) | φ ∈ C[M ]} = {φ ◦ g | φ ∈ C[M ]}, −

−

Ker(f ) = {ψ ∈ B[M ] | f (ψ) = ψ ◦ f ∈ ωA[M ] = A[ωM ]}. Let φ be a function in C[M ] such that φ(ωC ) ∩ ωM = ∅ (note that it is necessary to ωM 6= M ). Then, for every a ∈ A since g ◦ f (a) ∈ ωC and φ(ωC ) ∩ ωM = ∅, we obtain ε∗M (φ(g(f (a)))) 6= ωM . On the other hand, for −

every ψ ∈ Ker(f ) and every a ∈ A, ε∗M (ψ(f (a))) = ωM . So, by Lemma −

−

5.10 for φ ◦ g ∈ Im( g ) there is no member of Ker( g ) such that its class be equal to class of φ ◦ g. Therefore, in general the −[M ] functor is not exact. The same discussion is establish for the M [−] functor.

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Example 5.3. Consider the Hv -modules M , M1 and M2 as Example 1 f i and the sequence M → M1 → M1 where f (a) = 0, f (b) = 2 and i be f

i

identity. It is easy to see that the sequence M → M1 → M1 is exact. But the sequence −

−

f

i

M1 [M1 × M2 ] → M1 [M1 × M2 ] → M [M1 × M2 ] is not exact, because for φ ∈ M1 [M1 × M2 ] defined by φ(0) = (1,¯1), φ(1) = −

(2,¯ 1) and φ(2) = (1,¯ 2) there is no member of Ker ( f ) such that its class be equal to class of φ. In the following theorem we show that if the converse of Lemma 5.10 is establish, then the functors M [−] and −[M ] are exact. f

g

Theorem 5.11. Let A → B → C be an exact sequence of Hv -modules and strong Hv -homomorphisms. If the converse of Lemma 5.10 is establish, then the sequences −

−

g

f

−

−

C[M ] → B[M ] → A[M ]

(5.1)

g

f

M [A] → M [B] → M [C]

(5.2)

are exact sequences. −

We prove (2). The proof of (1) is similar. Suppose that h ∈ Im(f ).

Proof.

−

Then, there exists φ ∈ M [A] such that h =f (φ) = f ◦ φ ∈ M [B]. For every m ∈ M , f ◦ φ(m) ∈ Im(f ) and so, there exists bm ∈ Ker(g) such that ε∗B (f ◦ φ(m)) = ε∗B (bm ). Now, we define k ∈ M [B] by k(m) = bm . Since −

−

g (k) = gok ∈ M [ωC ] = ωM [C] , we obtain k ∈ Ker g . Finally, by the converse of Lemma 5.10 we have ε∗M [B] (h) = ε∗M [B] (k). −

−

Conversely, let k ∈ Ker( g ), then g (k) = g ◦ k ∈ ωM [C] = M [ωC ]. So, for all m ∈ M , g ◦ k(m) ∈ ωC and k(m) ∈ Ker(g). Then, there exists bm = f (a) ∈ Im(f ) for some a ∈ A such that ε∗B (bm ) = ε∗B (k(m)). We −

−

define ψ ∈ M [A] by ψ(m) = a and set φ = f ◦ ψ =f (ψ) ∈ Im(f ). Now, by the converse of Lemma 5.10 we obtain ε∗M [B] (k) = ε∗M [B] (φ).  Lemma 5.12. Let A, B and C are Hv -modules. Then, i

f

(1) ωA → A → B is exact if and only if f is weak-monic.

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j

(2) B → C → ωC is exact if and only if g is weak-epic. i

f

g

j

(3) ωA → A → B → C → ωC is exact if and only if f is weak-monic, g is w weak-epic and Im(f ) = Ker(g). Proof. (1) Suppose that the given sequence is exact. It is enough to show that Ker(f ) = ωA . Always, we have ωA ⊆ Ker(f ). On the other hand, if a ∈ Ker(f ), then there exists a1 ∈ Im(i) = ωA such that ε∗A (a) = ε∗A (a1 ) = ωA and so, a ∈ ωA . Therefore, Ker(f ) = ωA and f is weak-monic. Conversely, suppose that f is weak-monic. Then, Ker(f ) = ωA = Im(i) w and consequently Ker(f ) = Im(i). w (2) Suppose that the given sequence is exact. Then, Im(g) = Ker(j) and so, for every c ∈ Ker(j)(= C since ωωC = ωC ) there exists b ∈ B such that ε∗C (g(b)) = ε∗C (c). Therefore, g is weak-epic. Conversely, suppose that g is weak-epic. Then, for every c ∈ C(= Ker(j)) there exists b ∈ B such that ε∗C (g(b)) = ε∗C (c). On the other hand, for all g(b) ∈ Im(g) ⊆ C there exist some t ∈ B such that ε∗C (g(b)) = ε∗C (g(t)), where g(t) ∈ C = Ker(j) and consequently w Im(g) = Ker(j). (3) It follows from (1), (2) and the definition of exactness.  Lemma 5.13. Let f : A → B be a strong Hv -homomorphism of Hv modules. Then, f is weak-epic if and only if F is onto. Moreover f is weak-monic if and only if F is one to one. Finally, f is weak-isomorphism if and only if F is isomorphism. Proof. Suppose that f is weak-epic and ε∗B (b) ∈ B/ε∗B . Since f is weakepic, there exists a ∈ A such that ε∗B (f (a)) = ε∗B (b). But ε∗B (f (a)) = F (ε∗A (a)). So, F (ε∗A (a)) = ε∗B (b) and consequently F is onto. Conversely, let F is onto. Then, for every b ∈ B there exists ε∗A (a) ∈ A/ ∗ εA such that F (ε∗A (a)) = ε∗B (b). But F (ε∗A (a)) = ε∗B (f (a)). So, there exists a ∈ A such that ε∗B (f (a)) = ε∗B (b) and consequently f is weak-epic. The second part is proved in [41]. The third part is a obvious result.  i

f

g

j

Theorem 5.12. Let ωA → A → B → C → ωC be an exact sequence of Hv -modules and strong Hv -homomorphisms over an Hv -ring R. Then, I

F

G

J

0 = ωA /ε∗ωA → A/ε∗A → B/ε∗B → C/ε∗C → ωc /ε∗ωc = 0 is an exact sequence of R/γ ∗ -homomorphisms and R/γ ∗ -modules.

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It follows from Lemma 5.12, Lemma 5.13 and Theorem 4.8 of [41] f

g

F

G

that say if A → B → C is an exact sequence, then A/ε∗A → B/ε∗B → C/ε∗C is an exact sequence.  Theorem 5.13. (Five short lemma in Hv -modules) Let f

/A

ωA

h

g

k

 / A1

ωA1

/B

f1

 / B1

/C

/ ωC

l

g1

 / C1

/ ωC 1

be a commutative diagram of Hv -modules and Hv -homomorphisms over an Hv -ring R with both rows exact. Then, (1) If h and l are weak-monic, then k is weak-monic. (2) If h and l are weak-epic, then k is weak-epic. (3) If h and l are weak-isomorphism, then k is weak-isomorphism. Proof. (1) By Lemma 5.7 and Theorem 5.12 the following diagram of R/γ ∗ -modules and R/γ ∗ -homomorphisms is commutative with both rows exact 0 = ωA /ε∗ωA

/ A/ε∗A

F

H

0 = ωA1 /ε∗ωA1

/ B/ε∗B

G

K



/ A1 /ε∗A 1

/ C/ε∗ω = 0 C

L



F1

/ C/ε∗C

/ B1 /ε∗B 1

G1

 / C1 /ε∗C 1

/ ωC1 /ε∗ω = 0. C1

By Lemma 5.13, H and L are one to one R/γ ∗ -homomorphisms. Then, by five short lemma in modules K is one to one R/γ ∗ -homomorphism. So, by Lemma 5.13, k is weak-monic R-homomorphism. Alternative Proof. It is enough to show that Ker(k) = ωB . Always, we have ωB ⊆ Ker(k). On the other hand, suppose that b ∈ Ker(k). Then, k(b) ∈ ωB1 and so, g1 (k(b)) ∈ g1 (ωB1 ). Since g1 (ωB1 ) ⊆ ωC1 , we have g1 (k(b)) ∈ ωC1 . Since g1 ◦ k = l ◦ g and l is weak monic, we obtain w g(b) ∈ Ker(l) = ωC . Then, b ∈ Ker(g) = Im(f ) and consequently ε∗B (b) = ε∗B (f (a)) for some a ∈ A.

(5.3)

Since k is strong Hv -homomorphism, we have ε∗B1 (k(b)) = ε∗B1 (k(f (a))). Since k ◦ f = f1 ◦ h and b ∈ Ker(k), we obtain ε∗B1 (k(b)) = ε∗B1 (f1 (h(a))) = ωB1 . So, f1 (h(a)) ∈ ωB1 . Since f1 is weak-monic we obtain h(a) ∈ ωA1 and since h is weak-monic it follow that a ∈ ωA . So, f (a) ∈ f (ωA ) ⊆ ωB and

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by Equation (5.3) we obtain ε∗B (b) = ε∗B (f (a)) = ωB . Therefore, b ∈ ωB and prove is complete. (2) It is similar to (1). (3) It follows from (1) and (2).  Now, we prove the snake lemma and close this section. Theorem 5.14. (Snake lemma in Hv -modules) Let A

f

h

ωA1

 / A1

/B

g

k

f1

 / B1

/C

/ ωC

l

g1

 / C1

be a commutative diagram of Hv -modules and strong homomorphisms over an Hv -ring R with both exact rows. If l is weak-monic, then there exists an exact sequence as follows: α

β

Ker(h) → Ker(k) → Ker(l). Proof.

First we want to define α and β. We have Ker(h) = {a ∈ A | h(a) ∈ ωA1 }, Ker(k) = {b ∈ B | k(b) ∈ ωB1 }, Ker(l) = {c ∈ C | l(c) ∈ ωC1 }.

Now, for a ∈ Ker(h), f1 ◦ h(a) ∈ f1 (ωA1 ) ⊆ ωB1 . Since f1 ◦ h(a) = k ◦ f (a), we obtain f (a) ∈ Ker(k). Also, for b ∈ Ker(k), g1 ◦ k(b) ∈ g1 (ωB1 ) ⊆ ωC1 . Since g1 ◦ k(b) = l ◦ g(b), we obtain g(b) ∈ Ker(l). We define α by α(a) = f (a), for every a ∈ Ker(h) and β by β(b) = g(b), for every b ∈ Ker(k). Since the Ker(h), Ker(k) and Ker(l) are Hv submodules of A, B and C respectively and f , g are strong homomorphisms, it follows that α and β are strong homomorphisms. w We show that Im(α) = Ker(β). Let x ∈ Im(α) then, x = f (a) for some a ∈ Ker(h)(⊆ A). The first row is exact, so there exists b ∈ Ker(g) such that ε∗B (f (a)) = ε∗B (b), where g(b) ∈ ωC . Since l is weak-monic we have ker(l) = ωC ; but ωker(l) = ωωC = ωC and so β(b) = g(b) ∈ ωKer(l) . It is enough to show b ∈ Ker(k). Since ε∗B (f (a)) = ε∗B (b) and f (a) ∈ Im(α)(⊆ Ker(k)), we obtain b ∈ Ker(k). Conversely, let b ∈ Ker(β), then β(b) = g(b) ∈ ωKer(l) = ωC and b ∈ Ker(g). Since first row is exact, there exists f (a) ∈ Im(f ) for some a ∈ A such that ε∗B (b) = ε∗B (f (a)). It is enough to show a ∈ Ker(h). Since k is

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strong and diagram is commutative, we obtain ε∗B1 (k(b)) = ε∗B1 (k(f (a))) = ε∗B1 (f1 (h(a))). Since b ∈ Ker(β)(⊆ Ker(k)), it follows f1 (h(a)) ∈ ωB1 and h(a) ∈ Ker(f1 ). Since f1 is weak-monic (by exactness) we have Ker(f1 ) = ωA1 . Therefore, a ∈ Ker(h).  5.6

Shanuel’s lemma in Hv -modules

Vaziri and Ghadiri studied the concepts of star homomorphism, (star) isomorph sequences and star projective Hv -modules in order to find a generalization of Shanuel’s lemma. In this section, we present these concepts. The main reference is [109, 110]. Definition 5.7. A mapping f : M1 → M2 of Hv -modules M1 and M2 over an Hv -ring R is called a star homomorphism if for every x, y ∈ M1 and every r ∈ R: ε∗M2 (f (x + y)) = ε∗M2 (f (x) + f (y)) and ε∗M2 (f (rx)) = ε∗M2 (rf (x)); w w i.e. f (x + y) = f (x) + f (y) and f (rx) = rf (x). Every strong homomorphism is a star homomorphism but the converse is not true necessarily by the following example. Example 5.4. Let R be an Hv -ring. Consider the following Hv -modules on R: (1) M1 = {a, b} together with the following hyperoperations: ∗M1 a b

a b a b b a

and ·M1 : R × M1 → P ∗ (M1 ), (r,m1 )7→{a}

(2) M2 = {0, 1, 2} together with the following hyperoperations: ∗M2 0 1 2

0 1 0 1 1 {0, 2} 2 1

2 2 1 0

and ·M2 : R × M2 → P ∗ (M2 ) . (r,m2 )7→{0}

We obtain M2 /ε∗M2 = {ε∗M2 (0) = {0, 2}, ε∗M2 (1) = {1}}. If f : M1 → M2 defined by f (a) = 0 and f (b) = 1 then f is a star homomorphism but not strong homomorphism because f (b ∗M1 b) 6= f (b) ∗M2 f (b). Definition 5.8. Two mappings f, g : M → N on Hv modules are called weak equal if for every m ∈ M ; ε∗N (f (m)) = ε∗N (g(m)) and denote by

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w

f = g. The following diagram of Hv -modules and strong homomorphisms w is called star commutative if g ◦ f = h. A f

B

h

 /C

 g

Also, it is said commutative if for every a ∈ A, g ◦ f (a) = h(a). Definition 5.9. The sequences f

g

ωA → A → B → C → ωC and 0

f

0

0

g

0

0

ωA0 → A → B → C → ωC 0 are called isomorph (star isomorph) if there exist weak-isomorphisms (star 0 0 0 homomorphisms) α : A → A , β : B → B and γ : C → C such that the following diagram is commutative (star commutative): f

/A

ωA

/B

α

f

 / B0

0

/C

/ ωC

γ

β

 / A0

ωA0

g

g

0

 / C0

/ ω 0. C

Definition 5.10. An Hv -module P is called star projective if for every diagram of strong homomorphisms and Hv -modules as follows P f

M

g

 /N

/ ωN

that it’s row is exact, there exists a strong homomorphism φ : P → M such w that g ◦ φ = f . According to [41] for every strong homomorphism f : M → N there is the R/γ ∗ -homomorphism F : M/ε∗M → N/ε∗N of R/γ ∗ -modules defined by F (ε∗M (m)) = ε∗N (f (m)). Lemma 5.14. Let f : A → B be a strong homomorphism of Hv -modules. Then, f is weak-epic (weak-monic) if and only if F is onto (one to one). So, f is weak-isomorphism if and only if F is isomorphism.

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Proof. Suppose that f is weak-epic and ε∗B (b) ∈ B/ε∗B . Since f is weakepic, there exists a ∈ A such that ε∗B (f (a)) = ε∗B (b). But ε∗B (f (a)) = F (ε∗A (a)). So, F (ε∗A (a)) = ε∗B (b) and consequently F is onto. Conversely, let F is onto. Then, for every b ∈ B there exists ε∗A (a) ∈ A/ ε∗A such that F (ε∗A (a)) = ε∗B (b). But F (ε∗A (a)) = ε∗B (f (a)). So, there exists a ∈ A such that ε∗B (f (a)) = ε∗B (b) and consequently f is weak-epic. The proof of the remaining part is straightforward.  Theorem 5.15. (Shanuel’s lemma in Hv -modules) Let P1 and P2 are two star projective Hv -modules. Then the following exact sequences are star isomorph. ωK

/K

ωL

/L

f

f1

/ P1 / P2

g

g1

/M

/ ωM ,

(5.4)

/M

/ ωM .

(5.5)

Proof. Let γ : M → M be identity on M . Since P1 is a star projective Hv -module, there exists a strong homomorphism β : P1 → P2 such that for every p ∈ P1 ; ε∗M (g1 ◦ β(p)) = ε∗M (g(p)). Now, for every k ∈ K; f (k) ∈ P1 and then by exactness of sequence (5.4) we have β ◦ f (k) ∈ Ker(g1 ) and so by exactness of sequence (5.5) there exists lk ∈ L such that ε∗P2 (β(f (k))) = ε∗P2 (f1 (lk )). We define α : K → L by α(k) = lk . Suppose k1 , k2 ∈ K and r ∈ R, we have: ε∗P2 (β ◦ f (k1 + k2 )) = ε∗P2 (β(f (k1 )) + β(f (k2 ))) = ε∗P2 (βf (k1 )) ⊕ ε∗P2 (β(f (k2 ))) = ε∗P2 (f1 (lk1 )) ⊕ ε∗P2 (f1 (lk2 )) = ε∗P2 (f1 (lk1 ) + f1 (lk2 )) = ε∗P2 (f1 (lk1 + lk2 )) = ε∗P2 (f1 (α(k1 ) + α(k2 )) = F1 (ε∗L (α(k1 ) + α(k2 ))), on the other hand ε∗P2 (β ◦ f (k1 + k2 )) = {ε∗P2 (β(f (t)))| t ∈ k1 + k2 } = {ε∗P2 (f1 (lt ))| t ∈ k1 + k2 ; ε∗P2 (β(f (t))) = ε∗P2 (f1 (lt ))} = {ε∗P2 (f1 (α(t)))| t ∈ k1 + k2 } = ε∗P2 (f1 (α(k1 + k2 ))) = F1 (ε∗L (α(k1 + k2 ))). Thus F1 (ε∗L (α(k1 + k2 ))) = F1 (ε∗L (α(k1 ) + α(k2 ))). Now by Lemma 5.14, F1 is one to one and ε∗L (α(k1 + k2 )) = ε∗L (α(k1 ) + α(k2 )).

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Also, ε∗P2 (β ◦ f (rk1 )) = ε∗P2 (rβ(f (k1 )) = γ ∗ (r) ε∗P2 (β(f (k1 )) = γ ∗ (r) ε∗P2 (f1 (lk1 )) = ε∗P2 (rf1 (lk1 )) = ε∗P2 (rf1 (α(k1 ))) = ε∗P2 (f1 (rα(k1 ))) = F1 (ε∗L (rα(k1 )), on the other hand ε∗P2 (β(f (rk1 ))) = {ε∗P2 (β(f (t)))| t ∈ rk1 } = {ε∗P2 (f1 (lt ))| t ∈ rk1 ; ε∗P2 (β(f (t))) = ε∗P2 (f1 (lt ))} = {ε∗P2 (f1 (α(t)))| t ∈ rk1 } = ε∗P2 (f1 (α(rk1 ))) = F1 (ε∗L (α(rk1 ))). Thus F1 (ε∗L (α(rk1 ))) = F1 (ε∗L (rα(k1 )). Now by Lemma 5.14, F1 is one to one and ε∗L (α(rk1 )) = ε∗L (rα(k1 ) and α is a star homomorphism. One can check the star commutativity on these star homomorphisms.  Theorem 5.16. (1) Let /A

ωA

f

/B

g

γ

β

/ A1

ωA1

f1

 / B1

/C

g1

 / C1

be a star commutative diagram of Hv -modules and strong Hv homomorphisms with both exact rows. Then there exists a star homomorphism α : A → A1 such that star-commute the diagram. (2) Let A

f

α

 A1

/B

g

/C

/ ωC

/ C1

/ ωC1

β

f1

 / B1

g1

be a star commutative diagram of Hv -modules and strong homomorphisms with both exact rows. Then, there exists a star homomorphism γ : C → C1 such that star commute the diagram.

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Proof. (1) For every a ∈ A we have ε∗C1 (g1 ◦ β ◦ f (a)) = ε∗C1 (γ ◦ g ◦ f (a)). First row is exact and γ is strong homomorphism. Then g ◦ f (a) ∈ ωC and γ ◦ g ◦ f (a) ∈ ωC1 . So, β ◦ f (a) ∈ Ker(g1 ) and there exists a1 ∈ A1 such that ε∗B1 (β ◦ f (a)) = ε∗B1 (f1 (a1 )). Now we define α : A → A1 by α(a) = a1 . Similar to the proof of Theorem 5.15 one can show that α is a star homomorphism. Also, for every a ∈ A we have ε∗B1 (f1 ◦ α(a)) = ε∗B1 (f1 (a1 )) = ε∗B1 (β ◦ f (a)). (2) Since g is weak-epic for every c ∈ C there exists bc ∈ B such that ε∗C (c) = ε∗C (g(bc )). We define γ : C → C1 by γ(c) = g1 ◦ β(bc ). The remaining of proof is straight forward and similar to the proof of (1).  5.7

Product and direct sum in Hv -modules

In this section, we present the concepts of product and direct sum of Hv modules. The main reference is [110]. Definition 5.11. Let M be an Hv -module and H, K are Hv -submodules of M . M is said direct sum of H and K if H ∩ K ⊆ ωM and ε∗ (H + K) = ε∗ (M ). We denote it by H ⊕ K = M . Example 5.5. For every Hv -module M we have M = ωM ⊕ M . Example 5.6. Consider the following weak commutative Hv -group: ∗M 0 1 2 3 4 5 6

0 {0,1} {0,1 } 2 3 4 5 6

1 {0,1 } {0,1 } 2 3 4 5 6

2 2 2 {0,1} {5,6} {5,6 } {2,3,4 } {2,3,4}

3 3 3 {5,6} {0,1} {5,6 } {2,3,4} {2,3,4 }

4 4 4 {5,6 } {5,6} {0,1 } {2,3,4} {2,3,4}

5 5 5 {2,3,4 } {2,3,4} {2,3,4} 6 {0,1}

6 6 6 {2,3,4} {2,3,4} {2,3,4} {0,1} 5

One can check that R = (M, ∗M , .) is an Hv -ring where r1 .r2 = {0, 1} for every r1 , r2 ∈ R and M is an Hv -module over the Hv -ring R. Also, M/ε∗M = {ε∗M (0), ε∗M (2)} where ε∗M (0) = ωM = {0, 1, 5, 6},

ε∗M (2) = {2, 3, 4}.

Now, H = {0, 1, 2} and K = {0, 1, 5, 6} are Hv -submodules of M and H ⊕ K = M.

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Proposition 5.4. Let f : M → M be a strong homomorphism of Hv modules such that f 2 = f . Then, M is direct sum of Im(f ) and Ker(f ). Moreover f is identity on Im(f ) ∩ Ker(f ). Proof.

Let m ∈ Im(f ) ∩ Ker(f ) then m = f (m1 ) for some m1 in M

(5.6)

f (m) ∈ ωM .

(5.7)

and

2

By apply f on Eq. (5.6) we obtain f (m) = f (m1 ) = f (m1 ) = m as a member of ωM by Eq. (5.7). So, Im(f ) ∩ Ker(f ) ⊆ ωM and f is identity on Im(f ) ∩ Ker(f ). Now, for every m ∈ M we have: F (F (ε∗ (m))) = F (ε∗ (f (m))) = ε∗ (f 2 (m)) = ε∗ (f (m)) = F (ε∗ (m)). So Im(F )+Ker(F ) = M/ε∗M , since F is a R/γ ∗ -module such that F 2 = F . Therefore ε∗ (Im(f ) + Ker(f )) = ε∗ (M ).  Let {Mi }i∈I be a non-empty collection of Hv -modules. The product of this collection ui∈I {Mi } = {(xi )| xi ∈ M ; ∀ i ∈ I} with the following hyper operations is an Hv -module: (xi ) + (yi ) = {(zi )| zi ∈ xi + yi }, r(xi ) = {(wi )| wi ∈ rxi }. Lemma 5.15. Let ui∈I Mi be the product of non-empty collection of Hv modules. Then (1) Pk : uMi → Mk defined by Pk ((xi )) = xk is a strong homomorphism. φ ψ (2) For every exact sequence M1 → M → M2 the mapping: λ1 : M1 → M1 u M2 defined by λ1 (x) = (x, ψφ(x)) is a strong homomorphism. Also, λ2 : M2 → M1 u M2 defined by λ2 (x) = (a, x), where a is an arbitrary member of ωM1 , is a star homomorphism. In particular if there exists a t ∈ ωM1 such that t + t = t, then λ2 is a strong homomorphism. (3) Pk λk = IMk . Proof.

(1)

Pk ((xi ) + (yi )) = Pk ({(zi )| zi ∈ xi + yi }) = {zk | zk ∈ xk + yk }.

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On the other hand, Pk ((xi )) + Pk ((yi )) = xk + yk . Similarly, we obtain Pk (r(xi )) = rPk ((xi )). (2) We have S λ1 (x + y) = a∈x+y,b∈ψφ(x+y) (a, b) = (x, ψφ(x)) + (y, ψφ(y)) = λ1 (x) + λ1 (y). Similarly we obtain λ1 (rx) = rλ1 (x). Also, S ε∗ (λ2 (x + y)) = ε∗ ( a∈ωM ,b∈x+y (a, b)) 1

= ε∗ ((a1 , x) + (a1 , y)) where a1 ∈ ωM1 = ε∗ ((a1 , x)) ⊕ ε∗ ((a1 , y)) = ε∗ (λ2 (x)) ⊕ ε∗ (λ2 (y)) = ε∗ (λ2 (x) + λ2 (y)). Similarly ε∗ (λ2 (rx)) = ε∗ (rλ2 (x)). (3) The proof of this part is straightforward.



Theorem 5.17. Let {Mi } be a non-empty collection of Hv -modules. For every Hv -module X and every collection of strong homomorphisms {fi : X → Mi } there exists an unique strong homomorphism φ : X → uMi defined by φ(x) = (fi (x)) such that for every i ∈ I the following diagram is commutative.

φ

X Proof.

fi

uM = i Pi

 / Mi

The proof is straight forward.



We want to define the inverse of a weak-isomorphism to determine the conditions for split an exact sequence. Lemma 5.16. Let f : M → N be a weak-isomorphism. Then f −1 : N → M defined by f −1 (n) = mn for selected mn ∈ F −1 (ε∗N (n)), is a star homow w morphism such that f −1 ◦ f = IM and f ◦ f −1 = IN .

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Proof. Since f is a weak-isomorphism by Lemma 5.14, F is a isomorphism and have an inverse. For every n1 , n2 ∈ N we have f −1 (n1 + n2 ) = {mc | mc ∈ F −1 (ε∗N (c)), c ∈ n1 + n2 },

(5.8)

On the other hand f −1 (n1 ) + f −1 (n2 ) = mn1 + mn2 ⊆ F −1 (ε∗N (n1 )) + F −1 (ε∗N (n2 )) = F −1 (ε∗N (n1 + n2 )).

(5.9)

From Eq. (5.8) and Eq. (5.9) we obtain ε∗M (f −1 (n1 + n2 )) = ε∗M (f −1 (n1 ) + f −1 (n2 )) (notice that for every n1 , n2 ∈ N , n1 + n2 ⊆ ε∗N (n) for some n ∈ n1 + n2 ). Similarly, we obtain ε∗M (f −1 (rn)) = ε∗M (rf −1 (n)). Finally, for every m ∈ M we have f −1 ◦ f (m) ∈ F −1 (ε∗N (f (m))) = F −1 (F (ε∗M (m))) = ε∗M (m) and for every n ∈ N , f ◦ f −1 (n) = f (mn ), where mn ∈ F −1 (ε∗N (n)). But f (mn ) ∈ ε∗N (n).



Definition 5.12. Let f be a weak-isomorphism, the f −1 defined in Lemma 5.16 is called the inverse of f . It is clear that this inverse is not unique necessary. Theorem 5.18. Let M1 , M2 and M be three Hv -modules and the sequence φ

ψ

ωM1 →M1 → M → M2 →ωM2

(5.10)

is exact: 0 0 (1) If there exists a star homomorphism φ : M → M1 (ψ : M2 → M ) such 0 0 w w that φ φ = IM1 (ψψ = IM2 ), then the sequence (5.10) is star isomorph with the sequence λ

P

ωM1 →M1 →1 M1 u M2 →2 M2 →ωM2 .

(5.11)

(2) If the sequences (5.10) and (5.11) are isomorph, then there exist a 0 0 0 w star homomorphisms φ : M → M1 and ψ : M2 → M such that φ φ = IM1 , 0 w ψψ = IM2 .

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Proof. (1) We define α : M → M1 u M2 by α(x) = (φ (x), ψ(x)). It is easy to see that α is a star homomorphism. Since for every m1 ∈ M1 we 0 have φ φ(m1 ) ∈ ε∗M1 (m1 ) and ψφ(m1 ) ∈ ωM1 , the following diagram is star commutative with both exact rows. φ

/ M1

ωM1

1M1

ψ

λ1

 / M1 u M2

/ M2

/ ωM2 .

1M2

α

 / M1

ωM1

/M

P2

 / M2

/ ωM 2 0

Now let there exists the star homomorphism ψ : M2 → M such that w ψψ = IM2 . we define the mapping β : M1 × M2 → M by β((m1 , m2 )) = 0 mm1 ,m2 where mm1 ,m2 is a member of φ(m1 ) + ψ (m2 ) (according to choice axiom). we show that β is a star homomorphism. We have: 0 0 0 0 ε∗ (β((a1 , a2 )+(a1 , a2 ))) = ε∗ (β((t1 , t2 )), where t1 ∈ a1 +a1 and t2 ∈ a2 +a2 . and 0

0

0

0

0

0

0

ε∗ (β((a1 , a2 ))) ⊕ ε∗ (β((a1 , a2 ))) = ε∗ (φ(a1 ) + ψ (a2 )) ⊕ ε∗ (φ(a1 ) + ψ (a2 )) 0 0 0 0 = ε∗ (φ(a1 ) + ψ (a1 ) + φ(a2 ) + ψ (a2 )) 0 = ε∗ (φ(t1 )) ⊕ ε∗ (ψ (t2 ) 0 = ε∗ (φ(t1 ) + ψ (t2 )) = ε∗ (β((t1 , t2 )) 0

0

where t1 ∈ a1 + a1 and t2 ∈ a2 + a2 . So β is a star homomorphism. One can show that the following diagram is star commutative. ωM1

/ M1 O

φ

/M O

1 M1

ωM1

/ M1

ψ

/ M2 O 1 M2

β

λ1

/ M1 u M2

/ ωM2 .

P2

/ M2

/ ωM2

(2) By hypothesis there exist weak-isomorphisms α : M1 → M1 , β : M → M1 u M2 and γ : M2 → M2 such that commute the following diagram ωM1

/ M1

φ

α

ωM1

 / M1

/M

ψ

λ1

/ ωM

2

γ

β

 / M1 u M2

/ M2

P2

 / M2

/ ωM2

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By Lemma 5.16, there exists star homomorphism α−1 : M1 → M1 such that 0 0 w α−1 ◦ α = IM1 . Now, we define φ : M → M1 by φ = α−1 P1 β. Conse0 quently φ is a star homomorphism and w

0

φ φ = α−1 P1 βφ = α−1 P1 λ1 α = α−1 1M1 α = IM1 . Similarly, by hypothesis there exist weak-isomorphisms: α : M1 → M1 , β : M1 u M2 → M and γ : M2 → M2 such that the following diagram is commutative. / M1

ωM1

λ1

α

 / M1

ωM1

/ M1 u M2

P2

φ

/ ωM2

γ

β

 /M

/ M2

ψ

 / M2

/ ωM

2

By Lemma 5.16, there exists star homomorphism γ −1 : M2 → M2 such that 0 0 w γ ◦ γ −1 = IM2 Now, we define ψ : M2 → M by ψ = βλ2 γ −1 . Obviously 0 ψ is a star homomorphism and 0

w

ψψ = ψβλ2 γ −1 = γP2 λ2 γ −1 = γ1M2 γ −1 = IM2 .



An exact sequence in Theorem 5.18 is called a split sequence. 5.8

Fuzzy and intuitionistic fuzzy Hv -submodules

The concept of fuzzy modules was introduced by Negoita and Ralescu. Definition 5.13. Let M be a module over a ring R. A fuzzy set µ in M is called a fuzzy submodule of M if for every x, y ∈ M and r ∈ R the following conditions are satisfied: (1) µ(0) = 1; (2) min{µ(x), µ(y)} ≤ µ(x − y), for all x, y ∈ M ; (3) µ(x) ≤ µ(rx), for all x ∈ M and r ∈ R. In [25], Davvaz defined the concept of a fuzzy Hv -submodule of an Hv module which is a generalization of the concept of a fuzzy submodule. Definition 5.14. Let M be an Hv -module over an Hv -ring R and µ a fuzzy set in M . Then µ is said to be a fuzzy Hv -submodule of M if the following axioms hold: (1) min{µ(x), µ(y)} ≤ inf{µ(z) | z ∈ x + y}, for all x, y ∈ M ;

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(2) for all x, a ∈ M there exists y ∈ M such that x ∈ a + y and min{µ(a), µ(x)} ≤ µ(y); (3) for all x, a ∈ M there exists z ∈ M such that x ∈ z + a and min{µ(a), µ(x)} ≤ µ(z); (4) µ(x) ≤ inf{µ(z) | z ∈ r · x}, for all x ∈ M and r ∈ R. Theorem 5.19. For a fuzzy subset µ of an Hv -module M , the following statements are equivalent. (1) µ is a fuzzy Hv -submodule of M . (2) µt (t ∈ ImA) is an Hv -submodule of M . Proof.

The proof is similar to the proof of Theorem 4.51.



After the introduction of fuzzy sets by Zadeh, there have been a number of generalizations of this fundamental concept. The notion of intuitionistic fuzzy sets introduced by Atanassov is one among them. Definition 5.15. An intuitionistic fuzzy set A in a non-empty set X is an object having the form A = {(x, µA (x), λA (x)) | x ∈ X}, where the functions µA : X → [0, 1] and λA : X → [0, 1] denote the degree of membership (namely µA (x)) and the degree of nonmembership (namely λA (x)) of each element x ∈ X to the set A respectively, and 0 ≤ µA (x) + λA (x) ≤ 1 for all x ∈ X. For the sake of simplicity, we shall use the symbol A = (µA , λA ) for the intuitionistic fuzzy set A = {(x, µA (x), λA (x)) | x ∈ X}. Definition 5.16. Let A = (µA , λA ) and B = (µB , λB ) be intuitionistic fuzzy sets in X. Then (1) (2) (3) (4) (5) (6)

A ⊆ B if and only if µA (x) ≤ µB (x) and λA (x) ≥ λB (x), for all x ∈ X; Ac = {(x, λA (x), µA (x)) | x ∈ X}; A ∩ B = {(x, min{µA (x), µB (x)}, max{λA (x), λB (x)}) | x ∈ X}; A ∪ B = {(x, max{µA (x), µB (x)}, min{λA (x), λB (x)}) | x ∈ X}; A = {(x, µA (x), µcA (x)) | x ∈ X}; ♦A = {(x, λcA (x), λA (x)) | x ∈ X}.

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Biswas applied the concept of intuitionistic fuzzy sets to the theory of groups and studied intuitionistic fuzzy subgroups of a group. Now, we define an intuitionistic fuzzy submodule of a module. Definition 5.17. Let M be a module over a ring R. An intuitionistic fuzzy set A = (µA , λA ) in M is called an intuitionistic fuzzy submodule of M if (1) (2) (3) (4) (5) (6)

µA (0) = 1; min{µA (x), µA (y)} ≤ µA (x − y) for all x, y ∈ M , µA (x) ≤ µA (r · x), for all x ∈ M and r ∈ R; λA (0) = 0; λA (x − y) ≤ max{λA (x), λA (y)}, for all x, y ∈ M ; λA (r · x) ≤ λA (x), for all x ∈ M and r ∈ R.

Davvaz et al. applied the concept of intuitionistic fuzzy sets to Hv -modules. They introduced the notion of intuitionistic fuzzy Hv -submodules of an Hv -module and investigate some related properties. The main reference for this section is [40]. In what follows, let M denote an Hv -module over an Hv -ring R unless otherwise specified. We start by defining the notion of intuitionistic fuzzy Hv -submodules. Definition 5.18. An intuitionistic fuzzy set A = (µA , λA ) in M is called an intuitionistic fuzzy Hv -submodule of M if (1) min{µA (x), µA (y)} ≤ inf{µA (z) | z ∈ x + y}, for all x, y ∈ M ; (2) for all x, a ∈ M there exist y, z ∈ M such that x ∈ (a + y) ∩ (z + a) and min{µA (a), µA (x)} ≤ min{µA (y), µA (z)}; (3) µA (x) ≤ inf{µA (z) | z ∈ r · x}, for all x ∈ M and r ∈ R; (4) sup{λA (z) | z ∈ x + y} ≤ max{λA (x), λA (y)}, for all x, y ∈ M ; (5) for all x, a ∈ M there exist y, z ∈ M such that x ∈ (a + y) ∩ (z + a) and max{λA (y), λA (z)} ≤ max{λA (a), λA (x)}; (6) sup{λA (z) | z ∈ r · x} ≤ λA (x), for all x ∈ M and r ∈ R. Lemma 5.17. If A = (µA , λA ) is an intuitionistic fuzzy Hv -submodule of M , then so is A = (µA , µcA ). Proof. It is sufficient to show that µcA satisfies the conditions (4), (5), and (6) of Definition 5.18. For x, y ∈ M we have min{µA (x), µA (y)} ≤ inf{µA (z) | z ∈ x + y}

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and so min{1 − µcA (x), 1 − µcA (y)} ≤ inf{1 − µcA (z) | z ∈ x + y}. Hence, min{1 − µcA (x), 1 − µcA (y)} ≤ 1 − sup{µcA (z) | z ∈ x + y} which implies sup{µcA (z) | z ∈ x + y} ≤ 1 − min{1 − µcA (x), 1 − µcA (y)}. Therefore sup{µcA (z) | z ∈ x + y} ≤ max{µcA (x), µcA (y)}, and thus the condition (4) of Definition 5.18 is valid. Now, let a, x ∈ M. Then there exist y, z ∈ M such that x ∈ (a + y) ∩ (z + a) and min{µA (a), µA (x)} ≤ min{µA (y), µA (z)}. It follows that min{1 − µcA (a), 1 − µcA (x)} ≤ min{1 − µcA (y), 1 − µcA (z)} so that max{µcA (y), µcA (z)} ≤ max{µcA (a), µcA (x)}. Hence the condition (5) of Definition 5.18 is satisfied. For the condition (6), let x ∈ M and r ∈ R. Since µA is a fuzzy Hv -submodule of M , we have µA (x) ≤ inf{µA (z) | z ∈ r · x} and so 1 − µcA (x) ≤ inf{1 − µcA (z) | z ∈ r · x}, which implies that sup{µcA (z) | z ∈ r · x} ≤ µcA (x). Therefore the condition (6) of Definition 5.18 is satisfied.



Lemma 5.18. If A = (µA , λA ) is an intuitionistic fuzzy Hv -submodule of M , then so is ♦A = (λcA , λA ). Proof.

The proof is similar to the proof of Lemma 5.17.



Combining the above two lemmas it is not difficult to verify that the following theorem is valid. Theorem 5.20. A = (µA , λA ) is an intuitionistic fuzzy Hv -submodule of M if and only if A and ♦A are intuitionistic fuzzy Hv -submodules of M . Corollary 5.5. A = (µA , λA ) is an intuitionistic fuzzy Hv -submodule of M if and only if µA and λcA are fuzzy Hv -submodules of M .

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Definition 5.19. For any t ∈ [0, 1] and fuzzy set µ in M , the set U (µ; t) = {x ∈ M | µ(x) ≥ t} (respectively, L(µ; t) = {x ∈ M | µ(x) ≤ t}) is called an upper (respectively, lower) t-level cut of µ. Theorem 5.21. If A = (µA , λA ) is an intuitionistic fuzzy Hv -submodule of M , then the sets U (µA ; t) and L(λA ; t) are Hv -submodules of M for every t ∈ Im(µA ) ∩ Im(λA ). Proof. Let t ∈ Im(µA ) ∩ Im(λA ) ⊆ [0, 1] and let x, y ∈ U (µA ; t). Then µA (x) ≥ t and µA (y) ≥ t and so min{µA (x), µA (y)} ≥ t. It follows from the condition (1) of Definition 5.18 that inf{µA (z) | z ∈ x + y} ≥ t. Therefore z ∈ U (µA ; t) for all z ∈ x + y, and so x + y ⊆ U (µA ; t). Hence a + U (µA ; t) ⊆ U (µA ; t) and U (µA ; t) + a ⊆ U (µA ; t) for all a ∈ U (µA ; t). Now, let x ∈ U (µA ; t). Then there exist y, z ∈ M such that x ∈ (a + y) ∩ (z + a) and min{µA (x), µA (a)} ≤ min{µA (y), µA (z)}. Since x, a ∈ U (µA ; t), we have t ≤ min{µA (x), µA (a)} and so t ≤ min{µA (y), µA (z)}, which implies y ∈ U (µA ; t) and z ∈ U (µA ; t). This proves that U (µA ; t) ⊆ a + U (µA ; t) and U (µA ; t) ⊆ U (µA ; t) + a. Now, for every r ∈ R and x ∈ U (µA ; t) we show that r · x ⊆ U (µA ; t). Since A is an intuitionistic fuzzy Hv -submodule of M , we have t ≤ µA (x) ≤ inf{µA (z) | z ∈ r · x}. Therefore, for every z ∈ r · x we get µA (z) ≥ t which implies z ∈ U (µA ; t), so r · x ⊆ U (µA ; t). If x, y ∈ L(λA ; t), then max{λA (x), λA (y)} ≤ t. It follows from the condition (4) of Definition 5.18 that sup{λA (z) | z ∈ x + y} ≤ t. Therefore for all z ∈ x + y we have z ∈ L(λA ; t), so x + y ⊆ L(λA ; t). Hence for all a ∈ L(λA ; t) we have a + L(λA ; t) ⊆ L(λA ; t) and L(λA ; t) + a ⊆ L(λA ; t). Now, let x ∈ L(λA ; t). Then there exist y, z ∈ M such that x ∈ (a+y)∩(z+a) and max{λA (y), λA (z)} ≤ max{λA (a), λA (x)}. Since x, a ∈ L(λA ; t), we have max{λA (a), λA (x)} ≤ t and so max{λA (y), λA (z)} ≤ t. Thus y ∈ L(λA ; t) and z ∈ L(λA ; t). Hence L(λA ; t) ⊆ a+L(λA ; t) and L(λA ; t) ⊆ L(λA ; t)+a. Now, we show that r · x ⊆ L(λA ; t) for every r ∈ R and x ∈ L(λA ; t). Since A is an intuitionistic fuzzy Hv -submodule of M , we have sup{λA (z) | z ∈ r · x} ≤ λA (x) ≤ t. Therefore, for every z ∈ r · x we get λA (z) ≤ t, which implies z ∈ L(λA ; t), so r · x ⊆ L(λA ; t). 

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Theorem 5.22. If A = (µA , λA ) is an intuitionistic fuzzy set in M such that all non-empty level sets U (µA ; t) and L(λA ; t) are Hv -submodules of M , then A = (µA , λA ) is an intuitionistic fuzzy Hv -submodule of M . Proof. Assume that all non-empty level sets U (µA ; t) and L(λA ; t) are Hv -submodules of M . If t0 = min{µA (x), µA (y)} and t1 = max{λA (x), λA (y)} for x, y ∈ M , then x, y ∈ U (µA ; t0 ) and x, y ∈ L(λA ; t1 ). So x + y ⊆ U (µA ; t0 ) and x + y ⊆ L(λA ; t1 ). Therefore for all z ∈ x + y we have µA (z) ≥ t0 and λA (z) ≤ t1 , i.e., inf{µA (z) | z ∈ x + y} ≥ min{µA (x), µA (y)} and sup{λA (z) | z ∈ x + y} ≤ max{λA (x), λA (y)}, which verify the conditions (1) and (4) of Definition 5.18. Now, if t2 = min{µA (a), µA (x)} for x, a ∈ M , then a, x ∈ U (µA ; t2 ). So there exist y1 , z1 ∈ U (µA ; t2 ) such that x ∈ a + y1 and x ∈ z1 + a. Also we have t2 ≤ min{µA (y1 ), µA (z1 )}. Therefore the condition (2) of Definition 5.18 is verified. If we put t3 = max{λA (a), λA (x)} then a, x ∈ L(λA ; t3 ). So there exist y2 , z2 ∈ L(λA ; t3 ) such that x ∈ a + y2 and x ∈ z2 + a and we have max{λA (y2 ), λA (y2 )} ≤ t3 , and so the condition (5) of Definition 5.18 is verified. Now, we verify the conditions (3) and (6). Let t4 = µA (x) and t5 = λA (x) for some x ∈ M and let r ∈ R. Then x ∈ U (µA ; t4 ) and x ∈ L(λA , t5 ). Since U (µA ; t4 ) and L(λA , t5 ) are Hv -submodules of M , we get r · x ⊆ U (µA ; t4 ) and r · x ⊆ L(λA , t5 ). Therefore for every z ∈ r · x we have z ∈ U (µA ; t4 ) and z ∈ L(λA , t5 ) which imply µA (z) ≥ t4 and λA (z) ≤ t5 . Hence inf{µA (z) | z ∈ r · x} ≥ t4 = µA (x) and sup{λA (z) | z ∈ r · x} ≤ t5 = λA (x). This completes the proof.



Corollary 5.6. Let S be an Hv -submodule of an Hv -module M . If fuzzy sets µ and λ in M are defined by   α0 if x ∈ S, β0 if x ∈ S, µ(x) = λ(x) = α1 if x ∈ M \ S, β1 if x ∈ M \ S, where 0 ≤ α1 < α0 , 0 ≤ β0 < β1 and αi + βi ≤ 1 for i = 0, 1. Then A = (µ, λ) is an intuitionistic fuzzy Hv -submodule of M and U (µ; α0 ) = S = L(λ; β0 ).

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Corollary 5.7. Let χS be the characteristic function of an Hv -submodule S of M . Then A = (χS , χcS ) is an intuitionistic fuzzy Hv -submodule of M . Theorem 5.23. If A = (µA , λA ) is an intuitionistic fuzzy Hv -submodule of M , then µA (x) = sup{α ∈ [0, 1] | x ∈ U (µA ; α)} and λA (x) = inf{α ∈ [0, 1] | x ∈ L(λA ; α)} for all x ∈ M. Proof. Let δ = sup{α ∈ [0, 1] | x ∈ U (µA ; α)} and let ε > 0 be given. Then δ − ε < α for some α ∈ [0, 1] such that x ∈ U (µA ; α). This means that δ − ε < µA (x) so that δ ≤ µA (x) since ε is arbitrary. We now show that µA (x) ≤ δ. If µA (x) = β, then x ∈ U (µA ; β) and so β ∈ {α ∈ [0, 1] | x ∈ U (µA ; α)}. Hence µA (x) = β ≤ sup{α ∈ [0, 1] | x ∈ U (µA ; α)} = δ. Therefore µA (x) = δ = sup{α ∈ [0, 1] | x ∈ U (µA ; α)}. Now let η = inf{α ∈ [0, 1] | x ∈ L(λA ; α)}. Then inf{α ∈ [0, 1] | x ∈ L(λA ; α)} < η + ε for any ε > 0, and so α < η + ε for some α ∈ [0, 1] with x ∈ L(λA ; α). Since λA (x) ≤ α and ε is arbitrary, it follows that λA (x) ≤ η. In order to prove λA (x) ≥ η, let λA (x) = ζ. Then x ∈ L(λA ; ζ) and thus ζ ∈ {α ∈ [0, 1] | x ∈ L(λA ; α)}. Hence inf{α ∈ [0, 1] | x ∈ L(λA ; α)} ≤ ζ, i.e., η ≤ ζ = λA (x). Consequently λA (x) = η = inf{α ∈ [0, 1] | x ∈ L(λA ; α)}, which completes the proof.



Definition 5.20. A fuzzy set µ in a set X is said to have sup property if for every non-empty subset S of X, there exists x0 ∈ S such that µ(x0 ) = sup{µ(x)}. x∈S

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Proposition 5.5. Let M1 and M2 be two Hv -modules over an Hv -ring R and f : M1 → M2 be a surjection. If A = (µA , λA ) is an intuitionistic fuzzy Hv -submodule of M1 such that µA and λA have sup property, then (1) f (U (µA ; t)) = U (f (µA ); t), (2) f (L(λA ; t)) ⊆ L(f (λA ); t). Proof.

(1) We have y ∈ U (f (µA ); t) ⇔ f (µA )(y) ≥ t ⇔ sup {µA (x)} ≥ t x∈f −1 (y)

⇔ ∃x0 ∈ f −1 (y), µA (x0 ) ≥ t ⇔ ∃x0 ∈ f −1 (y), x0 ∈ U (µA ; t) ⇔ f (x0 ) = y, x0 ∈ U (µA ; t) ⇔ y ∈ f (U (µA ; t)). (2) We have y ∈ L(f (λA ); t) ⇒ f (λA )(y) ≤ t ⇒ sup {λA (x)} ≤ t x∈f −1 (y)

⇒ λA (x) ≤ t, for all x ∈ f −1 (y) ⇒ x ∈ L(λA ; t), for all x ∈ f −1 (y) ⇒ y ∈ f (L(λA ; t)).



Proposition 5.6. Let M1 and M2 be two Hv -modules over an Hv -ring R and f : M1 → M2 be a map. If B = (µB , λB ) is an intuitionistic fuzzy Hv -submodule of M2 , then (1) f −1 (U (µB ; t)) = U (f −1 (µB ); t), (2) f −1 (L(λB ; t)) = L(f −1 (λB ); t) for every t ∈ [0, 1]. Proof.

(1) We have x ∈ U (f −1 (µB ); t) ⇔ f −1 (µB )(x) ≥ t ⇔ µB (f (x)) ≥ t ⇔ f (x) ∈ U (µB ; t) ⇔ x ∈ f −1 (U (µB ; t)). (2) We have x ∈ L(f −1 (λB ); t) ⇔ f −1 (λB )(x) ≤ t ⇔ λB (f (x)) ≤ t ⇔ f (x) ∈ L(λB ; t) ⇔ x ∈ f −1 (L(λB ; t)).



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Definition 5.21. Let f be a map from a set X to a set Y . If B = (µB , λB ) is an intuitionistic fuzzy set in Y, then the inverse image of B under f is defined by: f −1 (B) = (f −1 (µB ), f −1 (λB )). It is easy to see that f −1 (B) is an intuitionistic fuzzy set in X. Corollary 5.8. Let M1 and M2 be two Hv -modules over an Hv -ring R and f : M1 → M2 be a strong epimorphism. If B = (µB , λB ) is an intuitionistic fuzzy Hv -submodule of M2 , then f −1 (B) is an intuitionistic fuzzy Hv -submodule of M1 . Proof. Assume that B = (µB , λB ) is an intuitionistic fuzzy Hv submodule of M2 . By Theorem 4.7, we know that the sets U (µB ; t) and L(λB ; t) are Hv -submodules of M2 for every t ∈ Im(µB ) ∩ Im(λB ). It follows from Proposition 3.6 that f −1 (U (µB ; t)) and f −1 (L(λB ; t)) are Hv submodules of M1 . Using Proposition 4.14, we have f −1 (U (µB ; t)) = U (f −1 (µB ); t), f −1 (L(λB ; t)) = L(f −1 (λB ); t). Now, the proof is completed.



Suppose γ ∗ (r) is the equivalence class containing r ∈ R, and ε∗ (x) the equivalence class containing x ∈ M . The kernel of the canonical map ϕ : M → M/ε∗ is is denoted by ωM . Definition 5.22. Let M be an Hv -module over an Hv -ring R and let A = (µA , λA ) be an intuitionistic fuzzy Hv -submodule of M . The intuitionistic ∗ fuzzy set A/ε∗ = (µA ε , λA ε∗ ) is defined as follows: ∗

µA ε : M/ε∗ → [0, 1]   sup {µA (a)} if ε∗ (x) 6= ωM ∗ µA ε (ε∗ (x)) = a∈ε∗ (x) 1 if ε∗ (x) = ωM and λA ε∗ : M/ε∗ → [0, 1] ( inf {λA (a)} if ε∗ (x) 6= ωM ∗ λA ε∗ (ε (x)) = a∈ε∗ (x) 0 if ε∗ (x) = ωM .

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In the following we show that ∗

0 ≤ µA ε (ε∗ (x)) + λA ε∗ (ε∗ (x)) ≤ 1, for all ε∗ (x) ∈ M/ε∗ . If ε∗ (x) = ωM , then the above inequalities are clear. Assume that x ∈ H and ε∗ (x) 6= ωM . Since 0 ≤ µA (a) and 0 ≤ λA (a) for all a ∈ ε∗ (x), we have 0 ≤ sup {µA (a)} + a∈ε∗ (x)

inf {λA (a)}

a∈ε∗ (x)

or ∗

0 ≤ µA ε (ε∗ (x)) + λA ε∗ (ε∗ (x)). On the other hand, we have µA (a) + λA (a) ≤ 1

or µA (a) ≤ 1 − λA (a),

∗

for all a ∈ ε (x), and so ∗

µA ε (ε∗ (x)) = sup {µA (a)} a∈ε∗ (x)

≤ sup {1 − λA (a)} a∈ε∗ (x)

=1−

inf {λA (a)}

a∈ε∗ (x)

= 1 − λA ε∗ (ε∗ (x)). ∗

Hence µA ε (ε∗ (x)) + λA ε∗ (ε∗ (x)) ≤ 1. Theorem 5.24. Let M be an Hv -module over an Hv -ring R and let µ be a ∗ fuzzy Hv -submodule of M . Then µA ε is a fuzzy submodule of the module M/ε∗ . Proof.

It is similar to the proof of Theorem 4.54.



Lemma 5.19. We have ε∗

(λcA )c = λA ε∗ . Proof.

If ε∗ (x) = ωM , then ε∗

ε∗

(λcA )c (ωH ) = 1 − (λcA )(ωM ) = 0 = λA ε∗ (ωM ). Now, assume that ε∗ (x) 6= ωM . Then ε∗

ε∗

(λcA )c (ε∗ (x)) = 1 − (λcA )(ε∗ (x)) = 1 − sup {λcA (a)} a∈ε∗ (x)

= 1 − sup {1 − λA (a)} a∈ε∗ (x)

=

inf {λA (a)}

a∈ε∗ (x)

= λA ε∗ (ε∗ (x)).



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Theorem 5.25. Let M be an Hv -module over an Hv -ring R and let A = (µA , λA ) be an intuitionistic fuzzy Hv -submodule of M . Then A/ε∗ = ∗ (µA ε , λA ε∗ ) is an intuitionistic fuzzy submodule of the fundamental module M/ε∗ . Proof.

It is straightforward.



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Chapter 6

Hyperalgebra and Lie-Santilli theory

6.1

Hv -vector space

In this section, we present the notions of Hv -vector space and Hv -Lie algebra. These concepts are introduced by Vougiouklis [119, 120, 136]. Definition 6.1. Let (F, +, ·) be an Hv -field, (V, +) a weak commutative Hv -group and there exists an external hyperoperation · : F × V → P ∗ (V ) (a, x) 7→ ax such that for all a, b ∈ F and x, y ∈ V we have (1) a(x + y) ∩ (ax + ay) 6= ∅, (2) (a + b)x ∩ (ax + bx) 6= ∅, (3) (ab)x ∩ a(bx) 6= ∅. Then V is called an Hv -vector space over F . In the case of an Hv -ring instead of Hv -field, we have an Hv -module. In the above cases the fundamental relation ε∗ is the smallest equivalence such that the quotient V /ε∗ is a vector space over the fundamental field F/γ ∗ . Indeed, similar to Hv -groups, Hv -rings and Hv -modules, we have the following theorem. Theorem 6.1. Let (V, +) be an Hv -vector space over the Hv -field F. Denote by ϑ the set of all expressions consisting of finite hyperoperations either on F and V or the external hyperoperation applied on finite sets of elements of F and V . We define the relation ε in V as follows: xεy ⇔ {x, y} ⊂ u where u ∈ ϑ. 237

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Then the relation ε* is the transitive closure of the relation ε. Proof. Let εb be the transitive closure of ε and denote εb(x) the class of the element x. First, we prove that the quotient set V /b ε is a vector space over the field F/γ ∗ . In V /b ε the sum ⊕ and the external product , using the γ ∗ classes in F , is defined in the usual manner for all A ∈ F and x, y ∈ V εb(x) ⊕ εb(y) = {b ε(z) | z ∈ εb(x) + εb(y)}, ∗ γ (a) εb(y) = {b ε(z) | z ∈ γ ∗ (a) · εb(y)}. Take x0 ∈ εb(x) and y 0 ∈ εb(y). Then, we have x0 εbx if and only if there exist x1 , ..., xm+1 with x1 = x0 , xm+1 = x and u1 , ..., um ∈ ϑ such that {xi , xi+1 ⊆ ui for i = 1, ..., m, and y 0 εby if and only if there exist y1 , ..., yn+1 with y1 = y 0 , yn+1 = y and v1 , ..., vm ∈ ϑ such that {yj , yj+1 ⊆ vj for j = 1, ..., n. From the above we obtain {xi , xi+1 } + y1 ⊆ ui + v1 , for i = 1, ..., m − 1,

(6.1)

xm+1 + {yj , yj+1 } ⊆ um + vj , for j = 1, ..., n.

(6.2)

Thus, the sums ui + v1 = ti , for i = 1, ..., m − 1 and um + vj = tm+j−1 , for j = 1, ..., n, are also the elements of ϑ. Therefore, tk ∈ ϑ, for all k ∈ {1, ..., m + n − 1}. Now, pick up any elements z1 , ..., zm+n such that zi ∈ xi + y1 , for i = 1, ..., m and zm+j ∈ xm+1 + yj+1 , for j = 1, ..., n, and by using Equations 6.1 and 6.2, we have {zk , zk+1 } ⊆ tk , for k = 1, ..., m + n − 1. Therefore, every element z1 ∈ x1 + y1 = a0 + b0 is εb equivalent to every element zm+n ∈ xm+1 + yn+1 = x + y. Thus, εb(x) + εb(y) is singleton. Hence, we can write εb(x) ⊕ εb(y) = εb(z), for all z ∈ εb(x) + εb(y). In a similar way, using the properties of γ ∗ in F , one can prove that γ ∗ (a) εb(x) = εb(z), for all z ∈ γ ∗ (a) · εb(x). The weak associativity and the weak distributivity in F and V guarantee that the associativity and distributivity in the quotient set V /b ε over F/γ ∗ . ∗ Therefore, V /b ε is a vector space over the field F/γ .

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Now, let σ be an equivalence relation in V such that V /σ is a vector space over F/γ ∗ . Denote σ(x) the class of x. Then, σ(x) ⊕ σ(y) and γ ∗ (a) σ(x) are singletons for all a ∈ F and x, y ∈ V , i.e., σ(x) ⊕ σ(y) = σ(z), for all z ∈ σ(x) + σ(y), γ ∗ (a) σ(x) = σ(z), for all z ∈ γ ∗ (a) · σ(x). Thus, we can write, for every a ∈ F , x, y ∈ V and A ⊆ γ ∗ (a), X ⊆ σ(x), X ⊆ σ(y), σ(x) ⊕ σ(y) = σ(x + y) = σ(X + Y ), γ ∗ (a) σ(x) = σ(ax) = σ(A · X). By the induction, we extend these relations on finite sums and external products. Thus, for every u ∈ ϑ, we have σ(x) = σ(u), for all x ∈ u. Consequently, for every x ∈ V , we obtain x0 ∈ ε(x) ⇒ x0 ∈ σ(x). But σ is transitively closed, so we obtain x0 ∈ εb(x) ⇒ x0 ∈ σ(x). That means that the relation εb is the smallest equivalence relation in V such that V /b ε is a vector space over the field F/γ ∗ , i.e., εb = ε∗ .  P Remark 6.1. Let (V, +) be an Hv -vector space over F . Denote ai vi where ai vi is either the the product ai vi if ai ∈ F or the element vi if ai does not appeared. Then, from the fundamental property it is immediate that P  P ε∗ ai vi = γ ∗ (ai )ε∗ (vi ). Denoting by β ∗ (v) the fundamental class of v in (V, +). Then, obviously, β ∗ (u) ⊆ ε∗ (v), for all v ∈ V . Too elements v1 , v2 ∈ V , which are not β ∗ -equivalent, can be ε∗ equivalent if they belong to a product av with a ∈ F and v ∈ V . Corollary 6.1. If for all a ∈ F and v ∈ V , there exists an element v ∈ V such that γ ∗ (a)β ∗ (v) ⊆ β ∗ (u), then ε∗ = β ∗ . Definition 6.2. Let [U ] be the set of elements which belong to the sets of P the form ai vi , for all ai ∈ F and vi ∈ U . If [U ] = V , then U generates V . If there is no element u ∈ U such that u ∈ [U \ {u}], then U is called linearly independent subset of V .

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Theorem 6.2. If the vectors ε∗ (v1 ), ..., ε∗ (vs ) are independent over F/γ ∗ , then for all v10 , ..., vs0 ∈ V , where vi0 ∈ ε∗ (vi ), i = 1, ..., s, are independent over F . Proof. Suppose that there exists vt0 ∈ [Ut ], where Ut = {v10 , ..., vs0 } \ {vt0 }. P Therefore there exists a0i s such that vt0 ∈ ai vi0 , where vi0 ∈ Ui . So, from P P ∗ Remark 6.1, it is obtained that ε∗ ai vi0 = γ (ai )ε∗ (vi0 ) which is a contradiction.  Definition 6.3. Let (L, +) be an Hv -vector space over the Hv -field (F, +, ·), φ : F → F/γ ∗ the canonical map and ωF = {x ∈ F | φ(x) = 0}, where 0 is the zero of the fundamental field F/γ ∗ . Similarly, let ωL be the core of the canonical map φ0 : L → L/ε∗ and denote by the same symbol 0 the zero of L/ε∗ . Consider the bracket (commutator) hhyperoperation: [ , ] : L × L → P (L) : (x, y) → [x, y]. Then L is an Hv -Lie algebra over F if the following axioms are satisfied. (L1) The bracket hyperoperation is bilinear, i.e., [λ1 x1 + λ2 x2 , y] ∩ (λ1 [x1 , y] + λ2 [x2 , y]) 6= ∅ [x, λ1 y1 + λ2 y2 ] ∩ (λ1 [x, y1 ] + λ2 [x, y2 ]) 6= ∅, for all x, x1 , x2 , y, y1 , y2 ∈ L, λ1 , λ2 ∈ F , (L2) [x, x] ∩ ωL 6= ∅, for all x ∈ L, (L3) ([x, [y, z]] + [y, [z, x]] + [z, [x, y]]) ∩ ωL 6= ∅, for all x, y, z ∈ L. This is a general definition thus one can use special cases in order to face problems in applied sciences.

6.2

e-hyperstructures

e-hyperstructures are a special kind of hyperstructures and, in what follows, we shall see that they can be interpreted as a generalization of two important concepts for physics: Isotopies and Genotopies. On the other hand, biological systems such as cells or organisms at large are open and irreversible because they grow. The representation of more complex systems, such as neural networks, requires more advances methods, such as hyperstructures. In this manner, e-hyperstructures can play a significant role for the representation of complex systems in physics and biology, such as nuclear fusion, the reproduction of cells or neural systems.

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These applications were investigated by Santilli and Vougiouklis and we mention here some of their results and examples (see [97], [95]). Firstly, we shall define and analyze several types of e-hyperstructures. Definition 6.4. A hypergroupoid (H, ·) is called an e-hypergroupoid if H contains a scalar identity (also called unit) e, which means that for all x ∈ H, x · e = e · x = x. In an e-hypergroupoid, an element x0 is called inverse of a given element x ∈ H if e ∈ x · x0 = x0 · x. Clearly, if a hypergroupoid contains a scalar unit, then it is unique, while the inverses are not necessarily unique. In what follows, we use some examples which are obtained as follows: Take a set where an operation “·” is defined, then we “enlarge” the operation putting more elements in the products of some pairs. Thus a hyperoperation “◦” can be obtained, for which we have x · y ∈ x ◦ y, for all x, y ∈ H. Recall that the hyperstructures obtained in this way are Hb -structures. Example 6.1. Consider the usual multiplication on the subset {1, −1, i, −i} of complex numbers. Then we can consider the hyperoperation ◦ defined in the following table: ◦ 1 1 1 −1 −1 i i −i −i

−1 i −i −1 i −i 1 −i {i, −i} −i −1 1 i {1, i} {−1, i}

Notice that we enlarged the products (−1)·(−i), (−i)·i and (−i)·(−i) by setting (−1)◦(−i)={i, −i}, (−i)◦i={1, i} and (−i)◦(−i)={−1, i}. We obtain an e-hypergroupoid, with the scalar unit 1. The inverses of the elements −1, i, −i are −1, −i, i respectively. Moreover, the above structure is an Hv -abelian group, which means that the hyperoperation ◦ is weak associative, weak commutative and the reproductive axiom holds. Example 6.2. Consider the set H={fi | i ∈ {1, 2, 3, 4, 5, 6}} of real functions, defined from the real open interval (0, 1) to (0, 1), where f1 (x) = x, f2 (x)=(1−x)−1 , f3 (x)=1−x−1 , f4 (x)=x−1 , f5 (x)=1−x, f6 (x)=x(1−x)−1 . Let the multiplication on H be the usual composition of functions. We can

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obtain a hyperoperation ◦, ◦ f1 f2 f1 f1 f2 f2 f2 f3 f3 f3 f1 f4 f4 f5 f5 f5 f6 f6 f6 f4

given f3 f3 f1 f2 f6 f4 f5

by the following table: f4 f5 f6 f4 f5 f6 {f6 , f5 } {f4 , f6 } {f5 , f4 } {f5 , f6 } {f6 , f4 } {f4 , f5 } f1 f2 f3 {f3 , f2 } f1 f2 {f2 , f3 } {f3 , f2 } f1

We obtain an e-hypergroupoid, with the scalar unit f1 . The inverses of the elements f2 , f3 , f4 , f5 , f6 are f3 , f2 , f4 , f5 , f6 respectively. Moreover, the above structure is an Hv -abelian group. Example 6.3. Consider now the finite noncommutative quaternion group Q = {1, −1, i, −i, j, −j, k, −k}, for which the multiplication is given by the following table: ◦ 1 −1 i −i j −j k −k 1 1 −1 i −i j −j k −k −1 −1 1 −i i −j j −k −k i i −i −1 1 k −k −j j −i −i i 1 −1 −k k j −j j j −j −k k −1 1 i −i −j −j j k −k 1 −1 −i i k k −k j −j −i i −1 1 −k −k k −j j i −i 1 −1 ¯ ¯ ¯ Denote i = {i, −i}, j = {j, −j} and k = {k, −k}. We can obtain a hyperoperation ◦, given by the following table: ◦ 1 −1 i −i j −j k −k 1 1 −1 i −i j −j k −k −1 −1 1 −i i −j j −k −k i i −i −1 1 k k¯ ¯j j ¯j −i −i i 1 −1 k¯ k j ¯i j j −j k¯ k −1 1 i −j −j j k k¯ 1 −1 ¯i i ¯ ¯ k k −k j j i i −1 1 −k −k k ¯j j i ¯i 1 −1 We obtain an e-hypergroupoid, with the scalar unit 1. The inverses of the elements −1, i, −i, j, −j, k, −k are −1, −i, i, −j, j, −k, k respectively. Moreover, the above structure is an Hv -abelian group, too.

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It is immediate the following basic result, that holds for all the above examples. Theorem 6.3. The weak associativity is valid for all Hb -structures with associative basic operations. We are interested now in another kind of an e-hyperstructure, which is the e-hyperfield. Definition 6.5. A set F , endowed with an operation +, which we call addition, and a hyperoperation ·, called multiplication, is said to be an e-hyperfield if the following axioms are valid. (1) (F, +) is an abelian group where 0 is the additive unit; (2) the multiplication · is weak associative; (3) the multiplication · is weak distributive with respect to +, i.e., for all x, y, z ∈ F , x(y + z) ∩ (xy + xz) 6= ∅, (x + y)z ∩ (xz + yz) 6= ∅; (4) 0 is an absorbing element, i.e., for all x ∈ F , 0 · x = x · 0 = 0; (5) there exists a multiplicative scalar unit 1, i.e., for all x ∈ F , 1 · x = x · 1 = x; (6) for every element x ∈ F there exists an inverse x−1 such that 1 ∈ x · x−1 ∩ x−1 · x. The elements of an e-hyperfield (F, +, ·) are called e-hypernumbers. Example 6.4. (1) Starting with the ring Z3 = {¯0, ¯1, ¯2}, we can obtain a hyperring by enlarging the product ¯2 ◦ ¯2 = {¯1} to ¯2 ◦ ¯2 = {¯1, ¯2}. In other words, we obtain the following table: ◦

¯0

¯1

¯2

¯0

¯0

¯0

¯0

¯1

¯0

¯1

¯2

¯2

¯0

¯2

{¯1, ¯2}

The above structure is an e-hyperfield. (2) In the above example, only a hyperproduct is not a singleton. These hyperstructures, for which only a hyperproduct is not a singleton, are called very thin and they are useful to the theory of representations of Hv -groups by hypermatrices, see [117]. Hence, a way to obtain

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a very thin hyperstructure is the following one: we take a classical structure and we choose two elements a, b, then we can enlarge the product a · b. Therefore, in order to obtain a very thin e-hyperfield we can take a field and enlarge only one product of two, nonzero and non-unit elements. This simple change of the operation leads to enormous changes to the algebraic hyperstructure, so it looks like a chain reaction in physics. (3) Another large class of e-hyperfields can be obtained by using Hb structures. For instance, we can take the field of real numbers R, or the field of complex numbers C or the field of quaternions Q and then we can enlarge all products of nonzero and nonunit elements by adding nonzero elements and so we obtain e-hyperfields. (4) We can use the above method starting from an e-hyperfield or a ring, not necessarily a field. For instance, we can take the ring Z6 = {¯ 0, ¯ 1, ¯ 2, ¯ 3, ¯ 4, ¯ 5} of integers modulo 6. We consider a hyperoperation ◦ given by the following table: ◦

¯ 0

¯1

¯2

¯3

¯4

¯5

¯ 0

¯ 0

¯ 0

¯0

¯0

¯0

¯0

¯ 1

¯ 0

¯ 1

¯2

¯3

¯4

¯5

¯ 2

¯ 0

¯ 2

¯4

{¯0, ¯1}

{¯2, ¯3}

{¯4, ¯5}

¯ 3

¯ 0

¯ 3

{¯0, ¯1}

{¯3, ¯2}

{¯0, ¯5}

{¯3, ¯4}

¯ 4

¯ 0

¯ 4

{¯2, ¯3}

{¯0, ¯5}

{¯4, ¯1}

¯2

¯ 5

¯ 0

¯ 5

{¯4, ¯5}

{¯3, ¯4}

¯2

¯1

Then (Z6 , +, ◦) is an e-hyperfield, for which the multiplication is not closed in Z6 − {¯ 0}. For the following example, we recall a P -hyperstructure notion. Let (G, ·) be a semigroup and P ⊆ G, P 6= ∅. The following hyperoperations are called P -hyperoperations: xP ∗ y = xP y, xPr∗ y = xyP, xPl∗ y = P xy, for all x, y ∈ G. If in a set they are defined P -hyperoperations, then we obtain P -hyperstructures. The P -hyperoperation P ∗ is associative, so (G, P ∗ ) is a semihypergroup. If P ⊆ Z(G), where Z(G) is the center of G, then the above three hyperoperations coincide. P -hyperoperations can be defined in groupoids or hypergroupoids as well, and so we obtain a large class of hyperstructures. We

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can also define P -hyperoperations in sets with partial operations. Moreover, in structures with more than one operations, we can define more P -hyperoperations. In a P -hypergroup the set of left or right units is P . −1 −1 The set of left inverses of x with respect to the unit p−1 is p−1 P 0 0 x and similarly the set of right inverses of x with respect to the unit p−1 0 is −1 −1 −1 P x p0 . Definition 6.6. An e-hypermatrix is a matrix with entries elements of an e-hyperfield. We can define the product of two e-matrices in an usual manner: the P elements of product of two e-matrices (aij ), (bij ) are cij = aik ◦bkj , where the sum of products is the usual sum of sets. If we consider the e-hyperfield given in Example 6.4(1), then we have:       ¯ ¯ ¯2 ◦ ¯2 + ¯1 ◦ ¯1 ¯2 ◦ ¯1 + ¯1 ◦ ¯1 2¯ 1 2¯ 1 ¯0 ¯ ◦ 1 ¯1 ¯ = 2¯ ◦ 2¯ + 0¯ ◦ 1¯ 2¯ ◦ 1¯ + 0¯ ◦ 1¯ 2   {¯1, ¯2} + ¯1 ¯2 + ¯1 = ¯ ¯ {1, 2} + ¯0 ¯2 + ¯0   {¯2, ¯0} ¯0 = ¯ ¯ ¯ {1, 2} 2         ¯0 ¯0 ¯0 ¯0 ¯2 ¯0 ¯2 ¯0 . , ¯¯ , ¯¯ , ¯¯ = 22 12 22 1¯ 2¯ Moreover, notice that the product of an e-hypernumber with an ehypermatrix is also a hyperoperation. For instance, again on the above hyperfield, we have     ¯ ¯ 2◦¯ 2¯ 2◦¯ 1 2¯ 1 ¯ 2◦ ¯ ¯ = ¯ ¯ ¯ ¯ 2◦2 2◦2 22                 ¯ ¯ ¯1 ¯2 ¯1 ¯2 ¯2 ¯2 ¯2 ¯2 ¯2 ¯2 ¯2 ¯2 1¯ 2 1¯ 2 = . ¯1 ¯ , 1 ¯2 ¯ , 2¯ 1¯ , 2¯ 2¯ , 1¯ 1¯ , 1¯ 2¯ , 2¯ 1¯ , 2¯ 2¯ 1 This remark is useful for the definition of an e-hypervector space. Definition 6.7. Let (F, +, ·) be an e-hyperfield. An ordered set a = (a1 , a2 , ..., an ) of n e-hypernumbers of F is called an e-hypervector and the e-hypernumbers ai , i ∈ {1, 2, ..., n} are called components of the e-hypervector a.

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Two e-hypervectors are equals if they have equal corresponding components. The hypersums of two e-hypervectors a, b is defined as follows: a + b = {(c1 , c2 , ..., cn ) | ci ∈ ai + bi , i ∈ {1, 2, ..., n}}. The scalar hypermultiplication of an e-hypervector a by an e-hypernumber λ is defined in a usual manner: λ ◦ a = {(c1 , c2 , ..., cn ) | ci ∈ λ · ai , i ∈ {1, 2, ..., n}}. The set F n of all e-hypervectors with elements of F , endowed with the hypersum and the scalar hypermultiplication is called n-dimensional e-hypervector space. The set of m × n hypermatrices is an mn-dimensional e-hypervector space. The next proposition can be easily verified. Proposition 6.1. Let F be an e-hyperfield and F n be its n-dimensional e-hypervector space. Then the following assertions hold: the additive unit is the zero e-hypervector 0 = (0, 0, ..., 0); λ ◦ (a + b) ∩ (λ ◦ a + λ ◦ b) 6= ∅, for all λ ∈ F and a, b ∈ F n ; (λ + α) ◦ a ∩ (λ ◦ a + α ◦ a) 6= ∅, for all λ, α ∈ F and a ∈ F n ; λ ◦ (α ◦ a) ∩ (λ · α) ◦ a 6= ∅, for all λ, α ∈ F and a ∈ F n ; 1 ◦ a = a, λ ◦ 0 = 0, for all λ ∈ F and a ∈ F n . S Notice that by (λ + α) · ai we intend t · ai , while λ · ai + α · ai

(1) (2) (3) (4) (5)

t∈λ+α

means that S

(x + y).

x∈λ·ai y∈α·ai

Definition 6.8. An e-hyperalgebra over an e-hyperfield (F, +, ·) is an ndimensional e-hypervector space F n , endowed with a multiplication of ehypervectors , such that (F n , +, ) is an e-hyperring and for all λ ∈ F and all x, y ∈ F n , we have λ ◦ (x y) = (λ ◦ x) y = x (λ ◦ y). The most important example of an e-hyperalgebra is the algebra of n×n square e-hypermatrices. As it is well know, Lie’s theory is at the foundation of all physical theories, including classical and quantum mechanics, particle physics, nuclear physics, superconductivity, chemistry, astrophysics, etc. Despite the mathematical and physical consistency, by no means Lie’s theory can represent

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the totality of systems existing in the universe. We conclude the presentation of e-hyperstructures with the definition of an e-hyper-Lie-algebra. Definition 6.9. Let (L, +) be an e-hypervector space over an e-hyperfield (F, +, ·). Consider any bracket or commutator hyperoperation: [ , ] : L × L → P ∗ (L) : (x, y) 7→ [x, y]. Then L is an e-hyper-Lie-algebra over F if the following axioms hold: (1) the bracket hyperoperation is bilinear, i.e. ∀x, x1 , x2 , y, y1 , y2 ∈ L, ∀α1 , α2 , β1 , β2 ∈ F, [α1 x1 + α2 x2 , y] ∩ (α1 [x1 , y] + α2 [x2 , y]) 6= ∅, [x, β1 y1 + β2 y2 ] ∩ (β1 [x, y1 ] + β2 [x, y2 ]) 6= ∅; (2) ∀x ∈ L, 0 ∈ [x, x]; (3) ∀x, y, z ∈ L, 0 ∈ ([x, [y, z]] + [y, [z, x]] + [z, [x, y]]). The most important thing in studying e-hyper-Lie-algebras is to check if a subset is closed under the Lie bracket. This is so, because the product of hypermatrices normally has an enormous number of elements. However, for some interesting subclasses it is easy to check if they are closed or not. Example 6.5. (1) Consider the Lie bracket of the two traceless e-hypermatrices, over the e-hyperfield given in Example 6.4(1):     ¯2 ¯2 ¯1 ¯0 A= ¯¯ , B= ¯¯ . 11 22 We obtain        ¯ ¯1 ¯0 ¯1 ¯0 ¯2 ¯2 2¯ 2 [A, B] = ¯ ¯ · ¯ ¯ − ¯ ¯ · ¯ ¯ 11 22 22 11     ¯ ¯2 + ¯0 ¯2 + ¯0 2 + {¯ 1, ¯2} ¯0 + {¯1, ¯2} = − ¯ ¯ ¯ + 2¯ 2 0¯ + 2¯ {1, 2} + ¯2 {¯1, ¯2} + ¯2     ¯2 ¯2 {¯ 0, ¯ 1} {¯1, ¯2} − = ¯ ¯2 1 {¯0, ¯1} {¯0, ¯1}   {¯ 1, ¯ 2} {¯2, ¯0} = ¯ ¯ ¯ ¯ . {1, 0} {2, 1} 

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We notice that the Lie bracket of A and B consists of 16 ehypermatrices and some of them are not traceless. For example,     ¯1 ¯0 ¯2 ¯2 , . 1¯ 1¯ 0¯ 2¯ Hence the set of traceless e-hypermatrices is not closed. (2) Let F be a field and P be a set such that 1 ∈ P ⊆ F − {0}. We define the following hyperoperation: xP ∗ y = xP y, for all x, y ∈ F − {0, 1} and xP ∗ y = xy otherwise. For instance, if we take the field Z7 of integers modulo 7 and the set P = {¯ 1, ¯ 3}, then the hyperoperation P ∗ is given by the following table: ¯ ¯ ¯2 ¯3 ¯4 ¯5 ¯6 0 1 P∗ ¯ 0

¯ 0

¯ 0

¯0

¯0

¯0

¯0

¯0

¯ 1

¯ 0

¯ 1

¯2

¯3

¯4

¯5

¯6

¯ 2

¯ 0

¯ 2

{¯4, ¯5}

{¯6, ¯4}

{¯1, ¯3}

{¯3, ¯2}

{¯5, ¯1}

¯ 3

¯ 0

¯ 3

{¯6, ¯4}

{¯2, ¯6}

{¯5, ¯1}

{¯1, ¯3}

{¯4, ¯5}

¯ 4

¯ 0

¯ 4

{¯1, ¯3}

{¯5, ¯1}

{¯2, ¯6}

{¯6, ¯4}

{¯3, ¯2}

¯ 5

¯ 0

¯ 5

{¯3, ¯2}

{¯1, ¯3}

{¯6, ¯4}

{¯4, ¯5}

{¯2, ¯6}

¯ 6

¯ 0

¯ 6

{¯5, ¯1}

{¯4, ¯5}

{¯3, ¯2}

{¯2, ¯6}

{¯1, ¯3}

Then, (Z7 , +, P ∗ ) is an e-hyperfield. Now, consider the e-hyperfield based on Z7 , where the multiplication is replaced by the P -hyperoperation given in the above table. Take the following e-hypermatrices:     ¯4 ¯5 ¯1 ¯2 A= ¯¯ , B= ¯¯ . 03 06 Then the Lie bracket is:     ¯ ¯4 ¯5 1¯ 2 [A, B] = ¯ ¯ , ¯ ¯ 03 06     ¯ ¯4 · ¯1 + ¯5 · ¯0 ¯4 · ¯2 + ¯5 · ¯3 1·¯ 4+¯ 2 · ¯0 ¯1 · ¯5 + ¯2 · ¯6 = ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ − ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 0·4+3·0 0·5+3·6 0 · 1 + 6 · 0} 0 · 2 + 6 · 3     ¯4 {¯2, ¯4, ¯6} {¯ 4 {¯ 3, ¯ 6} = ¯ ¯ ¯ − ¯ ¯ ¯ 0 {4, 5} 0 {4, 5}   ¯ 0 {¯ 1, ¯ 6, ¯ 4, ¯2, ¯0} = ¯ . ¯ 0 {0, ¯ 6, ¯1}

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The Lie bracket of A and B has 15 elements among them 5 are strictly upper triangular e-hypermatrices. Therefore the set of upper triangular e-hypermatrices is closed under the Lie bracket hyperoperation. Now, we connect the above e-hyperstructures to isotopies and genotopies. We give now an idea about these topics, which were constructed for physical needs. Isotopies can be traced back to the early stages of set theory, where two Latin squares were said to be isotopically related when they can be made to coincide via permutations. Since Latin squares can be interpreted as the multiplication table of quasigroups, the isotopies propagated to quasigroups and then to Jordan algebras. Santilli used the term isotopy from its Greak meaning of preserving the topology and interpreted them as axiompreserving. In fact, the new and old structures are indistinguishable at the abstract. Nowadays, the term “isotopies” denotes nonlinear, nonlocal and nonhamiltonian liftings of any given linear, local and Hamiltonian structure, which preserves linearity, locality and canonicity in generalized spaces over generalized fields. The main novelty of the isotopies studied by Santilli with respect to the preceding ones is the lifting of the trivial n-dimensional unit I = diag(1, 1, ..., 1) of a conventional theory into a nowhere singular, symmetric, real-valued, positive-defined and n-dimensional matrix: Iˆ = (Iˆi,j ) = (Iˆj,i ) = Iˆ−1 = (Iˆi,j )−1 = (Iˆj,i )−1 , i, j ∈ {1, 2, ..., n}, whose elements have a smooth but otherwise arbitrary functional dependence on the local coordinates x, their derivates x˙ , ¨x, ... with respect to an independent variable t and any needed additional local quantity, ˆ x˙ , ¨x, ...). Iˆ → I(x, The original theory is reconstructed in such a way to admit Iˆ as the new left and right unit. Thus, if (F, +, ·) is a field of characteristic zero, then we can construct an isofield Fˆ = (Fˆ , +, ◦), whose elements have the form a ˆ = a·ˆ 1, where a ∈ F and ˆ 1 is a positive-defined element generally outside F . The new multiplication ◦, called isomultiplication is defined as follows: a ˆ ◦ ˆb = a ˆ · ˆ1 · ˆb, for all a ˆ, ˆb. The element ˆ 1=ˆ 1−1 is the left and right unit of Fˆ . The structure (Fˆ , +, ◦) is a new field and it is called an isotope of F , while the lifting F → Fˆ is called an isotopy. For instance, we obtain the isofields (R, +, ◦) of isoreal numbers,

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(C, +, ◦) of isocomplex numbers, (Q, +, ◦) of isoquaternions. Notice that ˆ 1, ˆ 1/ˆ 1=ˆ 1. 1nˆ = |ˆ 1 ◦ {z ... ◦ ˆ 1} = ˆ n

Genotopies were introduced by Santilli from the Greak meaning of inducing topology and interpreted them as liftings of a given theory verifying certain axioms into a form which verifies broader axioms admitting the original ones as particular cases. The main difference between isotopies and the genotopies is that the isomultiplication of two isonumbers a ˆ, ˆb has no ordering, while for the genotopies one must assume an ordering. The multiplication of two quantities is ordered on the right and it is denoted by the symbol >, when the first quantity multiplies the second one on the right, while it is ordered on the left and denoted by the symbol , +, >) and (< F, +, and one for the multiplication on the left = (Fˆ > , +, >), a ˆ> = a ˆ·ˆ 1> and are called genonumbers on the right, where a ∈ F and ˆ1> ˆ called is a quantity generally outside F and Fˆ . The new multiplication >, genomultiplication on the right is defined as follows: ˆ · ˆb, for all a ˆ ˆb = a a ˆ> ˆ·Q ˆ, ˆb. ˆ −1 is the left and right unit 1ˆ> of Fˆ > . In other words, for The element Q > > ˆ a ˆ ˆ1> . all a ˆ ∈ Fˆ , ˆ 1> > ˆ> = a ˆ> = a ˆ> > ˆ is the image of Fˆ > = A genofield on the left < Fˆ = (< Fˆ , +, ˆ ˆ under the replacement of the genomultiplication on the right (F , +, >), ˆ with the genomultiplication on the left: >
m, we can consider the homogeneous system with respect to the ‘unknowns’ b1n , b2n , ..., bmn : cin =

n P

aik bkn = 0 for i = 1, 2, ..., m.

k=1

From which, since det(Am ) 6= 0, we obtain that b1n = b2n = ... = bmn = 0. Using this fact on the last equation, on the same unknowns, cnn = n P ank bkn = 1 we have 0=1, absurd.  k=1

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We recall the following definition and notations from Section 4.4. Let A = (aij ) ∈ Mm×n be matrix and s, t ∈ N be naturals such that 1 ≤ s ≤ m, 1 ≤ t ≤ n. Then we define the characteristic-like map cst from Mm×n to Ms×t by corresponding to A the matrix Acst = (aij ) where 1 ≤ i ≤ s, 1 ≤ j ≤ t. This map is called cut-projection of type st. In other words Acst is a matrix obtained from A by cutting the lines, with index greater than s, and columns, with index greater than t. We can use cut-projections on several types of matrices to define sums and products, however, in this case we have ordinary operations, not multivalued. In the same attitude we define hyperoperations on any type of matrices. Let A = (aij ) ∈ Mm×n be matrix and s, t ∈ N , such that 1 ≤ s ≤ m, 1 ≤ t ≤ n. We define the mod-like map st from Mm×n to Ms×t by corresponding to A the matrix Ast = (aij ) which has as entries the sets aij = {ai+κs,j+λt |1 ≤ i ≤ s, 1 ≤ j ≤ t, and κ, λ ∈ N, i+κs ≤ m, j+λt ≤ n}. We call this multivalued map helix-projection of type st. Thus Ast is a set of s × t-matrices X = (xij ) such that xij ∈ aij , for all i, j. Obviously Amn = A. We may define helix-projections on ‘matrices’ of which their entries are sets. Let A = (aij ) ∈ Mm×n be matrix and s, t ∈ N , such that 1 ≤ s ≤ m, 1 ≤ t ≤ n. Then it is clear that (Asn)st = (Amt)st = Ast. Let A = (aij ) ∈ Mm×n be matrix and s, t ∈ N , such that 1 ≤ s ≤ m, 1 ≤ t ≤ n. Then if Ast is not a set of matrices but one single matrix then A is called cut-helix matrix of type s × t. In other words, A is a helix matrix of type s × t, if Acst = Ast. Definition 6.23. Let A = (aij ) ∈ Mm×n and B = (bij ) ∈ Mu×v be matrices and s = min(m, u), t = min(n, u). We define a hyperoperation, called helix-addition or helix-sum, as follows: ⊕ : Mm×n × Mu×v → P ∗ (Ms×t ) : (A, B) → A ⊕ B = Ast + Bst = (aij ) + (bij ) ⊂ Ms×t , where (aij ) + (bij ) = {(cij ) = (aij + bij )|aij ∈ aij and bij ∈ bij }.

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Let A = (aij ) ∈ Mm×n and B = (bij ) ∈ Mu×v be two matrices and s = min(m, u). We define a hyperoperation, called helix-multiplication or helix-product, as follows: ⊗ : Mm×n × Mu×v → P ∗ (Mm×v ) : (A, B) → A ⊗ B = Ams · Bsv = (aij ) · (bij ) ⊂ Mm×v , where  P (aij ) · (bij ) = (cij ) = ( ait btj )|aij ∈ aij and bij ∈ bij . For the helix-multiplication we remark that we have A ⊗ B = Ams · Bsv so we have either Ams = A or Bsv = B, that means that the helixprojection was applied only in one matrix and only in the rows or in the columns. The commutativity is valid in the helix-addition. If the appropriate matrices in the helix-sum and in the helix-product are cut-helix, then the result is singleton. Remark 6.7. From the fact that the helix-product on non square matrices is defined, the definition of a Lie-bracket is immediate, therefore the helixLie Algebra is defined, as well. This algebra is an Hv -Lie Algebra where the fundamental relation * gives, by a quotient, a Lie algebra, from which a classification is obtained. In the following we restrict ourselves on the matrices Mm×n where m < n. Obviously we have analogous results in the case where m > n and for m = n we have the classical theory. In order to simplify the notation, since we have results on mod m, we use: For given κ ∈ N \ {0}, we denote by κ the remainder resulting from its division by m if the remainder is non zero, and κ = m if the remainder is zero. Thus a matrix A = (aκλ ) ∈ Mm×n , m < n is a cut-helix if we have aκλ = aκλ , for all κ, λ ∈ N \ {0}. Moreover, let us denote by Ic = (cκλ ) the cut-helix unit matrix which the cut matrix is the unit matrix Im . Therefore, since Im =(δκλ ), where δκλ is the Kroneckers delta, we obtain that, for all κ, λ, we have cκλ = δκλ . Proposition 6.5. For m < n in (Mm×n , ⊗) the cut-helix unit matrix Ic = (cκλ ), where cκλ = δκλ , is a left scalar unit and a right unit. It is the only one left scalar unit.

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Proof. If A, B ∈ Mm×n , then in the helix-multiplication, since m < n, we take helix projection of A, therefore the result A ⊗ B is singleton if A is a cut-helix matrix of type m × m. Moreover in order to have A ⊗ B = Amm · B = B, the Amm must be the unit matrix. Consequently Ic = (cκλ ), where cκλ = δκλ , for all κ, λ ∈ N \ {0}, is necessarily the left scalar unit element. Now we remark that it is not possible to have the same case for the right matrix B, therefore we have only to prove that Ic is a right unit but it is not a scalar, consequently it is not unique. Let A = (auv ) ∈ Mm×n and consider the hyperproduct A ⊗ Ic . In the entry κλ of this hyperproduct there are sets, for all 1 ≤ κ ≤ m, 1 ≤ λ ≤ n, of the form P P aκs csλ = aκs δsλ = aκλ 3 aκλ . Therefore, A ⊗ Ic 3 A, for all A ∈ Mm×n .



In the following examples we denote Eij any type of matrices which have the ij-entry 1 and in all the other entries we have 0. Construction 6.13. Consider the 2 × 3 matrices of the following form, for κ ∈ N, Aκ = E11 + κE21 + E22 + E23 , Bκ = κE21 + E22 + E23 . Then we obtain Aκ ⊗ Aλ = {Aκ+λ , Aλ+1 , Bκ+λ , Bλ+1 }. Similarly, we have Bκ ⊗ Aλ = {Bκ+λ , Bλ+1 },Aκ ⊗ Bλ = Bλ = Bκ ⊗ Bλ . Thus the set {Aκ , Bλ |κ, λ ∈ N} becomes an Hv -semigroup which is not COW because for κ 6= λ we have Bκ ⊗ Bλ = Bλ 6= Bκ = Bλ ⊗ B, however (Aκ ⊗ Aλ ) ∩ (Aλ ⊗ Aκ ) = {Aκ+λ , Bκ+λ } = 6 ∅, for all κ, λ ∈ N. All elements Bλ are right absorbing and B1 is a left scalar element, because B1 ⊗ Aλ = Bλ+1 and B1 ⊗ Bλ = Bλ . The element A0 is a unit. Construction 6.14. Consider the 2 × 3 matrices of the following form, for κ ∈ N, Aκλ = E11 + E13 + κE21 + E22 + λE23 . Then we obtain Aκλ ⊗ Ast = {Aκ+s,κ+t , Aκ+s,λ+t , Aλ+s,κ+t , Aλ+s,λ+t }. Moreover Ast ⊗ Aκλ = {Aκ+s,λ+s , Aκ+s,λ+t , Aκ+t,λ+s , Aκ+t,λ+t }, so Aκλ ⊗ Ast ∩ Ast ⊗ Aκλ = {Aκ+s,λ+t }, thus (⊗) is COW. The helix multiplication (⊗) is associative.

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Consider all m × n matrices with m ≤ n and we write these matrices as block matrices of the form A = (A|A0 ), where A be a square m × m matrix and A0 be of type m × (n − m). Denote Mm×n the set of all m × n matrices (with m ≤ n) such that in every A the square matrix A is invertible. Take any P ⊂ Mm×n and define a P -hyperoperation as follows: A◦ B = AP t B, for all A, B ∈ Mm×n , where P t is the set of all transpose matrices from the set P . Then the (Mm×n , +, ◦) becomes a multiplicative hyperring where all matrices of type Ae = (A−1 |0), for A ∈ P are left units. Indeed A◦e B = Ae P t B 3 Ae AB = Im×m B = B, for all B ∈ Mm×n . During last decades the hyperstructures have a variety of applications in other branches of mathematics and in many other sciences. These applications range from biomathematics -conchology, inheritance- and hadronic physics to mention but a few. The hyperstructures theory is closely related to fuzzy theory; consequently, hyperstructures can now be widely applicable in industry and production, too. A general construction based on the partial ordering of the Hv structures: Theorem 6.15. (The Main e-Construction). Given a group (G, ·), where e is the unit, then we define in G, a large number of hyperoperations (⊗) by extending (·) as follows: x ⊗ y = {xy, g1 , g2 , ...}, for all x, y ∈ G \ {e}, and g1 , g2 , ... ∈ G \ {e}. g1 , g2 ,... are not necessarily the same for each pair (x,y). Then (G, ⊗) becomes an Hv -group, actually is an Hb -group which contains the (G, ·). The Hv -group (G, ⊗) is an e-hypergroup. Moreover, if for each x,y such that xy = e, so we have x ⊗ y = xy, then (G, ⊗) becomes a strong e-hypergroup. Proof. The proof is immediate since we enlarge the results of the group by putting elements from G and applying the Little Theorem. Moreover one can see that the unit e is a unique scalar and for each x in G, there exists a unique inverse x−1 , such that 1 ∈ x · x−1 ∩ x−1 · x and if this condition is valid then we have 1 = x · x−1 = x−1 · x. So the hyperstructure (G, ⊗) is a strong e-hypergroup. 

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Chapter 7

Outline of applications and modeling

7.1

Chemical examples

Chemistry is the study of matter and of the changes matter undergoes. A chemical equation describes the products of a reaction that from the starting molecules or atoms. Chemistry seeks to predict the products that result from the reaction of specific quantities of atoms or molecules. Chemists accomplish this task by writing and balancing chemical equations. Symmetry is very important in chemistry researches and group theory is the tool that is used to determine symmetry. Classical algebraic structures (group theory) is a mathematical method by which aspects of a molecules symmetry can be determined. Algebraic hyperstructures are generalizations of classical algebraic structures. A motivation for the study of hyperstructures comes from chemical reactions. In [33], Davvaz and Dehghan-Nezhad provided examples of hyperstructures associated with chain reactions. In [36], Davvaz et al. introduced examples of weak hyperstructures associated with dismutation reactions. In [38], Davvaz et al. investigated the examples of hyperstructures and weak hyperstructures associated with redox reactions. In [2], Al Tahan and Davvaz presented three different examples of hyperstructures associated to electrochemical cells. In this section we review these examples. For more details we refer to the main references [2, 15, 31, 33–39].

7.1.1

Chain reactions

Chain reaction, in chemistry and physics, process yielding products that initiate further processes of the same kind, a self-sustaining sequence. Examples from chemistry are burning a fuel gas, the development of rancidity 271

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in fats, “knock” in internal-combustion engines, and the polymerization of ethylene to polyethylene. The best-known examples in physics are nuclear fissions brought about by neutrons. Chain reactions are in general very rapid but are also highly sensitive to reaction conditions, probably because the substances that sustain the reaction are easily affected by substances other than the reactants themselves. An atom or group of atoms possessing an odd (unpaired) electron is called radical. Radical species can be electrically neutral, in which case they are sometimes referred to as free radicals. Pairs of electrically neutral “free” radicals are formed via homolytic bond breakage. This can be achieved by heating in non-polar solvents or the vapor phase. At elevated temperature or under the influence ultraviolet light at room temperature, all molecular species will dissociate into radicals. Homolsis or homolytic bond fragmentation occurs when (in the language of Lewis theory) a two electron covalent bond breaks and one electron goes to each of the partner species. For example, chlorine, Cl2 , forms chlorine radicals (Cl• ) and peroxides form oxygen radicals. X—X −→ 2X • Cl—Cl −→ 2Cl• R—O—O–R −→ R—O• Radical bond forming reactions (radical couplings) are rather rare processes. The reason is because radicals are normally present at low concentrations in a reaction medium, and it is statistically more likely they will abstract a hydrogen, or undergo another type of a substitution process, rather than reacting with each other by coupling. And as radicals are uncharged, there is little long range columbic attraction between two radical centers. Radical substitution reactions tend to proceed as chain reaction processes, often with many thousands of identical propagation steps. The propensity for chain reactivity gives radical chemistry a distinct feel compared with polar Lewis acid/base chemistry where chain reactions are less common. Methane can be chlorinated with chlorine to give chloromethane and hydrogen chloride. The reaction proceeds as a chain, radical, substitution mechanism. The process is a little more involved, and three steps are involved: initiation, propagation and termination: (1) Cl2 −→ 2Cl• (1) is called Chain-initiating step. (2) Cl• + CH4 −→ HCl + CH3•

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Outline of applications and modeling

(3) CH3• + Cl2 −→ CH3 Cl + Cl• then (2), (3), (2), (3), etc, until finally: (2) and (3) are called Chain-propagating steps. (4) Cl• + Cl• −→ Cl2 or (5) CH3• + CH3• −→ CH3 CH3 or (6) CH3• + Cl• −→ CH3 Cl. (4), (5) and (6) are called Chain-terminating steps. First in the chain of reactions is a chain-initiating step, in which energy is absorbed and a reactive particle generated; in the present reaction it is the cleavage of chlorine into atoms (Step 1). There are one or more chainpropagating steps, each of which consumes a reactive particle and generates another; there they are the reaction of chlorine atoms with methane (Step 2), and of methyl radicals with chlorine (Step 3). A chlorine radical abstracts a hydrogen from methane to give hydrogen chloride and a methyl radical. The methyl radical then abstracts a chlorine atom (a chlorine radical) from Cl2 to give methyl chloride and a chlorine radical... which abstracts a hydrogen from methane... and the cycle continues... Finally there are chain-terminating steps, in which reactive particles are consumed but not generated; in the chlorination of methane these would involve the union of two of the reactive particles, or the capture of one of them by the walls of the reaction vessel. The halogens are all typical non-metals. Although their physical forms differ-fluorine and chlorine are gases, bromine is a liquid and iodine is a solid at room temperature, each consists of diatomic molecules; F2 , Cl2 , Br2 and I2 . The halogens all react with hydrogen to form gaseous compounds, with the formulas HF, HCl, HBr, and HI all of which are very soluble in water. The halogens all react with metals to give halides. : F..¨ - ..F¨ :,

¨ -..Cl ¨ :, : ..Cl

¨.. : Br

¨ - Br .. :,

: .I¨. -..I¨ :

The reader will find in [83] a deep discussion of chain reactions and halogens. In during chain reaction A2 + B2

Heat or Light

←→

2AB

there exist all molecules A2 , B2 , AB and whose fragment parts A• , B • in experiment. Elements of this collection can by combine with each other. All combinational probability for the set H = {A• , B • , A2 , B2 , AB} to do without energy can be displayed as in Table 7.1. Then, (H, ⊕) is an Hv -group [33]. Moreover, X = {A• , A2 } and Y = • {B , B2 } are only Hv -subgroups of (H, ⊕) [33].

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⊕ A• B• A2 B2 AB

2

A• , B • , AB A• , A2 A• , B • , B2 , AB A• , AB, A2 , B •

B• A• , B • , AB B • , B2 A• , B • , AB, A2 B • , B2 A• , B • , AB, B2

A2 A• , A2 A• , B • , AB, A2 A• , A2 A• , B • , A2 , B2 , AB A• , B • , A2 , AB

Table 7.2 ⊕ Ho I• H2 I2 HI

H• H• , H

2

H • , I • , HI H • , H2 H • , I • , I2 , HI H • , HI, H2 , I •

I• H • , I • , HI I • , I2 H • , I • , HI, I2 H • , I2 H • , I • , HI, I2

Chain reactions. B2 A• , B2 , B • , AB B • , B2 A• , B • , A2 , B2 , AB B • , B2 A• , B • , B2 , AB

AB A• , AB, A2 , B • A• , B • , AB, B2 A• , B • , A2 , AB A• , B • , B2 , AB A• , B • , A2 , B2 , AB

For H and I.

H2 H • H2 H • , I • , HI, H2 H • , H2 H • , I • , H2 , I2 , HI H • , I • , H2 , HI

I2

HI

H • , I2 , I • , HI I • , I2 H • , I • , H2 , I2 , HI H • , I2 H • , I • , H2 , HI

H • , HI, H2 , I • H • , I • , HI, I2 H • , I • , H2 , HI H • , I • , I2 , HI H • , I • , H2 , I2 , HI

If we consider A = H and B ∈ {F, CL, Br, I} (for example B = I), the complete reactions table becomes Table 7.2. 7.1.2

Dismutation reactions

In a redox reactions or oxidation-reduction reaction, electrons are transferred from one reactant to another. Oxidation refers to the loss of electrons, while reduction refers to the gain of electrons. A substance that has strong affinity for electrons and tends to extract them from other species is called an oxidizing agent or an oxidant. A reducing agent, or reductant, is a reagent that readily donates electrons to another species [101]. A half reaction is a reduction or an oxidation reaction. Two half-reactions are needed to form a whole reaction. Redox reactions have a number of similarities to acid-base reactions. Like acid-base reactions, redox reactions are a matched set; you don’t have an oxidation reaction without a reduction reaction happening at the same time. When the change in free energy (∆G) is negative, a process or chemical reaction proceeds spontaneously in the forward direction. When ∆G is positive, the process proceeds spontaneously in reverse. In electrochemical reactions ∆G = −nF E, where n, F and E are number of electrons transferred in the reaction, Faraday constant and cell potential, respectively [101]. The change in the oxidation state of a species lets you know if it has undergone oxidation or reduction. Oxidation is the process in which an atom undergoes an algebraic increase in oxidation number, and reduction is the process in which an atom undergoes an algebraic decrease in oxidation

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number. On this basis, oxidation-reduction is involved in the reaction; O2 + C −→ CO2 . In the reaction, oxidation number of the C atom increases from zero to +4 whereas, the oxidation number of O atom decreases from zero to −2. Furthermore, the total increase in the oxidation number equals to the total decrease in oxidation number [84]. Disproportionation or dismutation is used to describe two particular types of chemical reaction: 0 00 0 00 (1) A chemical reaction of the type 2A −→ A +A , where A, A and A are different chemical species [108]. Most but not all are redox reactions. For example 2H2 O −→ H3 O+ + OH − is a disproportionation, but is not a redox reaction. (2) A chemical reaction in which two or more atoms of the same element originally having the same oxidation state react with other chemical(s) or themselves to give different oxidation numbers. In another word, disproportionation is a reaction in which a species is simultaneously reduced and oxidized to form two different oxidation numbers. The reverse of disproportionation is called comproportionation. Comproportionation is a chemical reaction where two reactants, each containing the same element but with a different oxidation number, will form a product with an oxidation number intermediate of the two reactants. For example, an element tin in the oxidation states 0 and +4 can comproportionate to the state +2. The standard reduction potentials of all half reactions are: E ◦ Sn4+ /Sn2+ = 0.154 V, E ◦ Sn2+ /Sn = −0.136 V, E ◦ Sn4+ /Sn = 0.009 V . Therefore, the comproportionation reaction is spontaneous. Sn + Sn4+ −→ 2Sn2+ . All combinational probability for the set S = {Sn, Sn2+ , Sn4+ } to do without energy can be displayed as follows. The major products are written in Table 7.3. Then, (S, ⊕) is weak associative. Also, we can conclude that ({Sn, Sn2+ }, ⊕) is a hypergroup and ({Sn2+ , Sn4+ }, ⊕) is an Hv semigroup [36]. Chlorine gas reacts with dilute hydroxide to form chloride, chlorate and water. The ionic equation for this reaction is as follows [67]: 3Cl2 + 6OH − −→ 5Cl− + ClO3 − + 3H2 O As a reactant, the oxidation number of the elemental chlorine, chloride and chlorate are 0, -1 and +5, respectively. Therefore, chlorine has been oxidized to chlorate whereas; it has been reduced to chloride [67].

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Dismutation reactions Sn.

⊕

Sn

Sn2+

Sn4+

Sn

Sn

Sn, Sn2+

Sn2+

Sn2+

Sn, Sn2+

Sn2+

Sn2+ , Sn4+

Sn4+

Sn2+

Sn2+ , Sn4+

Sn4+

Table 7.4

Dismutation reactions In.

⊕

In

In+

In3+

In

In

In, In+

In, In3+

In+

In, In+

In, In3+

In+ , In3+

In3+

In, In3+

In+ , In3+

In3+

Indium has three oxidation states 0, +1 and +3. The standard reduction potentials of all half reactions are: E ◦ In3+ /In+ = −0.434 V, E ◦ In+ /In = −0.147 V, E ◦ In3+ /In = −0.338 V . According to the standard reduction potentials, disproportionation reaction of In+ is spontaneous. All combinational probability for the set S = {In, In+ , In3+ } to do without energy can be displayed as Table 7.4. Then, (S, ⊕) is weak associative. Clearly, ⊕ is commutative. Also, the reproduction axiom holds. Therefore, (S, ⊕) is a commutative Hv -group [36]. Vanadium forms a number of different ions including V, V 2+ , V 3+ , V O2+ and V O2 + . The oxidation states of these species are 0, +2, +3, +4 and +5, respectively. The standard reduction potentials of all corresponding half reactions are:

−0.236 0.041

V O2+

1.00

V O2+ 0.337 0.668 0.361

V 3+

−0.225

V 2+ −1.13

V

−0.838

All combinational probability for the set S = {V, V 2+ , V 3+ , V O2+ , V O2 + } to do without energy in acidic media can be displayed as Table 7.5. When the reactants are added in appropriate stoichiometric ratios. For example

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Outline of applications and modeling Table 7.5 ⊕

V

V

V

V 2+

V, V 2+

V

3+

V

2+

V

Vanadium. V 3+

V O2+

V O2 +

V, V 2+

V 2+

V 2+ , V 3+

V 3+

V 2+

V 2+ , V 3+

V 3+

V 3+ , V O2+

V

2+

2+ , V 3+

V

3+

V

3+ , V

O2+

V O2+

V O2+

V 2+ , V 3+

V 3+

V 3+ , V O2+

V O2+

V O2+ , V O2 +

V O2 +

V 3+

V 3+ , V O2+

V O2+

V O2+ , V O2 +

V O2 +

Table 7.6

The major products between all forms of vanadium.

⊕

V

V 2+

V 3+

V O2+

V O2 +

V

V

V, V 2+

V, V 2+

V 2+

V 2+ , V 3+

V 2+

V, V 2+

V 2+

V 2+ , V 3+

V 3+

V 3+ , V O2+

V 3+

V, V 2+

V 2+ , V 3+

V 3+

V 3+ , V O2+

V O2+

V 2+

V 3+

V 3+ , V O2+

V O2+

V O2+ , V O2 +

V 2+ , V 3+

V 3+ , V O2+

V O2+

V O2+ , V O2 +

V O2 +

V O2+ V O2 +

vanadium (V ) reacts with V O2 + as follows: 2V + 3V O2 + + 12H + −→ 5V 3+ + 6H2 O Then, (S, ⊕) is a hyperstructure. The hyperstructures ({V, V 2+ }, ⊕), ({V 2+ , V 3+ }, ⊕), ({V 3+ , V O2+ }, ⊕) and ({V O2+ , V O2+ }, ⊕) are hypergroups [36]. Moreover, we have: ({V, V 2+ }, ⊕) ∼ = ({V 2+ , V 3+ }, ⊕) ∼ = ({V 3+ , V O2+ }, ⊕) ∼ = ({V O2+ , V O2 + }, ⊕). The major products between all forms of vanadium are showed in Table 7.6. It is assumed the reactants are added together in 1 : 1 mole ratios. Then, (S, ⊕) is a hyperstructure. The hyperstructures ({V, V 2+ }, ⊕), ({V 2+ , V 3+ }, ⊕), ({V 3+ , V O2+ }, ⊕) and ({V O2+ , V O2+ }, ⊕) are hypergroups. Moreover, we have: ({V, V 2+ }, ⊕) ∼ = ({V 2+ , V 3+ }, ⊕) ∼ = ({V 3+ , V O2+ }, ⊕) ∼ = ({V O2+ , V O2 + }, ⊕).

7.1.3

Redox reactions

Redox (reduction-oxidation) reactions include all chemical reactions in which atoms have their oxidation state changed. This can be either a simple redox process, such as the oxidation of carbon to yield carbon dioxide

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(CO2 ) or the reduction of carbon by hydrogen to yield methane (CH4 ), or a complex process such as the oxidation of glucose (C6 H12 O6 ) in the human body through a series of complex electron transfer processes. Oxidation is the loss of electrons or an increase in oxidation state, and reduction is the gain of electrons or a decrease in oxidation state by an analyte (molecule, atom or ion). There can not be an oxidation reaction without a reduction reaction happening simultaneously. Therefore the oxidation alone and the reduction alone are each called a half-reaction, because two half-reactions always occur together to form a whole reaction [84]. Each half-reaction has a standard reduction potential (E 0 ), which is equal to the potential difference at equilibrium under standard conditions of an electrochemical cell in which the cathode reaction is the halfreaction considered, and the anode is a standard hydrogen electrode (SHE). 0 For a redox reaction, the potential of the cell is defined by: Ecell = 0 0 0 Ecathode − Eanode . If the potential of a redox reaction (Ecell ) is positive, this reaction will spontaneous [84]. For example, consider the redox reaction of Ag 2+ with Ag: Ag 2+ + Ag −→ Ag + . We can write two half-reactions for this reaction: (1) Ag 2+ + e −→ Ag + , (2) Ag −→ Ag + + e. 0 ) is 1.98 V (vs. SHE) and the E 0 The E 0 of the first reaction (Ecathode 0 of the second reaction (Eanode ) is 0.799 V (vs. SHE) [101]. Therefore, in 0 0 0 this case, the Ecell (Ecathode − Eanode = 1.181) is positive and the above 2+ redox reaction between Ag and Ag is spontaneous. Silver (Ag) is a transition metal and has a large number of applications in jewelry, electrical contacts and conductors, catalysis of chemical reactions, disinfectants and microbiocides. Silver plays no known natural biological role in humans and itself is not toxic, but most silver salts are toxic, and some may be carcinogenic. Ag can be in three oxidation state: Ag (0), Ag (I) and Ag (II). Among Ag (I) and Ag (II), Ag (I) is very well characterized and many simple ionic compounds are known containing Ag + . However, AgF2 is known which Ag has oxidation state of II in it. AgF2 is strongly oxidizing and a good fluorimating agent. But Ag (II) is more stable in complex forms. A number of Ag (II) complexes have been obtained by oxidation of Ag (I) salts is aqueous solution in the presences of the ligand. For example, [Ag (pyridine)4 ]2+ and [Ag (bi pyridine)2 ]2+

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Outline of applications and modeling Table 7.7 ⊕

Ag 2+

Ag 2+

Ag 2+

Ag +

279

Redox reactions Ag. Ag + Ag + ,

Ag + Ag + , Ag 2+ Ag

Davvaz-Vougiouklis

Ag 2+

Ag Ag +

Ag +

Ag, Ag +

Ag + , Ag

Ag

are quite stable. The +1 oxidation state is the best known oxidation state of silver. Ag + salts are generally insoluble in water with the exception of nitrate, fluoride and perchlorate. Most stable Ag (I) complexes have a linear structure [100]. As described above, Ag species with different oxidation state can react with themselves. All possible products for spontaneous reactions are presented in Table 7.7. Table 7.7 is isomorphic to Table 7.3 of dismutation reactions. Therefore, ⊕ is weak associative. Also, we conclude that ({Ag 2+ , Ag + }, ⊕) and ({Ag + , Ag}, ⊕) are hypergroups. Copper (Cu) is a ductile metal with very high thermal and electrical conductivity. It is used as a conductor of heat and electricity, a building material, and a constituent of various metal alloys. Cu can be in four oxidation state: Cu (0), Cu (I), Cu (II) and Cu (III). In nature, copper mainly is as CuF eS2 , with oxidation state of II for Cu. Also, Cu can be as Cu2 S or Cu2 O with the oxidation state of I. Pure copper is obtained by electrolytic refining using sheets of pure copper as cathode and impure copper as anode. In this process different ions of Cu, Cu (II) or Cu (I), reduced to Cu (0) at cathode. Cu (III) is generally uncommon, however some its complexes are known [100]. The standard reduction potential (E 0 ) for conversion of each oxidation state to another are: E 0 (Cu3+ /Cu2+ ) = 2.4 V , E 0 (Cu2+ /Cu+ )= 0.153 V , E 0 (Cu2+ /Cu)= 0.342 V and E 0 (Cu+ /Cu)= 0.521 V , where potentials are versus SHE [101]. According to these standard potentials, and similar to example of Ag, the following reactions are spontaneous: (1) Cu3+ + Cu+ −→ Cu2+ , (2) Cu3+ + Cu −→ Cu2+ + Cu+ . Therefore, all possible products in reactions between oxidation states of Cu which can be produced spontaneously are listed in Table 7.8.

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Redox reactions Cu.



Cu

Cu+

Cu2+

Cu3+

Cu

Cu

Cu, Cu+

Cu2+ , Cu

Cu2+ , Cu+

Cu+

Cu, Cu+

Cu+

Cu2+ , Cu+

Cu2+

Cu2+

Cu,

Cu2+ ,

Cu3+ Cu+ , Cu2+

Cu2+

Table 7.9 ⊗ Am

Am Am

Am2+ Am, Am2+

Cu+

Am2+

Am2+

Am3+ Am, Am3+ Am2+ , Am3+ Am4+ Am, Am4+

Cu2+ Cu2+ ,

Cu2+ , Cu3+

Cu3+

Cu3+

Redox reactions Am.

Am2+ Am,

Cu2+

Am3+

Am3+ Am,

Am3+

Am4+ Am, Am4+

Am2+ , Am3+

Am3+

Am3+

Am3+ , Am4+

Am3+ , Am4+

Am4+

In Table 7.8, the hyperoperation is weak associative. Hence, we have an Hv -semigroup. The hyperstructures ({Cu, Cu+ }, ), ({Cu, Cu2+ }, ), ({Cu+ , Cu2+ }, ) and ({Cu2+ , Cu3+ }, ) are hypergroups. Let H be a set with three elements. On H, we define the following hyperoperation: x ? y = {x, y}, for all x, y ∈ H. It is easy to see that ? is associative and so (H, ?) is a hypergroup. Now, we have ({Cu, Cu+ , Cu2+ }, ) ∼ = (H, ?). Note that ({Cu+ , Cu2+ , Cu3+ }, ) is not semihypergroup. Americium (Am) is a transuranic radioactive chemical element in actinide series. It have four oxidation states of 0, 2, 3 and 4. The standard reduction potential (E 0 ) for conversion of each oxidation state to another are: E 0 (Am4+ /Am3+ )= 2.6 V , E 0 (Am3+ /Am2+ )= −2.3 V , E 0 (Am3+ /Am)= −2.048 V and E 0 (Am2+ /Am)= −1.9 V , where potentials are versus SHE [101]. Therefore, the following reaction is spontaneous: Am4+ + Am2+ −→ Am3+ . Therefore, all possible combinations for different oxidation states of Am which can be produced without energy are presented in Table 7.9. Regarding to Table 7.9, similar to Table 7.8, we have ({Am, Am2+ , Am3+ }, ⊗) ∼ = (H, ?). Note that ({Am2+ , Am3+ , Am4+ }, ⊗) is not semihypergroup.

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Outline of applications and modeling Table 7.10

Redox reactions Au.

]

Au

Au+

Au

Au

Au, Au+

Au+ Au, Au+ Au2+

Au+

Au+ Au+ ,

Davvaz-Vougiouklis

Au2+

Au2+

Au3+

Au+

Au, Au3+

Au+ , Au2+ Au+ , Au3+ Au2+

Au3+ Au, Au3+ Au+ , Au3+ Au2+ , Au3+

Au2+ , Au3+ Au3+

Gold (Au) is a dense, soft, shiny, malleable and ductile metal and can be in four oxidation states of Au (0), Au (I), Au (II) and Au (III). Au (III) is common for gold compounds and exist as: Au2 O3 , AuF3 , AuCl3 , AuBr3 and Au (OH)3 . Au (I) is much less stable in solution and is stabilized in complexes [100]. The standard reduction potential (E 0 ) for conversion of each oxidation state to another are: E 0 (Au3+ /Au+ )= 1.401 V , E 0 (Au3+ /Au)= 1.498 V , E 0 (Au2+ /Au+ )= 1.8 V and E 0 (Au+ /Au)= 1.692 V , where potentials are versus SHE [101]. According to these standard potentials, the following reaction is spontaneous: Au2+ + Au −→ Au+ . Therefore, the major products in reactions between oxidation states of Au which can be produced spontaneously are listed in Table 7.10. The Hv -semigroups defined in Table 7.9 and Table 7.10 are isomorphic. 7.1.4

Galvanic cell

Chemical reactions involving the transfer of electrons from one reactant to another are called oxidation-reduction reactions or redox reactions. In a redox reaction, two half-reactions occur; one reactant (with less electronegativity) gives up electrons (undergoes oxidation) and another reactant (with higher electronegativity) gains electrons (undergoes reduction). For example, a piece of zinc going into a solution as zinc ions, with each Zn atom giving up 2 electrons, is an example of an oxidation half-reaction. Zn −→ Zn2+ + 2e− . In contrast, the reverse reaction, in which Zn2+ ions gain 2 electrons to become Zn atoms, is an example of a reduction half-reaction. Zn2+ + 2e− −→ Zn.

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A redox reaction results when an oxidation and a reduction half-reactions are combined to complete a transfer of electrons as in the following example: Zn + Cu2+ −→ Zn2+ + Cu. The electrons are not shown in the above redox reaction because they are neither reactants nor products but have simply been transferred from one species to another (from Zn to Cu2+ in this case). In this redox reaction, the Zn is referred to as the reducing agent because it causes the Cu2+ to be reduced to Cu. The Cu2+ is called the oxidizing agent because it causes the Zn to be oxidized to Zn2+ . A Galvanic cell or voltaic cell is a device in which a redox reaction spontaneously occurs and produces an electric current. In order for the transfer of electrons in a redox reaction to produce an electric current and be useful, the electrons are made to pass through an external electrically conducting wire instead of being directly transferred between the oxidizing and reducing agents. The design of a Galvanic cell allows this to occur. In a Galvanic cell, two solutions, one containing the ions of the oxidation halfreaction and the other containing the ions of the reduction half-reaction, are placed in separated compartments called half-cells. For each half-cell, the metal, which is called an electrode, is placed in the solution and connected to an external wire. The electrode at which oxidation occurs is called the anode (Zn in the above example) and the electrode at which reduction occurs is called the cathode (Cu in the above example). The two half-cells are connected by a salt-bridge that allows a current of ions from one halfcell to the other to complete the circuit of electron current in the external wires. When the two electrodes are connected to an electric load (such as a light bulb or voltmeter) the circuit is completed, the oxidation-reduction reaction occurs, and electrons move from the anode (−) to the cathode (+), producing an electric current. Galvanic cell consists of two half-cells, such that the electrode of one halfcell is composed of metal A (with larger electronegativity) and the electrode of the other half-cell is composed of metal B (with smaller electronegativity). The redox reactions for the two separate half-cells are given as follows: An+ + ne− −→ A, B −→ B m+ + me− . The two metals A and B can react with each other according to the following balanced equation: nB + mAn+ −→ mA + nB m+ .

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Outline of applications and modeling Table 7.11 A

B

An+

B m+

A

A

A, B

A, An+

A, B m+

B

A, B

B

B m+ , A

B, B m+

An+

A, An+

B m+ , A

An+

An+ , B m+

B m+ A, B m+ B m+ , B An+ , B m+

⊕1 a b c d

a a a, b a, c a, d

283

Galvanic cell.

⊕1

Table 7.12

Davvaz-Vougiouklis

B m+

Table for Theorem 7.1. b a, b b a, d b, d

c a, c a, d c c, d

d a, d b, d c, d d

Having the element Cu with greater electronegativity than that of Zn, we get that Zn + Cu2+ −→ Zn2+ + Cu is an example of a redox reaction occurring in a Galvanic cell. For more details about Galvanic cells, see [142]. Next, we present a commutative hyperstructure related to Galvanic cell and investigate its properties. We consider the set H = {A, B, An+ , B +m } and we define a hyperoperation ⊕1 on H as follows: x ⊕1 y is the result of a possible reaction between x and y in a Galvanic cell. If x and y do not react in a Galvanic cell then we set x ⊕1 y = {x, y}. All possible spontaneous redox reactions of {A, B, An+ , B +m } in a Galvanic cell are summarized in Table 7.11. In Table 7.11, if we change the names from A, B, An+ , B m+ to a, b, c, d respectively, then the following theorem holds. Theorem 7.1. Let H = {a, b, c, d}, ⊕1 be the hyperoperation on H and consider Table 7.12 corresponding to (H, ⊕1 ): Then (H, ⊕1 ) is a commutative Hv -semigroup. Proof. It is clear from the above table that (H, ⊕1 ) is a commutative hypergroupoid. We need to show that (H, ⊕1 ) is a weak associative hypergroupoid, i.e, x ⊕1 (y ⊕1 z) ∩ (x ⊕1 y) ⊕1 z 6= ∅ for all (x, y, z) ∈ H 3 . We have three cases for x; x = a or d, x = b and x = c:

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• Case x = a or d. We have that x ∈ x ⊕1 (y ⊕1 z) ∩ (x ⊕1 y) ⊕1 z 6= ∅. • Case x = b. We have that b ⊕1 (c ⊕1 c) = b ⊕1 c = {a, d} and that (b⊕c)⊕c = {a, d}⊕c = {a, c, d}. Thus, b⊕1 (c⊕1 c)∩(b⊕1 c)⊕1 c 6= ∅. Moreover, one can easily check that b ⊕1 (c ⊕1 z) ∩ (b ⊕1 c) ⊕1 z 6= ∅ and that b ⊕1 (y ⊕1 c) ∩ (b ⊕1 y) ⊕1 c 6= ∅. If y 6= c and z 6= c then b ∈ b ⊕1 (y ⊕1 z) ∩ (b ⊕1 y) ⊕1 z. • Case x = c. This case is similar to that of Case x = b.  Remark 7.1. Since a ⊕1 (b ⊕1 c) = {a, d} = 6 (a ⊕1 b) ⊕1 c = {a, c, d}, it follows that(H, ⊕1 ) is not a semihypergroup. Remark 7.2. (H, ⊕1 ) admits two identities; a and d. Moreover, a and d are strong identities. 7.1.5

Electrolytic cell

Voltaic cells are driven by a spontaneous chemical reaction that produces an electric current through an outside circuit. These cells are important because they are the basis for the batteries that fuel modern society. But they aren’t the only kind of electrochemical cells. The reverse reaction in each case is non-spontaneous and requires electrical energy to occur. It is possible to construct a cell that does work on a chemical system by driving an electric current through the system. These cells are called electrolytic cells (or reverse Galvanic cells), and operate through electrolysis. Electrolysis is used to drive an oxidation-reduction reaction in a direction in which it does not occur spontaneously by driving an electric current through the system while doing work on the chemical system itself, and therefore is non-spontaneous. Electrolytic cells, like Galvanic cells, are composed of two half-cells; one is a reduction half-cell, the other is an oxidation half-cell. The direction of electron flow in electrolytic cells, however, may be reversed from the direction of spontaneous electron flow in Galvanic cells, but the definition of both cathode and anode remain the same, where reduction takes place at the cathode and oxidation occurs at the anode. Because the directions of both half-reactions have been reversed, the sign, but not the magnitude, of the cell potential has been reversed. Electrolytic cells consist of two half-cells, such that the electrode of one half-cell is composed of metal A (with larger electronegativity) and the electrode of the other half-cell is composed of metal B (with smaller

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A

B

An+

B m+ An+ , B

A

A

A, B

B

A, B

B

An+ , B

B, B m+

An+

An+ , B m+

An+ , B m+

B m+

An+ A, An+ An+ , B An+ , B

285

Electrolytic cell.

A, An+

B m+

Davvaz-Vougiouklis

B m+ , B

electronegativity). The redox reactions for the two separate half-cells are given as follows: A −→ An+ + ne− , B m+ + me− −→ B. The two metals A and B can react with each other according to the following balanced equation: mA + nB m+ −→ nB + mAn+ . An example of a reaction in an Electrolytic cell is: Zn2+ + Cu −→ Zn + Cu2+ which is the reverse of the reaction described in Section 3. For more details about Electrolytic cells, see [142]. Next we present a hyperstructure related to Electrolytic cells and investigate its properties. We consider the set H = {A, B, An+ , B +m } and we define a hyperoperation ⊕2 on H as follows: x ⊕2 y is the result of a possible reaction between x and y in an Electrolytic cell. If x and y do not react in an Electrolytic cell then we set x ⊕2 y = {x, y}. All possible non-spontaneous redox reactions of {A, B, An+ , B +m } in an Electrolytic cell are summarized in Table 7.13. In Table 7.13, if we change the names from A, B, An+ , B m+ to a, b, c, d respectively, then the following theorem holds. Theorem 7.2. Let H = {a, b, c, d}, ⊕2 be the hyperoperation on H and consider Table 7.14 corresponding to (H, ⊕2 ). Then (H, ⊕2 ) is a commutative Hv -semigroup.

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Proof.

Davvaz-Vougiouklis

Table for Theorem 7.2.

a a a, b a, c b, c

b a, b b b, c b, d

c a, c b, c c c, d

d b, c b, d c, d d

Let f : (H, ⊕1 ) −→ (H, ⊕2 ) defined as follows: f (a) = b, f (b) = a, f (c) = d and f (d) = c.

It is easy to see that f is an isomorphism and thus, (H, ⊕1 ) ∼ = (H, ⊕2 ). The latter and Theorem 7.1 imply that (H, ⊕2 ) is a commutative Hv -semigroup.  Remark 7.3. (H, ⊕2 ) admits two identities; b and c. Moreover, b and c are strong identities. 7.1.6

Galvanic/Electrolytic cells

In this section, we present a commutative hyperstructure related to Galvanic/Electrolytic cells and investigate its properties. We consider the set H = {A, B, An+ , B +m } and we define a hyperoperation ⊕ on H as follows: x⊕y is the result of a possible reaction between x and y in either a Galvanic cell or in an Electrolytic cell. If x and y neither react in a Galvanic cell nor in an Electrolytic cell then we set x ⊕ y = {x, y}. All possible spontaneous/non-spontaneous redox reactions of {A, B, An+ , B +m } in a Galvanic/Electrolytic cell are summarized in Table 7.15. Table 7.15 ⊕

A

Galvanic/Electrolytic cell. B

An+

B m+ An+ , B

A

A

A, B

A, An+

B

A, B

B

A, B m+

B, B m+

An+

An+ , B m+

An+ , B m+

B m+

An+ A, An+ A, B m+ B m+

An+ , B

B m+ , B

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a a a, b a, c b, c

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Table of Theorem 7.3. b a, b b a, d b, d

c a, c a, d c c, d

d b, c b, d c, d d

Remark 7.4. We can define (H, ⊕) as follows:   x ⊕1 y, if x ⊕2 y = {x, y}; x ⊕ y = x ⊕2 y, if x ⊕1 y = {x, y};  {x, y}, if x ⊕1 y = x ⊕2 y. In Table 7.15, if we change the names from A, B, An+ , B m+ to a, b, c, d respectively, then the following theorem and propositions hold. Theorem 7.3. Let H = {a, b, c, d}, ⊕ be the hyperoperation on H and consider Table 7.16 corresponding to (H, ⊕). Then (H, ⊕) is a commutative Hv -semigroup. Proof. It is clear from the above table that (H, ⊕) is a commutative hypergroupoid. We need to show that (H, ⊕) is a weak associative hypergroupoid. Let (x, y, z) ∈ H 3 . We have four cases for x; x = a, x = b, x = c and x = d: • Case x = a. We have that a ⊕ (d ⊕ d) = a ⊕ d = {b, c} and that (a ⊕ d) ⊕ d = {b, c} ⊕ d = {b, c, d}. Thus, a ⊕ (d ⊕ d) ∩ (a ⊕ d) ⊕ d 6= ∅. Moreover, one can easily check that a ⊕ (d ⊕ z) ∩ (a ⊕ d) ⊕ z 6= ∅ and that a ⊕ (y ⊕ d) ∩ (a ⊕ y) ⊕ d 6= ∅. If y 6= d and z 6= d then a ∈ a ⊕ (y ⊕ z) ∩ (a ⊕ y) ⊕ z. • Case x = b. We have that b ⊕ (c ⊕ c) = b ⊕ c = {a, d} and that (b ⊕ c) ⊕ c = {a, d} ⊕ c = {a, c, d}. Thus, b ⊕ (c ⊕ c) ∩ (b ⊕ c) ⊕ c 6= ∅. Moreover, one can easily check that b ⊕ (c ⊕ z) ∩ (b ⊕ c) ⊕ z 6= ∅ and that b ⊕ (y ⊕ c) ∩ (b ⊕ y) ⊕ c 6= ∅. If y 6= c and z 6= c then b ∈ b ⊕ (y ⊕ z) ∩ (b ⊕ y) ⊕ z. • Case x = c. We have that c ⊕ (b ⊕ b) = c ⊕ b = {a, d} and that (c ⊕ b) ⊕ c = {a, d} ⊕ c = {b, c, d}. Thus, c ⊕ (b ⊕ b) ∩ (c ⊕ b) ⊕ b 6= ∅. Moreover, one can easily check that c ⊕ (b ⊕ z) ∩ (c ⊕ b) ⊕ z 6= ∅ and that c ⊕ (y ⊕ b) ∩ (c ⊕ y) ⊕ b 6= ∅. If y 6= b and z 6= b then c ∈ c ⊕ (y ⊕ z) ∩ (c ⊕ y) ⊕ z.

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• Case x = d. We have that d ⊕ (a ⊕ a) = a ⊕ d = {b, c} and that (d ⊕ a) ⊕ a = {b, c} ⊕ a = {a, b, c}. Thus, d ⊕ (a ⊕ a) ∩ (d ⊕ a) ⊕ a 6= ∅. Moreover, one can easily check that d ⊕ (a ⊕ z) ∩ (d ⊕ a) ⊕ z 6= ∅ and that d ⊕ (y ⊕ a) ∩ (d ⊕ y) ⊕ a 6= ∅. If y 6= a and z 6= a then d ∈ d ⊕ (y ⊕ z) ∩ (d ⊕ y) ⊕ z.  Remark 7.5. Every element in (H, ⊕) is idempotent. This is trivial from chemical point of view as no reaction exists in an electrochemical cell between two identical elements, so, the element is unchanged. Proposition 7.1. (H, ⊕) is not a quasi-hypergroup nor a semihyperegroup. Proof. Since d is not an element in a ⊕ H, it follows that (H, ⊕) is not a quasi-hypergroup. Having a ⊕ (d ⊕ d) = {b, c} = 6 (a ⊕ d) ⊕ d = {b, c} ⊕ d = {b, c, d} implies that (H, ⊕) is not a semihypergroup.  Proposition 7.2. (H, ⊕) does not admit an identity element. Proof. Since d, c, b, a are not elements of a ⊕ d, b ⊕ c, c ⊕ b, d ⊕ a, it follows that none of our elements is an identity.  Remark 7.6. Proposition 7.2 implies that there exist no element x ∈ H (in a Galvanic/Electrolytic cell) such that the following reaction occurs for all y ∈ H and some x ∈ H: x + y −→ y + z. Remark 7.7. Remark 7.2, Theorem 7.2 and Proposition 7.2 imply that (H, ⊕) is not isomorphic neither to (H, ⊕1 ) nor to (H, ⊕2 ). Proposition 7.3. There are only two Hv -subsemigroups of (H, ⊕) up to isomorphism. Proof. It is easy to see that ({a}, ⊕) and ({a, b}, ⊕) are the only two Hv -subsemigroups of (H, ⊕) up to isomorphism. Moreover, ({a}, ⊕) and ({a, b}, ⊕) are hypergroups.  Definition 7.1. Let (H, ◦) be an Hv -semigroup and A be a non-empty subset of H. A is a complete part of H if for any natural number n and for all hyperproducts P ∈ HH (n), the following implication holds: A ∩ P 6= ∅ =⇒ P ⊆ A.

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Proposition 7.4. (H, ⊕) has no proper complete parts. Proof. Let A 6= ∅ be a complete part of (H, ⊕). We consider the following cases for A: • Case a ∈ A. Having a ∈ a ⊕ x, x ∈ a ⊕ x for all x ∈ {a, b, c} imply that x ∈ a ⊕ x ⊆ A. We get now that b ∈ A. Since b ∈ b ⊕ d and d ∈ b ⊕ d, it follows that d ∈ b ⊕ d ⊆ A. Thus, A = H. • Case b ∈ A. Having b ∈ b ⊕ a implies that a ∈ b ⊕ a ⊆ A. The latter implies that a ∈ A and thus A = H by the first case. • Case c ∈ A. Having c ∈ c ⊕ a implies that a ∈ c ⊕ a ⊆ A. The latter implies that a ∈ A and thus A = H by the first case. • Case d ∈ A. Having d ∈ c ⊕ d implies that c ∈ c ⊕ d ⊆ A. The latter implies that c ∈ A and thus A = H by the previous case. Therefore, (H, ⊕) has no proper complete parts.



Proposition 7.5. (H, ⊕) has a trivial fundamental group. Proof. Since {a, b} ⊆ a ⊕ b, it follows that aβ2 b. Similarly, we obtain aβ2 c, bβ2 d, cβ2 d. Having β ∗ the transitive closure of β, one can easily see that xβ ∗ y for all (x, y) ∈ H 2 . Thus, |H/β ∗ | = 1. 

7.2 7.2.1

Biological examples Inheritance examples

Scientific studies of inheritance began in 1866 with the experiments of Gregor Mendel. Mendel worked out the mathematical rules for the inheritance of characteristics in the garden pea. The significance of his work did not become widely appreciated until 1900. He discovered the principles of heredity by crossing different varieties of pea plants and analyzing the transmission pattern of traits in subsequent generations. Mendel began by studying monohybrid crosses, those between parents that differed in a single characteristic. Mendel’s approach to the study of heredity was effective for several reasons. The foremost was his choice of an experimental subject, the pea plant, Pisum sativum, which offered obvious advantages for genetic investigations. It is easy to cultivate, and Mendel had a monastery garden and a greenhouse at his disposal. Peas grow relatively rapidly, completing an entire generation in a single growing season

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[51, 108]. Mendel started with 34 varieties of peas and spent 2 years selecting those varieties that he would use in his experiments. In [37], Davvaz et al. considered specific examples of simple and incomplete inheritance and relate them to hyperstructures by studying only the monohybrid and dihybrid cases. In [1], Al Tahan and Davvaz discussed simple and incomplete inheritances for the n-hybrid case with n ≥ 1. So that the examples presented in [37] can be considered as special cases of our work. The aim of our paper is to provide examples about different types of autosomal inheritance (Mendelian and Non-Mendelian inheritance) and relate them to hyperstructures and to generalize the work done in [37]. Also, in [59], Ghadiri et al. analyzed the second generation genotypes of monohybrid and a dihybrid with a mathematical structure. They used the concept of Hv -semigroup structure in the F2 -genotypes with cross operation and proved that this is an Hv -semigroup. They determined the kinds of number of the Hv -subsemigroups of F2 -genotypes. Here are some examples of mono- and dihybrid crosses for Mendelian and Neomendelian inheritance. In the following examples, “parents” is denoted by P , “filial generation” by F and “mating” by ⊗. The main references for this section are [1, 37, 59]. Example 7.1. The Monohybrid Cross in the Pea Plant Mendel began by studding monohybrid crosses those between parents that differed in a single characteristic. In one experiment, Mendel crossed a pea plant homozygous for round seeds with one that was homozygous for wrinkled seeds [41, 108]. The first generation of the cross was the P (parental) generation. After crossing the two varieties in the P generation, Mendel observed the offspring that resulted from the cross. The results of this experiment can be summarized in the following way: P : Round (RR genotype) ⊗ Wrinkled (rr genotype) ↓ F1 : All Round (Rr genotype) and F1 ⊗ F1 : Round (Rr genotype) ⊗ Round (Rr genotype) ↓ F2 : Round (RR genotype), Round (Rr genotype), Wrinkled (rr genotype). Round is denoted by R, and Wrinkled by W . The process can be describe in Table 7.17.

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Table 7.17 ⊗ R W R R, W R W R W

Table 7.18 ⊗ T D T T, D T D T D

Considering H = {R, W }, it is easy to see that (H, ⊗) is a hypergroup. In other experiment, Mendel crossed a pea plant homozygous for Tall with one that was homozygous for Short [41, 6]. The results are similar to those of the previous example: P : Tall (T T genotype) ⊗ Short (tt genotype) ↓ F1 : All Tall (T t genotype) and F1 ⊗ F1 : F2 :

Tall (T t genotype) ⊗ Tall (T t genotype) ↓ Tall (T T genotype), Tall (T t genotype), Short (tt genotype).

Tall is denoted by T , and Dwarf by D. Hence, we have Table 7.18. Considering H = {T, D}, it is easy to see that (H, ⊗) is a hypergroup. Example 7.2. The Dihybrid Cross in the Pea Plant In addition to his work on monohybrid crosses, Mendel crossed varieties of peas that differed in two characteristics (dihybrid crosses) [41, 108]. For example, he had a homozygous variety of peas that produced round seeds and tall plants. Another homozygous variety produced wrinkled seeds and short plants. When he crossed the two plants, all the F 1 progenies had round seeds and tall plants. For example: P : Round, Tall (RRT T genotype) ⊗ Wrinkled, Short (rrtt genotype) ↓ F1 : All Round, Tall (RrT t genotype)

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B

C

D

A A B C D A B C D A B C D A B C D

B A B C D

C A B C D A B C D

D A B C D

A B C D

C D

C D

B D

C D

D

B D

B D

and F1 ⊗ F1 : Round, Tall (RrT t genotype) ⊗ Round, Tall (RrT t genotype) ↓ F2 : Round, Tall (RRT T genotype) Round, Short (RRtt and Rrtt genotypes) Wrinkled, Tall (rrT T and rrT t genotypes) Wrinkled, Short (rrtt genotype). Tall and Round are denoted by A, Tall and Wrinkled by B, Dwarf and Round by C, finally Dwarf and Wrinkled by D. Hence, we have Table 7.19. Considering H = {A, B, C, D}, (H, ⊗) is a hypergroup. Obviously, H0 = {C, D} is a subhypergroup of H. Example 7.3. Flower Color (Mirabilis Jalapa) Inheritance in the Four-o’clock Plant Mirabilis jalapa (The four o’clock flower) which is the most commonly grown ornamental species of Mirabilis is available in a range of colors. The plant produces fragrant flowers in a range of colors from white to red over a course of a few months. It is a multi-branched perennial plant in southern and warm western regions, and an annual plant in cooler northern regions of its native tropical South America. It has been naturalized in many parts of the world [101].

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Table 7.20 ⊗ R P W R R R, P P P R, P R, P, W R, P W P R, P W

Using four-o’clock plants, a plant with red flower petals can be crossed with another that has white flower petals; the offspring will have pink flower petals. If these pink-flowered F1 plants are crossed, F2 plants appear in a ratio of 1:2:1, with red, pink, or white flower petals respectively. The pinkflowered plants are heterozygotes that have a petal color between the red and the white colors of the homozygotes. In this case, one allele (R1 ) specifies a red pigment color, and another allele specifies no color (R2 ; the flower petals have a white background [51, 108]. If Red and White-flowered plants cross, then the following results obtain: P : Red (R1 R1 genotype) ⊗ White (R2 R2 genotype) ↓ F1 : All Pink (R1 R2 genotype) and F1 ⊗ F1 : Pink (R1 R2 genotype) ⊗ Pink (R1 R2 genotype) ↓ F2 : Red (R1 R1 genotype), Pink (R1 R2 genotype), White (R2 R2 genotype). Red is denoted by R, Pink by P , and White by W . Thus, we have Table 7.20. Considering H = {R, P, W }, (H, ⊗) is an Hv -semigroup. For example: R ⊗ (R ⊗ W ) = R ⊗ P = {R, P } (R ⊗ R) ⊗ W = R ⊗ W = P. Example 7.4. Coat Color of Shorthorn Cattle Inheritance The Shorthorn breed of cattle originated in the northeast of England in the late 18th century. The breed was developed as a dual purpose, suitable for both dairy and beef production. All Shorthorn cattle are red, white or roan although roan cattle are preferred by some. Also, completely white animals are not common [111]. When homozygous red-haired cattle are crossed with the homozygous white-haired type, the F1 has reddish grey

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A Walk Through Weak Hyperstructures: Hv -Structures Table 7.21 ⊗ R G W R R R, G G G R, G R, G, W G, W W G G, W W

hair and is designated as “roan”. It must be noted that there is no mixture of red and grey pigments in a roan. But some hair is all red, some other all white, so that the final result is a reddish grey coat color [108]. Thus, let two shorthorn cattle (Red and White hairs) cross. Then, we have: P : Red (r1 r1 genotype) ⊗ White (r2 r2 genotype) ↓ F1 : All Roan (r1 r2 genotype) and F1 ⊗ F1 : Roan (r1 r2 genotype) ⊗ Roan (r1 r2 genotype) ↓ F2 : Red (r1 r1 genotype), Roan (r1 r2 genotype), White (r2 r2 genotype). Red-haired is denoted by W , Reddish grey-haired (Roan) by G, and Whitehaired by W . Considering H = {R, G, W }, (H, ⊗) is an Hv -semigroup (see Table 7.21). Example 7.5. ABO Blood Group Inheritance In 1900, the Austrian physician Karl Landsteiner realized that human blood was of different types, and that only certain combinations were compatible [57]. The International Society of Blood Transfusion (ISBT) recognizes 285 blood group antigens of which 245 are classified as one of 29 blood group systems (http://blood.co.uk/ibgrl/). Each blood group system represents either a single gene or a cluster of two or three closely linked genes of related sequence with little or no recognized recombination occurring between them. Consequently, each blood group system is a genetically discrete entity. Blood groups are inherited from both parents. The ABO blood type is controlled by a single gene (the ABO gene) with three alleles: I A , I B and i. The gene encodes glycosyltransferase that is an enzyme that modifies the carbohydrate content of the red blood cell antigens. The gene is located on the long arm of the ninth chromosome (9q34) [41, 108].

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Table 7.22 ⊗

O

O

O

A

O A

B

O B

AB

A B

A O A O A AB A B O AB A B

B O B AB A B O

AB A B AB A B

O B

AB A B

AB A B

AB A B

People with blood type A have antigen A on the surfaces of their blood cells, and may be of genotype I A I A or I A i. People with blood type B have antigen B on their red blood cell surfaces, and may be of genotype I B I B or I B i. People with the rare blood type AB have both antigens A and B on their cell surfaces, and are genotype I A I B . People with blood type O have neither antigen, and are genotype ii. A type A and a type B couple can also have a type O child if they are both heterozygous (I A i and I B i, respectively). Considering H = {O, A, B, AB}, (H, ⊗) is an Hv -semigroup (see Table 7.22). For example: (O ⊗ B) ⊗ AB = {O, B} ⊗ AB = {A, B, AB} O × (B ⊗ AB) = O ⊗ {A, B, AB} = {O, A, B}. If H0 = {O, A} and H1 = {O, B}, then (H0 , ⊗) and (H1 , ⊗) are hypergroups. ABO blood type is often further differentiated by a + or −, which refers to another blood group antigen called the Rh factor. In this system, the Rh+ phenotype (D allele) is dominant on the Rh− phenotype (d allele). The antigen was originally identified in rhesus monkeys, hence the name [108]. For example: P : Rh+ (DD genotype) ⊗ Rh− (dd genotype) ↓ F1 : Rh+ (Dd genotype)

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A Walk Through Weak Hyperstructures: Hv -Structures Table 7.23 ⊗ Rh+ Rh− Rh+ Rh+ , Rh− Rh+ , Rh− Rh− Rh+ , Rh− Rh−

and F1 ⊗ F1 : F2 :

Rh+ (Dd genotype) ⊗ Rh+ (Dd genotype) ↓ Rh+ (Dd genotype), Rh+ (Dd genotype), Rh− (dd genotype).

The different crosses of Rhesus system are indicated in Table 7.23. Considering H = {Rh+ , Rh− }, (H, ⊗) is a hypergroup. Table 7.24 indicates different crosses of two blood antigenic phenotypes together (ABO and Rhesus systems). For example: P : F1 : O +

O+ (iiDd genotype) ⊗ O+ (iiDd genotype) ↓ (iiDD genotype), O+ (iiDd genotype), O− (iidd genotype).

Considering H = {O+ , O− , A+ , A− , B + , B − , AB + , AB − }, (H, ⊗) is an Hv -semigroup. If H1 = {O+ , O− }, H2 = {O− , A− }, H3 = + − + − {O , O , A , A } and H4 = {O+ , O− , B + , B − }, then (H1 , ⊗), (H2 , ⊗), (H3 , ⊗) and (H4 , ⊗) are commutative Hv -subgroups. Example 7.6. M N Blood Group Inheritance M N blood is distinct from the better-known ABO blood groups, but the principle is the same. Blood is typed according to what type(s) of antigen (a cellular product that induces antibody formation in a foreign host) is/are found on the surface of the red blood cells. Within the M N blood groups, there are two antigens, M and N , whose production is determined by a gene with two alleles, LM and LN . LM confers the ability to produce the M antigen, while LN confers the ability to produce the N antigen. Individuals who have genotype LM LM will only have the M antigen on their red cells, and will be of type M . Individuals with genotype LN LN will only have the N antigen on their red cells, and will be of type N . Heterozygotes (LM LN ) produce both antigens, and are of type M N [41, 108]. The following example is suggestive: P : M (LM LM genotype) ⊗ N (LN LN genotype) ↓ F1 : All M (LM LN genotype)

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Outline of applications and modeling Table 7.24 ⊗

O+

O−

O+

O+ O−

O+ O−

O−

O+ O−

O−

A+

A+ A− O+ O−

A−

A+ A− O+ O−

A+ A− O+ O−

A+ A− O+ O−

A− O−

A+ A− O+ O−

B+

B+ B− O+ O−

B−

B+ B− O+ O−

B− O−

AB +

A+ A− B+ B−

A+ A− B+ B−

−

A+ A− B+ B−

A− B−

AB

A+ A+ A− O+ O− A+ A− O+ O−

B+ B− O+ O−

AB + AB − A+ A− B+ B− O+ O− AB + AB − A+ A− B+ B− O+ O− AB + AB − A+ A− B+ B− AB + AB − A+ A− B+ B−

A− A+ A− O+ O− A− O−

A+ A− O+ O−

A− O−

B+ B+ B− O+ O− B+ B− O+ O− AB + AB − A+ A− B+ B− O+ O− AB + AB − A+ A− B+ B− O+ O−

AB + AB − A+ A− B+ B− O+ O−

B+ B− O+ O−

AB − A− B− O−

B+ B− O+ O−

AB + AB − A+ A− B+ B−

AB + AB − A+ A− B+ B− AB + AB − A+ A− B+ B−

AB − A− B−

B− B+ B− O+ O−

AB + A+ A− B+ B− A+ A− B+ B−

AB − A+ A− B+ B−

AB + AB − A+ A− B+ B− O+ O−

AB + AB − A+ A− B+ B−

AB + AB − A+ A− B+ B−

AB − A− B− O−

AB + AB − A+ A− B+ B−

AB − A− B−

B+ B− O+ O−

AB + AB − A+ A− B+ B−

AB + AB − A+ A− B+ B−

B− O−

AB + AB − A+ A− B+ B−

AB − A− B−

B− O−

AB + AB − A+ A− B+ B− AB − A− B−

AB + AB − A+ A− B+ B− AB + AB − A+ A− B+ B−

A− B−

AB + AB − A+ A− B+ B− AB − A− B−

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A Walk Through Weak Hyperstructures: Hv -Structures Table 7.25 ⊗ M MN N M M M, M N MN M N M, M N M, M N, N N, M N N MN N, M N N

Table 7.26 ∗ µ(O, O)(x) µ(O, A)(x) µ(O, B)(x) µ(O, AB)(x) µ(A, O)(x) µ(A, A)(x) µ(A, B)(x) µ(A, AB)(x) µ(B, O)(x) µ(B, A)(x) µ(B, B)(x) µ(B, AB)(x) µ(AB, O)(x) µ(AB, A)(x) µ(AB, B)(x) µ(AB, AB)(x)

x=O 1 1/4 1/4 0 1/4 2/16 1/16 0 1/4 1/16 2/16 0 0 0 0 0

x=A 0 3/4 0 1/2 3/4 14/16 3/16 4/8 0 3/16 0 1/8 1/2 4/8 1/8 1/4

x=B 0 0 3/4 1/2 0 0 3/16 1/8 3/4 3/16 14/16 4/8 1/2 1/8 4/8 1/4

x = AB 0 0 0 0 0 0 9/16 3/8 0 9/16 0 3/8 0 3/8 3/8 1/2

and F1 ⊗ F1 : M (LM LN genotype) ⊗ M (LM LN genotype) ↓ F2 : M (LM LM genotype), M N (LM LN genotype), N (LN LN genotype). The different crosses of M N system are indicated in Table 7.25. Considering H = {M, M N, N }, (H, ⊗) is an Hv -semigroup. Now, ABO Blood Group Inheritance is considered as a probabilistic hypergroupoid. The probabilities are indicated in Table 7.26. This table shows the calculated proportions of phenotypic classes expected from ABO system mating. For example, the A phenotype can arise from either of two genotypes, I A I A or I A i, and the frequency of each genotype is 1/2. Thus, if a person with blood type A mates with a person with blood type O, they can have type A or type O children. The occurrence probability of each of

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these events is: P : 1/2I A I A ⊗ ii ↓ F1 : 1/2I A i (A phenotype) or 1/2I A i ⊗ ii ↓ F1 : 1/2 × 1/2 = 1/4I A i (A phenotype, 1/2 × 1/2 = 1/4ii (O phenotype). P :

Thus, the probability of A phenotype children is 1/2 + 1/4 = 3/4, and the probability of O phenotype children is 1/4. 7.2.2

Examples of different types of non-Mendelian inheritance

In [37], some examples of simple inheritance and incomplete inheritance were discussed. In this section, we study some examples of five different types of non-Mendelian inheritance (Epistasis, Supplementary gene, Inhibitory gene, Complementary gene, Supplementary and complementary gene) and relate them to hypergroup theory. In the Mendelian inheritance, the presence of the dominant allele A over the recessive allele a, correspondb and b ing to the phenotypes A a respectively, in the genotype of an organism b (and its absence in the genotype implies implies that its phenotype is A that its phenotype is b a). The examples presented in [1] are not equivalent to any of those presented by the authors in [37]. The main reference for this paragraph and the next paragraphs is [1]. Example 7.7. Epistasis of dominant gene in the coat color of dogs. There are two allelomorphic pairs which may be named Aa and Bb, A and B are dominant over a and b respectively. They interact as follows: AxBy and Axbb have phenotype white, aaBy has phenotype black and aabb has phenotype brown. Here x = A or a and y = B or b. The results of this experiment can be summarized in the following way: P: AABB ⊗ aabb F1 : AaBb and F1 ⊗ F1 : AaBb ⊗ AaBb F2 : White, Black, Brown.

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A Walk Through Weak Hyperstructures: Hv -Structures Table 7.27 ⊗ A1 A2 A3 A1 H H H A2 H A2 , A 3 A2 , A 3 A3 H A2 , A 3 A3

White is denoted by A1 , Black by A2 and Brown by A3 . Considering H = {A1 , A2 , A3 }. We present (H, ⊗) in Table 7.27. Since no new elements are presented in the above table and H is found in each row and column of the table then (H, ⊗) is a hypergroupoid that satisfies the reproduction axiom. Moreover, having our table symmetric implies that (H, ⊗) is commutative. It is easy to see that (H, ⊗) is associative. Thus, (H, ⊗) is a commutative hypergroup. Moreover, it is obvious from the table that A3 is a unique identity of (H, ⊗), A3 is the only idempotent element of (H, ⊗) and that H = A21 is a single power cyclic hypergroup with period 2. Remark 7.8. Example 7.7 is not an example of the codominance inheritance which is discussed in ([37], p.183). Our example is not equivalent to that of [37]; the set of phenotypes was proved to be an Hv -semigroup in [37] whereas in our example it is a cyclic hypergroup. Example 7.8. Supplementary gene, The anthocyanin pigmentation of flowers. The red-type anthocyanin color of many flowers is caused by two alleles which may be termed as Aa and Bb. In the snapdragon (Antirrhinum) flower: AxBy is the genotype of magenta flower, Axbb is the genotype of ivory flower and aaBy, aabb are the genotypes of white flower where x = A or a and y = B or b. The results of this experiment can be summarized in the following way: P: AABB ⊗ aabb F1 : AaBb and F1 ⊗ F1 : AaBb ⊗ AaBb F2 : Magneta, Ivory, White. Magneta is denoted by B1 , Ivory by B2 and White by B3 . Considering K = {B1 , B2 , B3 }. We present (K, ⊗) in Table 7.28. Since no new elements are presented in Table 7.28 and K is found in each row and column of the table then (K, ⊗) is a hypergroupoid that satisfies the

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Table 7.28 ⊗ B1 B2 B1 K K B2 K B2 , B3 B3 K K

B3 K K B3

reproduction axiom. Moreover, having our table symmetric implies that (K, ⊗) is commutative. It is easy to see that (K, ⊗) is associative. Thus, (K, ⊗) is a commutative hypergroup. It is obvious from the table that B3 is the unique identity and unique idempotent of (K, ⊗) and that K = B12 is a single power cyclic hypergroup with period 2. Remark 7.9. The two hypergroups (H, ⊗) and (K, ⊗) defined in Examples 7.7 and 7.8 are non isomorphic hypergroups. If (H, ⊗) and (K, ⊗) are isomorphic then there exists a bijective function f : H −→ K satisfying f (x ⊗ y) = f (x) ⊗ f (y) for all x, y ∈ H. It is easy to see that f (A3 ) = B3 and thus f (A2 ) = B2 , f (A1 ) = B1 or f (A2 ) = B1 , f (A1 ) = B2 . If f (A2 ) = B2 , f (A1 ) = B1 then f (A2 ⊗ A3 ) = {f (A2 ), f (A3 )} = {B2 , B3 } and f (A2 ) ⊗ f (A3 ) = B2 ⊗ B3 = K. The latter two non equal expressions imply that f is not an isomorphism. If f (A2 ) = B1 , f (A1 ) = B2 then f (A1 ⊗ A1 ) = f (H) = K and f (A1 ) ⊗ f (A1 ) = B2 ⊗ B2 = {B2 , B3 }. The latter two non equal expressions imply that f is not an isomorphism. Example 7.9. Inhibitory gene, Rice leaf. In some rice variety the presence of the gene P causes its leaves to be colored deep purple. But if a gene I is present then the purple color is inhibited and the leaf becomes normal green. The I gene may be considered as epistatic over P . They interact as follows: The genotypes IxP y, Ixpp, iipp correspond to green and the genotype iiP y corresponds to purple where x = I or i and y = P or p. The results of this experiment can be summarized in the following way: P: IIP P ⊗ iipp F1 : IiP p and F1 ⊗ F1 : IiP p ⊗ IiP p F2 : Green, Purple. Green is denoted by A1 and Purple by A2 .

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A Walk Through Weak Hyperstructures: Hv -Structures Table 7.29 ⊗ A1 A1 L A2 L

A2 L L

Table 7.30 ⊗ A1 A1 M A2 M

A2 M M

Considering L = {A1 , A2 }. We present (L, ⊗) in Table 7.29. It is obvious from Table 7.29 that (L, ⊗) is a cyclic hypergroup with finite period equal two. Example 7.10. Complementary gene, Some rice variety. In some rice varieties, the following interaction of two pairs of genes has been noted: AxBy = red grain and aaBy, Axbb, aabb = grey grain. P: AABB ⊗ aabb F1 : AaBb and F1 ⊗ F1 : AaBb ⊗ AaBb F2 : Red, Grey. Red is denoted by A1 and Grey by A2 . Considering M = {A1 , A2 }. We present (M, ⊗) in Table 7.30. It is obvious from Table 7.30 that (M, ⊗) is a cyclic hypergroup with finite period equal two and it is isomorphic to (L, ⊗) defined in Example 7.9. Example 7.11. Supplementary and complementary gene, Seed-coat color. We consider an interesting case of gene interaction involving three pairs of genes (A, B and C) instead of two. It has been noted that seed-coat color in maize is controlled by three pairs of genes. C and B are two complementary genes which have no action independently but together cause a brownish green colouration of the stem while the grain remains colorless. If the gene A is added to C and B some plant parts as well as the grain-coat become purple by the development of anthocyanin. We may suppose that the anthocyanin pigment is developed in two stages: C and B are complementary in forming the brown pigment and when A is supplemented the

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Table 7.31 ⊗ A1 A1 N A2 N

A2 N N

full anthocyanin pigment is developed. A is, therefore, a supplementary gene. AxByCz = purple seed and {AxBycc, AxbbCz, aaByCz, aabbCz, aaBycc, Axbbcc, aabbcc} = colorless seed. P: AABBCC ⊗ aabbcc F1 : AaBbCc and F1 ⊗ F1 : AaBbCc ⊗ AaBbCc F2 : Purple, Colorless. Purple is denoted by A1 and Colorless by A2 . Considering N = {A1 , A2 }. We present (N, ⊗) in Table 7.31. It is obvious from Table 7.31 that (N, ⊗) is a cyclic hypergroup with finite period equal two and it is isomorphic to (L, ⊗) and (M, ⊗) defined in Examples 7.9 and 7.10. Remark 7.10. Examples 7.9, 7.10 and 7.11 are not equivalent to the Mendelian examples discussed in [37]. In our examples, we got total hypergroups which is not the case in [37]. 7.2.3

Hyperstructures in second generation genotype

In this section we generalize the results in [37] regarding the simple and incomplete dominance (Blending inheritance) by doing hypothetical crosses of homozygous with independent number of alleles. First, we present results for the hypothetical cross of simple dominance of n different traits given by A1 A2 . . . An ⊗ a1 a2 . . . an . Next, we present results for the hypothetical cross of incomplete dominance of n different traits given by B 1 B 2 . . . Bn ⊗ B 1 B 2 . . . B n . Finally, we present results for the hypothetical cross of simple and incomplete dominance combined together of m + n different traits given by

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A Walk Through Weak Hyperstructures: Hv -Structures Table 7.32 ⊗

c1 A

c2 A

c1 H A

H

c2 H {A c2 } A

B1 B2 . . . Bn A1 A2 . . . Am ⊗ B1 B2 . . . Bn a1 a2 . . . am . Throughout this section, for i = 1, 2, . . . , n, we denote by Ai the dominant allele over the recessive allele ai and by Bi and Bi the codominance alleles (the cross of 2 homozygous with different phenotypes leads to the presence of a new phenotype in the offspring). 7.2.4

The hypothetical cross of n different traits, case of simple dominance

The case of simple dominance can be given by: A1 A2 . . . An ⊗ a1 a2 . . . an where n ≥ 1. We consider first results for the Monohybrid cross (n = 1) that differs in a single trait; a homozygous dominant parent (A1 A1 ) ⊗ a homozygous recessive parent (a1 a1 ). The results of this hypothetical experiment can be summarized in the following way: P: A1 A1 ⊗ a1 a1 F1 : A1 a1 and F1 ⊗ F1 : A1 a1 ⊗ A1 a1 c1 (of genotype A1 x1 ), A c2 (of genotype a1 a1 ). F2 : A Where x1 = A1 or a1 . c1 , A c2 }. Then (H, ⊗) is a regular and single Proposition 7.6. Let H = {A power cyclic hypergroup with unique identity. Proof. We present (H, ⊗) in Table 7.32. Since no new elements are presented in Table 7.32 and H is found in each row and column of the table then (H, ⊗) is a hypergroupoid that satisfies the reproduction axiom. Moreover, having our table symmetric implies that (H, ⊗) is commutative. It is easy to see that (H, ⊗) is associative. Thus, (H, ⊗) is a commutative hypergroup.

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c2 is a unique identity of (H, ⊗) and It is obvious from the table that A 2 c that H = A1 is a single power cyclic hypergroup with period 2. Since (H, ⊗) admits an identity and each element in it admits at least one inverse −1 −1 c1 c1 , A c2 } and A c2 c2 ), it follows that (H, ⊗) is a regular (A = {A = A hypergroup.



Proposition 7.7. (H, ⊗) has no proper linear subsets. Proof. Let M be a proper linear subset of (H, ⊗); i.e M 6= H. Then c1 is not x ⊗ y and x/y are subsets of M for all x, y ∈ M . We have that A c c c an element of M else H = A1 ⊗ A1 ⊂ M . Also, A2 is not an element of M c2 /A c2 ⊂ M . else H = A  Proposition 7.8. (H, ⊗) has one proper subhypergroup. Proof. It is easy to see that the only subhypergroups of (H, ⊗) are (H, ⊗) c2 }, ⊗). and ({A  We consider now results for the Dihybrid cross (n = 2) that differs in two traits; a homozygous dominant parent (A1 A1 A2 A2 ) ⊗ a homozygous recessive parent (a1 a1 a2 a2 ). The results of this hypothetical experiment can be summarized in the following way: P: A1 A1 A2 A2 ⊗ a1 a1 a2 a2 F1 : A1 a1 A2 a2 and F1 ⊗ F1 : A1 a1 A2 a2 ⊗ A1 a1 A2 a2 c1 (of genotype A1 x1 A2 x2 ), A c2 (of genotype A1 x1 a2 a2 ), A c3 (of F2 : A c4 (of genotype a1 a1 a2 a2 ). genotype a1 a1 A2 x2 ), A Where x1 = A1 or a1 and x2 = A2 or a2 . c1 , A c2 , A c3 , A c4 }. Then (H, ⊗) is a regular and Proposition 7.9. Let H = {A single power cyclic hypergroup with unique identity. Proof. We present (H, ⊗) in Table 7.33. Since no new elements are presented in Table 7.33 and H is found in each row and column of the table then (H, ⊗) is a hypergroupoid that satisfies the reproduction axiom. Moreover, having our table symmetric implies that (H, ⊗) is commutative. It is easy to see that (H, ⊗) is associative. Thus, (H, ⊗) is a commutative hypergroup.

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A Walk Through Weak Hyperstructures: Hv -Structures Table 7.33 c1 ⊗ A

c2 A

c3 A

c1 H A

H

H

H

H

c2 , A c4 A

c2 H A c2 , A c4 A c3 H A

H

c4 A

c3 , A c4 A c3 , A c4 A

c4 H A c2 , A c4 A c3 , A c4 A

c4 A

c4 is a unique identity of (H, ⊗) and It is obvious from the table that A 2 c1 is a single power cyclic hypergroup with period 2. Since that H = A (H, ⊗) admits an identity and each element in it admits at least one inverse c4 ∈ (A c1 ⊗ A c4 ) ∩ (A c2 ⊗ A c4 ) ∩ (A c3 ⊗ A c4 ) ∩ (A c4 ⊗ A c4 )), it follows that (H, ⊗) (A is a regular hypergroup.  c4 is the only idempotent element in (H, ⊗). It is easy to see that the A Proposition 7.10. (H, ⊗) has no proper linear subsets. Proof. Let M be a proper linear subset of (H, ⊗); i.e M 6= H. Then c1 x ⊗ y and x/y are subsets of M for all x, y ∈ M . We have that H = x ⊗ A c c for all x ∈ H. We get now that A1 ∈ M as A1 ∈ x/y for all x, y ∈ M . c1 ∈ M , it follows that H = A c1 ⊗ A c1 ⊂ M . Since A  Proposition 7.11. (H, ⊗) has only 2 proper subhypergroups up to isomorphism. Proof. It is easy to see that the only subhypergroups of (H, ⊗) are H, c2 }, {A c2 , A c4 } and {A c3 , A c4 }. Having that {A c2 , A c4 } and {A c3 , A c4 } isomor{A phic implies that we have two proper subhypergroups.  We consider now the n−hybrid cross (n ≥ 3) that differs in n traits; a homozygous dominant parent (A1 A1 A2 A2 . . . An An ) ⊗ a homozygous recessive parent (a1 a1 a2 a2 . . . an an ). The results of this hypothetical experiment can be summarized in the following way: P: A1 A1 A2 A2 . . . An−1 An−1 An An ⊗ a1 a1 a2 a2 . . . an an F1 : A1 a1 A2 a2 . . . An an and F1 ⊗ F1 : A1 a1 A2 a2 . . . An an ⊗ A1 a1 A2 a2 . . . An an c1 (of genotype A1 x1 A2 x2 . . . An xn ), A c2 (of genotype F2 : A c A1 x1 A2 x2 . . . An−1 xn−1 an an ), . . ., Ak (of genotype a1 a1 a2 a2 . . . an an ). Where k = 2n is the number of different phenotypes, xi = Ai or ai .

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c1 , A c2 , . . . , A ck }. Then (H, ⊗) is a regular and Theorem 7.4. Let H = {A single power cyclic hypergroup with unique identity. Proof.

The proof is similar to that of Propositions 7.6 and 7.9. It is easy 2 c1 and A ck is a unique identity of (H, ⊗). to see that H = A  Proposition 7.12. (H, ⊗) has no proper linear subsets. Proof.

The proof is the same as that of Proposition 7.10.



Proposition 7.13. The only idempotent element in (H, ⊗) is the identity element. ck is an idempotent in Proof. It is easy to see that the identity A cr be an idempotent in (H, ⊗) with corresponding genotype (H, ⊗). Let A 2 cr = A cr implies that xi = yi for all i = 1, . . . , n. x1 y1 . . . xn yn . Having A The latter implies that xi = yi = Ai or ai . If there exists i ∈ [1, n] such that xi = yi = Ai then the genotype x1 y1 . . . Ai ai . . . xn yn corresponds to 2 cr which the phenotype Ar . It is easy to see that x1 y1 . . . ai ai . . . xn yn ∈ A 2

cr = A cr . contradicts our assumption that A



c1 , A c2 , A c3 , A c4 , Example 7.12. Trihybrid case, n = 3. Let H = {A c5 , A c6 , A c7 , A c8 } be the set of different phenotypes in F2 generation of A c1 corresponds to the genotype A1 x1 A2 x2 A3 x3 , the trihybrid case and A c c3 corresponds to the A2 corresponds to the genotype A1 x1 A2 x2 a3 a3 , A c4 corresponds to the genotype A1 x1 a2 a2 a3 a3 , genotype A1 x1 a2 a2 A3 x3 , A c5 corresponds to the genotype a1 a1 A2 x2 A3 x3 , A c6 corresponds to the A c genotype a1 a1 A2 x2 a3 a3 , A7 corresponds to the genotype a1 a1 a2 a2 A3 x3 c8 corresponds to the genotype a1 a1 a2 a2 a3 a3 and xi = Ai or ai and A for i = 1, 2, 3. Since (H, ⊗) is commutative then we can represent it by Table 7.34. c2 , A c4 , A c6 , A c8 }, L = {A c3 , A c4 , A c7 , A c8 } and M = Here, K = {A c c c c {A5 , A6 , A7 , A8 }. c8 },{A c4 , A c8 }, {A c6 , A c8 }, {A c7 , A c8 }, K, L and M It is easy to see that {A c4 , A c8 }, {A c6 , A c8 }, are the only proper subhypergroups of (H, ⊗). Having {A c c {A7 , A8 } isomorphic hypergroups and K, L and M isomorphic hypergroups implies that (H, ⊗) has only 3 proper subhypergroups (up to isomorphism).

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c1 A

c1 A

H

c2 A c3 A

c3 A

c4 A

c5 A

H K

H

H

H

K L

L

c6 A

c7 A

c8 A

H

H

H

H

H

K

H

K

H

H

L

L

c4 A

c4 , A c8 H A

K

L

c4 , A c8 A

c5 A

M

M

M

M

M

c6 , A c8 A

c6 A c7 A c8 A

7.2.5

c2 A

c6 , A c8 A

c7 , A c8 A c7 , A c8 A c8 A

The hypothetical cross of n different traits, case of incomplete dominance

The case of incomplete dominance can be given by: B1 B2 . . . Bn ⊗ B1 B2 . . . Bn with n ≥ 1. We consider first results for the Monohybrid cross (n = 1) that differs in a single trait; a homozygous parent (B1 B1 ) ⊗ a homozygous parent (B1 B1 ). The results of this hypothetical experiment can be summarized in the following way: P: B1 B1 ⊗ B1 B1 F1 : B1 B1 and F1 ⊗ F1 : B1 B1 ⊗ B1 B1 c1 (of genotype B1 B1 ), A c2 (of genotype B1 B1 ), F2 : A c A3 (of genotype B1 B1 ). c1 , A c2 , A c3 }. Then (H, ⊗) is a cyclic Hv Proposition 7.14. Let H = {A semigroup with identity and two idempotent elements. Proof. We present (H, ⊗) in Table 7.35. Since no new elements are present in Table 7.35 and the table is symmetric then (H, ⊗) is a commu2 c2 implies that H is cyclic of period tative hypergroupoid. Having H = A c2 is the unique identity of (H, ⊗) and that A c1 two. It is easy to see that A c3 are idempotents. and A  Remark 7.11. (H, ⊗) is not a hypergroup since H does not appear in the first row of Table 7.35.

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Table 7.35 ⊗

c1 A

c2 A

c3 A

c1 A

c1 A

c1 , A c2 A

c2 A

H

c2 , A c3 A

c2 , A c3 A

c3 A

c2 A c1 , A c2 A c3 A

c2 A

Remark 7.12. An example of the monohybrid case is studied in ([37], p. 183). Proposition 7.15. There is only one hypergroup contained in (H, ⊗) (up to isomorphism). c1 } and {A c3 } are the only hypergroups contained Proof. It is clear that {A in H and they are isomorphic.  We consider now results for the dihybrid cross (n = 2) that differs in 2 traits; a homozygous parent (B1 B1 B2 B2 ) ⊗ a homozygous parent (B1 B1 B2 B2 ). The results of this hypothetical experiment can be summarized in the following way: P: B1 B1 B2 B2 ⊗ B1 B1 B2 B2 F1 : B1 B1 B2 B2 and F1 ⊗ F1 : B1 B1 B2 B2 ⊗ B1 B1 B2 B2 c3 (of c1 (of genotype B1 B1 B2 B2 ), A c2 (of genotype B1 B1 B2 B2 ), A F2 : A c c genotype B1 B1 B2 B2 ), A4 (of genotype B1 B1 B2 B2 ), A5 (of genotype c6 (of genotype B1 B1 B2 B2 ), A c7 (of genotype B1 B1 B2 B2 ), B1 B1 B2 B2 ), A c c A8 (of genotype B1 B1 B2 B2 ) and A9 (of genotype B1 B1 B2 B2 ). c1 , A c2 , . . . , A c9 }. Then (H, ⊗) is a cyclic Proposition 7.16. Let H = {A Hv -group with identity and four idempotent elements. Proof. It is easy to prove that (H, ⊗) is an Hv -semigroup. Having H = 2 c5 implies that H is cyclic of period two. Also, we can easily show that A c5 is the unique identity of (H, ⊗) and that A c1 , A c3 , A c7 and A c9 are the A only idempotents in (H, ⊗).



We consider now results for the n− hybrid cross (n ≥ 3) that differs in n traits; a homozygous parent (B1 B1 B2 B2 . . . Bn Bn ) ⊗ a homozygous parent (B1 B1 B2 B2 . . . Bn Bn ). The results of this hypothetical experiment can be summarized in the following way:

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P: B1 B1 B2 B2 . . . Bn Bn ⊗ B1 B2 B2 . . . Bn Bn F1 : B1 B1 B2 B2 . . . Bn Bn and F1 ⊗ F1 : B1 B1 B2 B2 . . . Bn Bn ⊗ B1 B1 B2 B2 . . . Bn Bn c1 (of genotype B1 B1 B2 B2 . . . Bn Bn ), A c2 (of genotype F2 : A [ B1 B1 . . . Bn−1 Bn−1 Bn Bn ), . . .) A k−1 (of genotype B1 B1 . . . Bn−1 ck (of genotype B1 B1 B2 B2 . . . Bn Bn ). Bn−1 Bn Bn ) and A It is easy to see that we have k = 3n different phenotypes. Let r = cr is of genotype B1 B1 B2 B2 . . . Bn Bn . [1, k] such that A

k+1 2

∈

c1 , A c2 , . . . , A ck } and k = 3n . Then (H, ⊗) Proposition 7.17. Let H = {A is a cyclic Hv -semigroup with identity and 2n idempotent elements. Proof. Let X, Y , Z ∈ H such that x1 x01 . . . xn x0n , y1 y10 . . . yn yn0 , z1 z10 . . . zn zn0 are their corresponding genotypes. Since the phenotype responsible for x1 z10 . . . xn zn0 ∈ ((X ⊗Y )⊗Z)∩(X ⊗(Y ⊗Z)) and no new phe2 cr notypes appear in (H, ⊗) then (H, ⊗) is an Hv -semigroup. Having H = A cr is the unique implies that H is cyclic of period two. It is easy to see that A c identity of (H, ⊗) and that Ai of genotypes x1 y1 x2 y2 . . . xn yn with xi = yi for all i = 1, 2 . . . , n are the only idempotents in (H, ⊗). It can be easily shown that the number of such element is 2n .  7.2.6

The hypothetical cross of m + n different traits, case of simple and incomplete dominance combined together

The case of combination of simple and incomplete dominance can be given by: B1 B2 . . . Bn A1 A2 . . . Am ⊗ B1 B2 . . . Bn a1 a2 . . . am with m, n ≥ 1. We consider first results for the cross (m = n = 1) that differs in two traits; a homozygous parent (B1 B1 A1 A1 ) ⊗ a homozygous parent (B1 B1 a1 a1 ). The results of this hypothetical experiment can be summarized in the following way: P: B1 B1 A1 A1 ⊗ B1 B1 a1 a1 F1 : B1 B1 A1 a1 and F1 ⊗ F1 : B1 B1 A1 a1 ⊗ B1 B1 A1 a1

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c1 (of genotype B1 B1 A1 y1 ), A c2 (of genotype B1 B1 a1 a1 ), A c3 (of F2 : A c4 (of genotype B1 B1 a1 a1 ),A c5 (of genotype B1 genotype B1 B1 A1 y1 ), A c6 (of genotype B1 B1 a1 a1 ). B1 A1 y1 ) and A Where y1 = A1 or a1 . c1 , A c2 , . . . , A c6 }. Then (H, ⊗) is a cyclic Proposition 7.18. Let H = {A Hv -semigroup with identity and two idempotent elements. 2

c3 and that A c3 is an identity of Proof. It is easy to see that H = A c c (H, ⊗). Moreover, A2 and A6 are the only idempotents in (H, ⊗).  c1 , A c2 } and {A c5 , A c6 } are two hypergroups contained in (H, ⊗). {A Example 7.13. M N blood group inheritance with the Rh factor. The blood group can be differentiated by a + and − which refers to a blood group antigen called the Rh factor. In this system, the Rh+ (which can be considered as A1 ) is dominant over the Rh− which can be considered as a1 . Furthermore, in the M N system, there are two antigens M and N , whose production is determined by a gene with two alleles, LM and LN that produce the M and N antigens respectively. Individuals having the LM LM genotype will have only the M antigen on their red cells and will be of type M , those having the LN LN genotype will have only the N antigen on their red cells and will be of type N . Hetrozygous (with LM LN genotype) will have both antigens on their red cells and will be of type M N . After denoting M by B1 and N by B1 , we can apply Proposition 7.18 to this blood system. We consider results for the cross that differs in m + n traits; a homozygous parent (B1 B1 . . . Bn Bn A1 A1 . . . Am Am ) ⊗ a homozygous parent (B1 B1 . . . Bn Bn a1 a1 . . . am am ). The results of this hypothetical experiment can be summarized in the following way: P: B1 B1 . . . Bn Bn A1 A1 . . . Am Am ⊗ B1 B1 . . . Bn Bn a1 a1 . . . am am F1 : B1 B1 . . . Bn Bn A1 a1 . . . Am am and F1 ⊗ F1 : B1 B1 . . . Bn Bn A1 a1 . . . Am am ⊗ B1 B1 . . . Bn Bn A1 a1 . . . Am am c1 (of genotype B1 B1 . . . Bn Bn A1 y1 . . . Am ym ), A c2 (of genotype F2 : A ck (of genotype B1 B1 B1 . . . Bn Bn A1 y1 . . . Am−1 ym−1 am am ), . . . and A B1 . . . Bn Bn a1 a1 . . . am am ). Where k = 2m 3n is the number of different phenotypes and yi = Ai or ai for i = 1, . . . , m.

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c1 , A c2 , . . . , A ck }. Then (H, ⊗) is a cyclic Hv Theorem 7.5. Let H = {A semigroup with identity and 2n idempotent elements. Proof. The proof of (H, ⊗) is an Hv -semigroup is similar to that of Proposition 7.17. cr is the phenotype corresponding to Suppose that r ∈ [1, k] such that A 2 cr and the genotype B1 B1 . . . Bn Bn A1 a1 . . . Am am . It is clear that H = A cr is an identity of (H, ⊗). Moreover, the only idempotents elements of that A (H, ⊗) are those having genotypes of the form x1 x1 . . . xn xn a1 a1 . . . am am where xi = Bi or Bi . It can be easily shown that the number of such elements is 2n .  7.3

Physical examples

In the elementary particle physics, a fundamental particle is known as a particle which have no substructure and it is one of the basic building blocks of the universe from which all other particles are made. Nowadays, the Standard Model (SM) of elementary particles is known to be a well established theory to describe the elementary particles and the interacting forces between them [86]. In the SM, the Quarks, Leptons and Gauge Bosons are introduced as the elementary particles. This model contains six types of quarks, known as flavors: Up, Down, Charm, Strange, Bottom and Top plus their corresponding antiparticles which is known as antiquark. Since the quarks are never found in isolation, therefore quarks combine to form composite particles which are called Hadrons and they appear into two families: Baryons (made of three quarks) and Mesons (made of one quark and one antiquark). In the SM, gauge bosons consist of the photons(γ), gluons(g), W ± and Z bosons act as carriers of the fundamental forces of nature [60]. The third group of the elementary particles are Leptons. In the SM, there are six flavors of leptons along with six antileptons. Leptons are an important part of the SM, especially the electrons which are one of the components of atoms. Since the leptons can be found freely in the universe and they are one of the important groups of the elementary particles. A motivation for the study of hyperstructures comes from physical phenomenon as the nuclear fission. This motivation and the results were presented by S. Hoˇskov´ a, J. Chvalina and P. Raˇckov´a (see [72], [73]). In [48], the authors provided, for the first time, a physical example of hyperstructures associated with the elementary particle physics, Leptons. They have considered this important group of the elementary particles and shown that

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Table 7.36 Leptons classification to three generations. Q stands for charge in unit of the electron charge. Le , Lµ and Lτ stand for the electronic, muonic and tauonic numbers, respectively. Classify Leptons Q Le Lµ Lτ

First Generation e νe e+ νe −1 0 +1 0 1 1 −1 −1 0 0 0 0 0 0 0 0

Second Generation µ νµ µ+ νµ −1 0 +1 0 0 0 0 0 1 1 −1 −1 0 0 0 0

Third Generation τ ντ τ+ ντ −1 0 +1 0 0 0 0 0 0 0 0 0 1 1 −1 −1

this set along with the interactions between its members can be described by the algebraic hyperstructures. The main reference for this section is [48]. 7.3.1

Leptons

In the SM, leptons form three generations. The first generation includes the electronic leptons which are electron (e), electron neutrino (νe ), positron (e+ ) and electron antineutrino (ν e ). The second generation contains the muonic leptons, i.e. muon (µ), muon neutrino (νµ ), antimuon (µ+ ) and muon antineutrino (ν µ ). The third generation comprising the tauonic leptons which are tau (τ ), tau neutrino (ντ ), antitau (τ + ) and tau antineutrino (ν τ ). In the leptons group the electron, muon and tau have the electric charge Q = −1 (the charge of a particle is expressed in unit of the electron charge). According to the definition of antiparticle, the electric charge of positron, antimuon and antitau is Q = +1 but the antineutrinos are neutral as well as neutrinos. The main differences between the neutrinos and antineutrinos are in the other quantum numbers such as leptonic numbers [60, 62]. In the SM, leptonic numbers are assigned to the members of every generation of leptons. Electron and electron neutrino have an electronic number of Le = 1 while muon and muon neutrino have a muonic number of Lµ = 1 and tau and tau neutrino have a tauonic number of Lτ = 1. Antileptons have their respective generation’s leptonic numbers of −1. These numbers are classified in Table 7.36. In every interaction, the leptonic numbers should be conserved. Conservation of the leptonic numbers implies that leptons and antileptons must be created in pairs of a single generation. For example, the following processes are allowed under conservation of the electronic and munic numbers, respectively: e + νe → e + νe µ + νµ → µ + νµ .

(7.1)

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In this work we use a new notation for the outgoing productions. For example, we write down: e + νe → {e, νe }. In other interactions, outgoing particles might be different, therefore all leptonic numbers must be checked. In the following interaction, the conservation of the electronic and muonic numbers implies to have two outgoing modes: e + νµ → e + νµ ,

µ + νe .

(7.2)

According to the introduced notation we write down e + νµ → {e, νµ , µ, νe }. The conservation of the electric charges and the leptonic numbers are required to occur a leptonic interaction. Considering these conservation rules, for the electron-positron interaction the interacting modes are: e + e+ → e + e+ , +

µ + µ+ , +

τ + τ +, +

νe + ν e ,

νµ + ν µ ,

ντ + ν τ .

+

We write: e + e → {e, e , µ, µ , τ, τ , νe , ν e , νµ , ν µ , ντ , ν τ } = L. Other interactions between the members of the leptons group are shown in Table 7.37. To arrange this table we avoided writing the repeated symbols. For example, in the productions of the electron-electron interaction we only write e instead of e + e. All interactions shown in Table 7.37 are in the first order. It means in higher orders other particles can be produced that we do not consider them. For example in the electron-electron scattering (Muller scattering) one or several photons might be appeared in productions of the interaction, i.e. e + e → e + e + γ or in the electron-positron scattering (Bhabha scattering [14]) we can have: e+e+ → e+e+ +γ. There also exist other processes that we do not consider them in this work. For example: e + e+ → γ + γ, e + e+ → W − + W + , τ + τ + → Z 0 + Z 0 and so on. 7.3.2

The algebraic hyperstructure of Leptons

In this section, by considering the definitions presented in Section 2, we investigate that the Leptons along with the interactions arranged in Table 7.37 found a hyperstructure. If we assume, L is the set of Leptons and hyperoperation ⊗ is the Leptonic interactions arranged in Table 7.37, then (L, ⊗) is a commutative Hv -group. In order to show the weak associativity of this hyperstructure, consider the following example: νµ ⊗ (¯ νe ⊗ e+ ) = νµ ⊗ {¯ νe , e+ } = {e+ , µ, ν¯e , νµ }, (νµ ⊗ ν¯e ) ⊗ e+ = {e+ , µ, ν¯e , νµ } ⊗ e+ = {e+ , µ, ν¯e , νµ }.

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Outline of applications and modeling Table 7.37 ⊗

e

νe

+ e

Interaction between leptons are shown. ν ¯e e µ τ ν ¯e ν ¯µ ν ¯τ

νµ

+ µ

e µ

e µ νe νµ

e + µ ν ¯µ νe

e ν ¯µ

µ

ν ¯µ

τ

+

ντ

τ

e τ

e τ νe ντ

e + τ ν ¯τ νe

ν ¯τ

e νe

L

e νe

νe

+ e + µ + τ νe νµ ντ

L

e µ νe νµ

νe νµ

+ µ νe

e + µ ν ¯µ νe

e τ νe ντ

νe ντ

τ

e+

L

+ e + µ + τ νe νµ ντ

+ e

+ e ν ¯e

+ e µ ν ¯e νµ

+ e νµ

+ e + µ

+ e + µ ν ¯e ν ¯µ

+ e τ ν ¯e ντ

+ e ντ

+ e + τ

+ e + τ ν ¯e ν ¯τ

ν ¯e

e µ τ ν ¯e ν ¯µ ν ¯τ

L

+ e ν ¯e

ν ¯e

µ ν ¯e

+ e µ ν ¯e νµ

+ e + µ ν ¯e ν ¯µ

ν ¯e ν ¯µ

τ ν ¯e

+ e τ ν ¯e ντ

+ e + τ ν ¯e ν ¯τ

ν ¯e ν ¯τ

µ

e µ

e µ νe νµ

+ e µ ν ¯e νµ

µ ν ¯e

µ

µ νµ

L

e µ τ ν ¯e ν ¯µ ν ¯τ

µ τ

µ τ νµ ντ

µ + τ ν ¯τ νµ

µ ν ¯τ

νµ

e µ νe νµ

νe νµ

+ e νµ

+ e µ ν ¯e νµ

µ νµ

νµ

+ e + µ + τ νe νµ ντ

L

µ τ νµ ντ

νµ ντ

+ τ νµ

µ + τ ν ¯τ νµ

µ+

e + µ ν ¯µ νe

+ µ νe

+ e + µ

L

+ e + µ + τ νe νµ ντ

+ µ

+ µ ν ¯µ

+ µ τ ν ¯µ ντ

+ µ ντ

+ µ + τ

+ µ + τ ν ¯µ ν ¯τ

ν ¯µ

e ν ¯µ

e + µ ν ¯µ νe

+ e + µ ν ¯e ν ¯µ

ν ¯e ν ¯µ

e µ τ ν ¯e ν ¯µ ν ¯τ

L

ν ¯µ µ+

ν ¯µ

τ ν ¯µ

+ µ τ ν ¯µ ντ

+ µ + τ ν ¯µ ν ¯τ

ν ¯µ ν ¯τ

τ

e τ

e τ νe ντ

+ e τ ν ¯e ντ

τ ν ¯e

µ τ

µ τ νµ ντ

+ µ τ ν ¯µ ντ

τ ν ¯µ

τ

τ ντ

L

e µ τ ν ¯e ν ¯µ ν ¯τ

ντ

e τ νe ντ

νe ντ

+ e ντ

+ e τ ν ¯e ντ

µ τ νµ ντ

νµ ντ

+ µ ντ

+ µ τ ν ¯µ ντ

+

τ ν ¯e ν ¯τ

µ + τ ν ¯τ νµ

ν ¯e ν ¯τ

µ ν ¯τ

e

νe

e

τ+

e + τ ν ¯τ νe

ν ¯τ

e ν ¯τ

νe

+ e + τ

e + τ ν ¯τ νe

+ e + τ ν ¯e ν ¯τ

τ

+

e

+

+ µ ν ¯e ν ¯µ

e

+

τ ντ

ντ

+ e + µ + τ νe

e + τ ν ¯τ νe

L

νµ ντ

τ νµ

+ µ + τ

+ µ + τ ν ¯µ ν ¯τ

L

+ e + µ + τ νe νµ ντ

µ + τ ν ¯τ νµ

+ µ + τ ν ¯µ ν ¯τ

ν ¯µ ν ¯τ

e µ τ ν ¯e ν ¯µ ν ¯τ

L

+

+ νe

e ν ¯τ

+

+ τ ν ¯τ

+ τ ν ¯τ

ν ¯τ

τ

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As it is seen, νµ ⊗(¯ νe ⊗e+ )∩(νµ ⊗ ν¯e )⊗e+ 6= ∅. To investigate the condition of weak associativity of this hyperstructure we used MAPLE 14 software. This property is established for all members. Since, there is an antiparticle for every element in the Leptons set which their interactions can produce all Leptons, then the reproduction axiom holds automatically. In other words: µ ⊗ L = τ ⊗ L = νµ ⊗ L = · · · = L. All the elements of (L, ⊗) are idempotents. In the next step, we summarize some results on this hyperstructure in the following. If (L, ⊗) is the above Hv -group, then the following statements hold: (i) There is not any Hv -subgroups of orders 5, 7, 8, 9, 10 and 11 for (L, ⊗). (ii) All the Hv -subgroups of order 1 (L1i , i = 1, . . . , 12) are: {e}, {e+ }, {µ}, {µ+ }, {τ }, {τ + }, {¯ νe }, {¯ νµ }, {¯ ντ }, {νe }, {νµ }, {ντ }, respectively. (iii) All the Hv -subgroups of order 2 (L2i , i = 1, . . . , 30) are: {e, µ}, {e, τ }, {e, ν¯µ }, {e, ν¯τ }, {e, νe }, {e+ , µ+ }, {e+ , τ + }, {e+ , νµ }, {e+ , ντ }, {µ, τ }, {µ, ν¯e }, {µ, ν¯τ }, {µ, νµ }, {µ+ , τ + }, {µ+ , ν¯µ }, {µ+ , νe }, {µ+ , ντ }, {τ, ν¯e }, {τ, ν¯µ }, {τ, ντ }, {τ + , ν¯τ }, {τ + , νe }, {τ + , νµ }, {¯ νe , ν¯µ }, {¯ νe , ν¯τ }, {¯ νµ , ν¯τ }, {νe , νµ }, {νe , ντ }, {νµ , ντ }, {e+ , ν¯e }, respectively. (iv) All the Hv -subgroups of order 3 (L3i , i = 1, . . . , 16) are: {e, µ, τ }, {e, µ, ν¯τ }, {e, τ , ν¯µ }, {e+ , µ+ , τ + }, {e+ , µ+ , ντ }, {e+ , τ + , νµ }, {e+ , νµ , νe }, {µ, τ , ν¯e }, {µ, ν¯e , ν¯τ }, {µ+ , τ + , νe }, {µ+ , νe , ντ }, {τ , ν¯e , ν¯µ }, {e, ν¯µ , ν¯τ }, {τ + , νe , νµ }, {¯ νe , ν¯µ , ν¯τ }, {νe , νµ , ντ }, respectively. (v) All the Hv -subgroups of order 4 (L4i , i = 1, . . . , 9) are: {e, µ, νe , νµ }, {e, µ+ , ν¯µ , νe }, {e, τ , νe , ντ }, {e, τ + , ν¯τ , νe }, {e+ , µ+ , ν¯e , ν¯µ }, {µ, τ , νµ , ντ }, {µ, τ + , ν¯τ , νµ }, {µ+ , τ , ν¯µ , ντ }, {µ+ , τ + , ν¯µ , ν¯τ }, respectively. (vi) All the Hv -subgroups of order 6 (L6i , i = 1, . . . , 8) are: (1) (2) (3) (4) (5) (6)

L61 L62 L63 L64 L65 L66

= {e+ , µ+ , τ + , νe , νµ , ντ } with the multiplicative Table 7.38. = {e, µ, τ, νe , νµ , ντ } with the multiplicative Table 7.39. = {e, µ, τ + , ν¯τ , νe , νµ } with the multiplicative Table 7.40. = {e, µ+ , τ, ν¯µ , νe , ντ } with the multiplicative Table 7.41. = {e, µ+ , τ + , ν¯µ , ν¯τ , νe } with the multiplicative Table 7.42 = {e+ , µ+ , τ, ν¯e , ν¯µ , ντ } with the multiplicative Table 7.43.

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(7) L67 = {e+ , µ+ , τ + , ν¯e , ν¯µ , ν¯τ } with the multiplicative Table 7.44. (8) L68 = {e, µ, τ, ν¯e , ν¯µ , ν¯τ } with the multiplicative Table 7.45. There exist two classes of non-isomorphic Hv -subgroups of order 6, i.e., ∼ L6 , (1) L61 = 8 (2) L62 ∼ = L63 ∼ = L64 ∼ = L65 ∼ = L66 ∼ = L67 .

Multiplicative table for L61 = {e+ , µ+ , τ + , νe , νµ , ντ }.

Table 7.38 e+

µ+

τ+

e+

e+

e+ , µ +

e+ , τ +

µ+

e+ , µ+

µ+

µ+ , τ +

τ+

τ + , e+

µ+ , τ +

νe

e+ , µ+ , τ + , νe , νµ , ντ

νµ

e+ , νµ

ντ

, e+

ντ

Table 7.39

νµ

ντ

e+ , νµ

e+ , ντ

µ+ , νe

e+ , µ + , τ + , νe , νµ , ν τ

µ+ , ντ

τ+

τ + , νe

τ + , νµ

e+ , µ+ , τ + , νe , νµ , ντ

µ+ , νe

τ + , νe

νe

νe , ν µ

νe , ντ

e+ , µ+ , τ + , νe , νµ , ν τ

νµ , τ +

νµ , ν e

νµ

νµ , ν τ

µ+ , ν

e+ , µ + , τ + , νe , νµ , ν τ

ντ , ν e

ντ , ν µ

ντ

τ

νe e+ , µ+ , τ + , νe , νµ , ν τ

Multiplicative table for L62 = {e, µ, τ, νe , νµ , ντ }.

e

µ

τ

νe

e

e

e, µ

e, τ

e, νe

µ

e, µ

µ

µ, τ

τ

e, τ

τ, µ

τ

νe

e, νe

e, µ, νe , ν µ

e, τ, νe , ν τ µ, τ, νµ , ν τ

νµ ντ

e, µ, νe , ν µ e, τ, νe , ν τ

νµ , µ µ, τ, νµ , ν e

ντ , τ

νµ e, µ, νe , νµ

ντ e, τ, νe , ν τ µ, τ, νµ , ν τ

e, µ, νe , ν µ e, τ, νe , ν τ

µ, τ, νµ , ν τ

τ, ντ

νe

νe , νµ

νe , ν τ

νµ , ν e

νµ

νµ , ν τ

ντ , ν e

ντ , ν µ

ντ

µ, νµ

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A Walk Through Weak Hyperstructures: Hv -Structures

Multiplicative table for L63 = {e, µ, τ + , ν¯τ , νe , νµ }.

Table 7.40 e

µ

e

e

e, µ

µ

e, µ

µ

e, τ + ,

µ, τ +

ν¯τ , νe

ν¯τ , νµ

ν¯τ

e, ν¯τ

νe νµ

τ+

ν¯τ

νe

e, ν¯τ

e, νe

µ, ν¯τ

e, µ, νe , ν µ

µ, νµ

τ+

τ + , ν¯τ

τ + , νe

τ + , νµ

ν¯τ , µ

ν¯τ , τ +

ν¯τ

e, τ + , ν¯τ , νe

µ, τ + , ν¯τ , νµ

e, νe

e, µ, νe , ν µ

τ + , νe

νe

νe , νµ

e, µ, νe , ν µ

νµ , µ

νµ , τ +

νµ , ν e

νµ

µ+ e, µ+ , ν¯µ , νe

e e

e

µ+

e, µ+ , ν¯µ , νe

τ

ν¯µ

νe

e, τ

e, ν¯µ

e, νe

µ+

µ+ , ν¯µ

µ+ , νe

e, τ

µ+ , τ, ν¯µ , ντ

τ

τ, ν¯µ

ν¯µ

e, ν¯µ

ν¯µ , µ+

ν¯µ , τ

ν¯µ

νe

νe , e

νe , µ +

e, τ, νe , ντ

ντ

e, τ, νe , ν τ

µ+ , ντ

ντ , τ

e, µ+ , ν¯µ , νe µ+ , τ, ν¯µ , ντ

e

τ+

τ

µ+ , τ ν¯µ , ντ

Table 7.42

µ+

e, τ + , ν¯τ , νe µ, τ + , ν¯τ , νµ

e e, µ+ , ν¯µ , νe e, τ + , ν¯τ , νe

µ+ , ντ

µ+ , τ, ν¯µ , ντ

νe

νe , ν τ

ντ , ν e

ντ

τ, ντ

Multiplicative table for L65 = {e, µ+ , τ + , ν¯µ , ν¯τ , νe }. µ+ e, µ+ , ν¯µ , νe

τ+ e, τ + , ν¯τ , νe

ν¯µ e, ν¯µ

ν¯τ

νe

e, ν¯τ

e, νe

µ+ , τ + ,

µ+

µ+ , τ +

µ+ , ν¯µ

τ + , µ+

τ+

µ+ , τ + , ν¯µ , ν¯τ

τ + , ν¯τ

ν¯µ , ν¯τ

ν¯µ , e

ν¯µ , µ+

µ+ , τ + , ν¯µ , ν¯τ

ν¯µ

ν¯µ , ν¯τ

ν¯τ

ν¯τ , e

µ+ , τ + , ν¯µ , ν¯τ

ν¯τ , τ +

ν¯µ , ν¯τ

ν¯τ

e, µ+ ,

e, τ + , ν¯τ , νe

νe , e

ντ e, τ, νe , ν τ

e, τ, νe , ν τ e, µ+ , ν¯µ , νe

ν¯µ

νe

νµ e, µ, νe , νµ

Multiplicative table for L64 = {e, µ+ , τ, ν¯µ , νe , ντ }.

Table 7.41

e

τ+ e, τ + , ν¯τ , νe µ, τ + , ν¯τ , νµ

νe , µ+

νe , τ +

ν¯µ , νe

µ+ , νe τ + , νe e, µ+ , ν¯µ , νe e, τ + , ν¯τ , νe νe

Davvaz-Vougiouklis

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Outline of applications and modeling

Multiplicative table for L66 = {e+ , µ+ , τ, ν¯e , ν¯µ , ντ }.

Table 7.43

e+

e+

µ+

e+

e+ , µ +

µ+

µ+ , e +

µ+

τ

e+ , τ, ν¯e , ντ

ν¯e

ν¯e , e+

µ+ , τ, ν¯µ , ντ e+ , µ + , ν¯e , ν¯µ

ν¯µ

e+ , µ + , ν¯e , ν¯µ

ντ

ντ , e+

Table 7.44

ν¯µ , µ+ ντ , µ+

τ e+ , τ, ν¯e , ντ µ+ , τ, ν¯µ , ντ

e+ , µ + , ν¯e , ν¯µ

µ+ , ν¯µ

µ+ , ντ

τ

τ, ν¯e

τ, ν¯µ

τ, ντ

ν¯e , τ

ν¯e

ν¯e , ν¯µ

ν¯µ , τ

ν¯µ , ν¯e

ν¯µ

e+ , τ,

µ+ , τ,

ν¯e , ντ

ν¯µ .ντ

ντ , τ

ν¯e e+ , ν¯e

µ+

τ+

ν¯e

e+

e+

e+ , µ +

e+ , τ +

e+ , ν¯e

µ+

µ+ , e +

µ+

µ+ , τ +

τ+

τ + , e+

τ + , µ+

τ+

ν¯e

ν¯e , e+

e+ , µ+ , ν¯e , ν¯µ

e+ , τ + , ν¯e , ν¯τ µ+ , τ + , ν¯µ , ν¯τ

ν¯µ ν¯τ

ντ e+ , ν

τ

e+ , τ, ν¯e , ντ µ+ , τ, ν¯µ .ντ ντ

Multiplicative table for L67 = {e+ , µ+ , τ + , ν¯e , ν¯µ , ν¯τ }.

e+

e+ , µ+ , ν¯e , ν¯µ e+ , τ + , ν¯e , ν¯τ

ν¯µ e+ , µ + , ν¯e , ν¯µ

ν¯µ , µ+ µ+ , τ + , ν¯µ , ν¯τ

ν¯τ , τ +

ν¯µ e+ , µ+ , ν¯e , ν¯µ

ν¯τ e+ , τ + , ν¯e , ν¯τ µ+ , τ + , ν¯µ , ν¯τ

e+ , µ+ , ν¯e , ν¯µ e+ , τ + , ν¯e , ν¯τ

µ+ , τ + , ν¯µ , ν¯τ

τ + , ν¯τ

ν¯e

ν¯e , ν¯µ

ν¯e , ν¯τ

ν¯µ , ν¯e

ν¯µ

ν¯µ , ν¯τ

ν¯τ , ν¯e

ν¯τ , ν¯µ

ν¯τ

µ+ , ν¯µ

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Table 7.45

Multiplicative table for L68 = {e, µ, τ, ν¯e , ν¯µ , ν¯τ }.

e

µ

τ

e

e

e, µ

e, τ

µ

e, µ

µ

µ, τ

τ

τ, e

µ, τ

ν¯e

e, µ, τ, ν¯e , ν¯µ , ν¯τ

ν¯µ

ν¯τ

ν¯e e, µ, τ, ν¯e , ν¯µ , ν¯τ

ν¯µ

ν¯τ

e, ν¯µ

e, ν¯τ

µ, ν¯e

e, µ, τ, ν¯e , ν¯µ , ν¯τ

µ, ν¯τ

τ

τ, ν¯e

τ, ν¯µ

e, µ, τ, ν¯e , ν¯µ , ν¯τ

µ, ν¯e

τ, ν¯e

ν¯e

ν¯e , ν¯µ

ν¯e , ν¯τ

e, ν¯µ

e, µ, τ, ν¯e , ν¯µ , ν¯τ

ν¯µ , τ

ν¯µ , ν¯e

ν¯µ

ν¯µ , ν¯τ

ν¯τ , e

µ, ν¯τ

e, µ, τ, ν¯e , ν¯µ , ν¯τ

ν¯τ , ν¯e

ν¯τ , ν¯µ

ν¯τ

Davvaz-Vougiouklis

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Bibliography

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[95] R.M. Santilli, Isotopic, genotopic and hyperstructural liftings of Lie’s theory and their isoduals, Algebras, Groups and Geometries, 15 (1998), 473-498. [96] R.M. Santilli, Hadronic Mathematics, Mechanics and Chemistry, Volumes I, II, III, IV and V, International Academic Press, USA, 2007. [97] R.M. Santilli and T. Vougiouklis, Isotopies, genotopies, hyperstructures and their applications, New frontiers in Hyperstructures, Hadronic, (1996), 1-48. [98] R.M. Santilli and T. Vougiouklis, Lie-admissible hyperalgebras, Italian J. Pure Appl. Math., 31 (2013), 239-254. [99] R.M. Santilli and T. Vougiouklis, Hyperstructures in Lie-Santilli admissibility and iso-theories, Ratio Matematica, 33 (2017), 151-165. [100] M.S. Sethi and M. Satake, Chemistry of Transition Elements, South Asian publishers, New Delhi, 1988. [101] D.A. Skoog, D.M. West and F.J. Hollers, Fundamentals of Analytical Chemistry, 5th edit. (1988), p 282. [102] S. Spartalis, On reversible Hv -groups, Algebraic hyperstructures and applications (Iasi, 1993), 163-170, Hadronic Press, Palm Harbor, FL, (1994). [103] S. Spartalis, On the number of Hv -rings with P-hyperoperations, Combinatorics (Acireale, 1992), Discrete Math. 155 (1996), no. 1-3, 225-231. [104] S. Spartalis, Homomorphisms on (H, R)-Hv -rings, Sixth Int. Congress on AHA (1996), Prague. Check Rep., Democ. Univ. Press, 133-138. [105] S. Spartalis, Quotients of P -Hv -rings, New frontiers in hyperstructures (Molise, 1995), 167-177, Ser. New Front. Adv. Math. Ist. Ric. Base, Hadronic Press, Palm Harbor, FL, 1996. [106] S. Spartalis, A. Dramalides and T. Vougiouklis, On Hv -group rings, Algebra Group Geom., 15(1) (1998), 47-54. [107] M.S. Tallini, Hypervector spaces, Proc. Fourth Int. Congress on Algebraic Hyperstructures and Applications (AHA 1990), World Scientific, (1991) 167-174. [108] [108] R.H. Tamarin, Principles of Genetics, Seventh Edition, The McGrawHill Companies, 2001. [109] Y. Vaziri, M. Ghadiri and B. Davvaz, The M[-] and -[M] functors and five short lemma in Hv -modules, Turk. J. Math., 40 (2016), 397-410. [110] Y. Vaziri and M. Ghadiri, Schanuel’s lemma, the snake lemma, and product and direct sum in Hv -modules, Turk. J. Math., 41 (5), 1121-1132. [111] S. Vougioukli, Hv -vector spaces from helix hyperoperations, Int. J. Math. Anal. N. S., 1(2) (2009), 109-120. [112] T. Vougiouklis, Cyclicity in a special class of hypergroups, Acta Univ. Carolin.Math. Phys., 22(1) (1981), 3-6. [113] T. Vougiouklis, Representations of hypergroups. Hypergroup algebra, Hypergroups, other multivalued structures and their applications (Italian) (Udine, 1985), 59-73, Univ. Studi Udine, Udine, 1985. [114] T. Vougiouklis, Generalization of P -hypergroups, Rend. Circ. Mat. Palermo (2), 36(1) (1987), 114-121. [115] T. Vougiouklis, Representations of hypergroups by hypermatrices, Riv. Mat. Pura Appl., 2 (1987), 7-19.

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[116] T. Vougiouklis, Groups in hypergroups, Annals Discrete Math., 37 (1988), 459-468. [117] T. Vougiouklis, The very thin hypergroups and the S-construction, Combinatorics’88, Vol. 2 (Ravello, 1988), 471-477, Res. Lecture Notes Math., Mediterranean, Rende, 1991. [118] T. Vougiouklis, The fundamental relation in hyperrings. The general hyperfield, Algebraic hyperstructures and applications (Xanthi, 1990), 203-211, World Sci. Publishing, Teaneck, NJ, 1991. [119] T. Vougiouklis, Hv -vector spaces, Algebraic hyperstructures and applications (Iasi, 1993), 181-190, Hadronic Press, Palm Harbor, FL, 1994. [120] T. Vougiouklis, Hyperstructures and Their Representations, Hadronic Press Monographs in Mathematics. Hadronic Press, Inc., Palm Harbor, FL, 1994. [121] T. Vougiouklis, A new class of hyperstructures, J. Combin. Inform. System Sci., 20(1-4) (1995), 229-235. [122] T. Vougiouklis, Hv -groups defined on the same set, Discrete Math., 155 (1996) 259-265. [123] T. Vougiouklis, On Hv -fields, Proc. of 6th AHA Congress, Prague, 1996, Democritus Un. Press (1997), 151-159. [124] T. Vougiouklis, Enlarging Hv -structures, Algebras and Combinatorics (Hong Kong, 1997), 455-463, Springer, Singapore, 1999. [125] T. Vougiouklis, On Hv -rings and Hv -representations, Combinatorics (Assisi, 1996). Discrete Math., 208/209 (1999), 615-620. [126] T. Vougiouklis, On hyperstructures obtained by attaching elements, Constantin Carath´eodory in his . . . origins (Vissa-Orestiada, 2000), 197-206, Hadronic Press, Palm Harbor, FL, 2001. [127] T. Vougiouklis, A hyperoperation defined on a groupoid equipped with a map, Ratio Mathematica, 1 (2005), 25-36. [128] T. Vougiouklis, The ∂ hyperoperation, Structure elements of hyperstructures, 53–64, Spanidis, Xanthi, 2005. [129] T. Vougiouklis, A hyperoperation defined on a groupoid equipped with a map, Ration Matematica, 1 (2005), 25-36. [130] T. Vougiouklis, The hyper theta-algebras, Advances in Algebra, 1 (2008), 67-78. [131] T. Vougiouklis, Hv -fields and Hv -vector spaces with ∂-operations, 6th Pan. Conf. Algebra, Number Theory, Thessaloniki, Greece, (2006), 95-102. [132] T. Vougiouklis, ∂-operations and Hv -fields, Acta Math. Sinica, 24(7) (2008), 1067-1078. [133] T. Vougiouklis, The relation of the theta-hyperoperation (∂) with the other classes of hyperstructures, Journal of Basic Science, 4(1) (2008), 135-145. [134] T. Vougiouklis, Bar and Theta Hyperoperations, Ratio Mathematica, 21 (2011), 27-42. [135] T. Vougiouklis, The e-hyperstructures, Journal of Mahani Mathematical Research Center, 1 (2012), 13-28. [136] T. Vougiouklis, The Lie-hyperalgebras and their fundamental relations, Southeast Asian Bull. Math., 37(4) (2013), 601-613. [137] T. Vougiouklis, Hv -fields, h/v-fields, Ratio Matematica, 33 (2017), 181-201.

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[138] T. Vougiouklis and P. Kambaki, Algebraic models in applied research, Jordan Journal of Mathematics and Statistics (JJMS), 1(1) (2008), 81-90. [139] T. Vougiouklis and S. Vougiouklis, The helix hyperoperations, Ital. J. Pure Appl. Math., 18 (2005), 197-206. [140] M. Yazer, M. Olsson and M. Palcic, The cis-AB blood group phenotype: fundamental lessons in glycobiology, Transfus Med. Rev., 20(3) (2006), 20717. [141] Zadeh, L.A., Fuzzy sets, Inform. Control., 8 (1965), 338-353. [142] S. Zumdahl, Chemistry, Seventh edition, Houghton Mifflin Compan, New York, 2007.

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Index

(H, R) − Hv -ring, 160 Hv -Lie algebra, 240 Hv -field, 124, 139 small non-degenerate, 141 Hv -group, 69 Hb -group, 72 cyclic, 71 feebly quasi-canonical feebly canonical, 81 minimal, 75 reversible left, 81 right, 81 single-power cyclic with infinite period, 71 Hv -group ring, 173 Hv -homomorphism, 122, 180 strong, 180, 193 Hv -ideal, 166 bilaterally, 180 fuzzy left, 183 regular, 190 right, 183 left, 180 right, 180 Hv -isolated ideal, 167 Hv -matrix, 263 represention, 263 Hv -module, 193 of fractions, 196 Hv -near ring, 178

distributive, 178 left, 178 right, 178 zero-symmetric, 178 Hv -prime ideal, 168 Hv -ring, 115 commutative, 115 dual, 115 dual, 115 multiplicative, 125, 135 of fractions, 165 Hv -semi-near-ring, 157 Hv -semigroup, 69 left reproductive, 107 right reproductive, 107 Hv -structure very thin, 139 Hv -subgroup, 70 Hv -vector space, 237 P -hyperalgebra general matrix, 266 P -hyperoperation, 45, 244 generalization, 251 Pe -hyperoperation, 252 R-hypermodule, 61 R-module, 34 homomorphism, 37 unitary, 35 S-hyperoperation S-construction, 176 b-∂-hyperoperation, 154 e-hyper-Lie-algebra, 247 329

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330

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e-hyperalgebra, 246 e-hyperfield, 243 e-hypergroupoid, 241 e-hypermatrix, 245 e-hypernumber, 243 component, 245 e-hyperstructure, 240 e-hypervector, 245 e-hypervector space, 245 n-dimentional, 246 f -partition, 156 h/v-field attached, 140 h/v-group, 139 h/v-matrix, 263 represention, 263 t-level cut lower, 229 upper, 229 addition, 17 admissible Jordan, 257 algebra, 254 helix-Lie, 268 Jordan, 256 associative, 1 Θ-weak, 171 general, 5 weak associative, 69 attach, 140 automorphism, 29, 71 basis, 42 chain initiating, 272 propagating, 273 closed subset multiplicatively strong, 163 commutative strongly commutative, 69 weak commutative, 69 comproportionation, 275 congruent to, 8

convolution, 172 core, 182 coset left, 8 right, 8 cut-addition, 132 cut-helix matrix, 267 cut-multiplication, 133 cut-projective, 267 cyclic single-power, 71 dihybrid cross pea plant, 291 dimensional finite, 42 direct limit, 200 direct product external, 189 direct system, 200 dismutation, 275 disproportionation, 275 distributive strongly, 57 division ring, 22 electrolytic cell, 284 element attach, 140 idempotent, 71 infinite order, 9 inverse, 241 left absorbing-like, 44 order, 9 right absorbing-like, 44 single, 79, 139 zero, 17, 44 endomorphism, 29 epimorphism, 29, 37 equations chemical, 271 exact, 40 sequence, 40 factor through, 38 field, 23

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Index

four o’clock flower, 292 functor, 203 natural, 203 fuzzy Hv -homomorphism, 103 Hv -subgroup, 96 Hv -submodule, 225 left reproduction axiom, 96 right reproduction axiom, 96 anti Hv -subgroup, 99 closed, 99 grade, 94 inclusion homomorphism, 103 left closed, 99 max, 186 right closed, 99 strong homomorphism, 103 subgroup, 95 submodule, 225 subset, 94 Galois Field, 23 Galvanic cell, 281 general enlargement, 251 genofield on the left, 250 genotopies, 250 genotype second generation, 303 good representation, 263 group, 4 abelian, 6 automorphism, 16 commutative, 6 cyclic, 6 fundamental, 54 homomorphic, 13 homomorphism, 13 kernel, 13 inner automorphism, 16

integers modulo n, 6 isomorphic, 13 isomorphism, 13 order, 6 product, 9 quaternion, 6 quotient, 11 group ring, 20 groupoid, 1 halogens, 273 heart, 194 height, 102 helix-addition, 133 helix-multiplication, 268 helix-projection, 132 helix-sum, 133 homomorphism, 37, 50, 67 Hv -homomorphism, 70 good, 60 good homomorphism, 50 inclusion, 122 inclusion homomorphism, 50, 70 natural, 29 strong, 122 strong homomorphism, 71 weak, 122 weak homomorphism, 70 hyperalgebra, 254 Jordan, 256 hypergroup, 46 P -hypergroup, 46 canonical, 55 heart, 54 regular, 46 total, 46 hypergroupoid, 43 Hv -algebra, 173 hyperideal, 56, 59 left, 56 right, 56 hypermodule, 61 generated, 63 multiplicative, 63 quotient, 63 hyperoperation

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332

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A Walk Through Weak Hyperstructures: Hv -Structures

P -hyperoperation, 145 S-hyperoperation, 176 boundary, 92 braket, 240 greater than, 71 noise, 107 smaller than, 71 total, 176 hyperproduct, 80, 92 semidirect, 81 hyperring, 60 Krasner, 55 commutative, 55 with unit element, 55 multiplicative, 57 unitary, 58 hyperstructure very thin, 243 ideal, 26 fuzzy, 183 left, 183 right, 183 maximal, 31 prime, 32 proper, 26 trivial, 26 identity, 1, 5, 44 weak, 57 image, 38, 95 inverse, 95 inclusion associativity, 112 on the left, 112 on the right, 112 inclusion representation, 263 incomplete dominance, 308 inducing topology, 250 Inheritance, 289 inheritance ABO blood group, 294 M N blood group, 296 coat color of shorthorn cattle, 293 non-Mendelian, 299 integral domain, 22

intuitionistic fuzzy Hv -submodule, 227 intuitionistic fuzzy set, 226 intuitionistic fuzzy submodule, 227 inverse, 5, 44 inverse class, 140 isoP -Hv -number, 252 isomorphic, 29, 122, 193 isomorphism, 29, 71, 180 isotopically related, 249 Isotopies, 249 isotopy, 249 kernel, 30, 38, 181, 194 Leptons, 313 Lie hyperalgebra, 254 Lie-admissible, 256 linear combination, 41 linearly dependent, 42 linearly independent, 42, 239 map canonical, 11 characteristic-like, 131 cut-projection, 131 mapping natural, 168 membership degree, 94 function, 94 module cyclic, 36 quotient, 35 sum, 36 zero, 35 monohybrid cross pea plant, 290 monoid, 1 monomorphism, 37 multiplication, 17 near-ring fundamental, 182 operation

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333

Index

binary, 1 period of generator, 71 polynomial, 18 preserving the topology, 249 produce Cartesian diagonal, 119 product Cartesian, 189 direct, 2 quasihypergroup, 46 radical, 33 reactions chain, 271 dismutation, 274 redox, 277 relation ε, 193 chain, 180 equivalence fundamental, 54 regular, 51 regular on the left, 51 regular on the right, 51 strongly regular, 51 strongly regular on the left, 51 strongly regular on the right, 51 fundamental, 68, 76, 124, 135, 182, 193 reproduction axiom, 46 reversible left completely, 82 right completely, 82 ring, 17 Boolean, 18 characteristic, 24 commutative, 17 fundamental, 124 homomorphism, 29 of fractions, 19 of Gaussian integers, 18

Davvaz-Vougiouklis

of power series, 19 of real quaternions, 21 quotient, 28 with unit element, 17 semigroup, 1 bicyclic, 2 commutative, 1 finite, 1 semihypergroup, 43 commutative, 43 finite, 43 subsemihypergroup, 47 sequence exact, 195 spliti, 225 set of fundamental maps, 171 partial identities, 50 skew field, 22 subgroup, 7 cyclic, 7 generated by a subset, 8 image, 14 index, 8 inverse image, 14 maximal, 8 normal, 10 proper, 7 trivial, 7 subhypergroup, 47 closed, 48 closed on the left, 48 closed on the right, 48 conjugable, 48 conjugable on the left, 48 conjugable on the right, 48 invertible, 48 invertible on the left, 48 invertible on the right, 48 ultraclosed, 48 ultraclosed on the left, 48 ultraclosed on the right, 48 weak normal, 180 subhypermodule, 62, 63

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normal, 62, 63 subhyperring, 59 normal, 56 submodule, 36 cyclic, 36 generated by, 36 subset multiplicative, 19 subspace, 41 sup property, 231 support, 102 table Cayley, 2 multiplication, 2

unit scalar, 115 unit class, 140 uniting elements, 130 unitize pair of matrices, 266 vector space, 40 weak distributive, 115 epic, 195 equal, 195 isomorphism, 195 monic, 195 zero-divisor, 22

union ∂-hyperoperation, 154

Davvaz-Vougiouklis