POLYGROUP THEORY AND RELATED SYSTEMS 9814425303, 9789814425308

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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

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

POLYGROUP THEORY AND RELATED SYSTEMS Copyright © 2013 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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ISBN 978-981-4425-30-8

Printed in Singapore.

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Preface

The theory of groups is the oldest branch of ordinary algebra. The concept of a hypergroup which is a generalization of the concept of a group, first was introduced by Marty. Indeed, hypergroups represent a natural extension of groups. In a group, the composition of two elements is an element, while in a hypergroup, the composition of two elements is a set. Application of hypergroups have mainly appeared in special subclasses. For example, polygroups which are certain subclass of hypergroups are used to study color algebra and combinatorics. The idea to write this book, and more important the desire to do so, is a direct outgrowth of a course I gave in the department of mathematics at Yazd University. One of my main aims was to present an introduction to this recent progress in the theory of polygroups and related systems. I have tried to keep the preliminaries down to a bare minimum. The book covers most of the mathematical ideas and techniques required in the study of polygroups. The presented book is composed by five chapters. The first chapter contains a fairly detailed discussion of the basic ideas underlying the theory of groups. The second chapter is about hypergroups, their history and some basic results on some important classes of hypergroups such as complete hypergroups, join spaces and canonical hypergroups. The following chapters are about polygroups. In third chapter, the concept of polygroups and some examples are presented. Then, fundamental relations defined on polygroups, isomorphism theorems, permutation polygroups, polygroup hyperrings, solvable polygroups and nilpotent polygroups are studied. The fourth chapter deal with the concepts of weak algebraic hyperstructures; and the notions of Hv -groups (weak hypergroups) and weak polygroups are studied. In the last chapter, we present some combinatorial aspects of

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polygroups such as chromatic polygroups. A large number of people have influenced the writing of this book. I wish to thank Professor Comer, Professor Corsini, Professor De Salvo, Professor Freni, Professor Jantosciak, Professor Leoreanu-Fotea, Professor Spartalis, Professor Vougiouklis, Professor Zahedi and my PhD students who helped me directly or indirectly. The final word belongs to my wife. She deserves an accolade for her patient during the (seemingly interminable) time the book was being written. Finally may I say that I hope that (most days) you enjoy reading this book as much as I (most days) enjoyed writing it.

Bijan Davvaz Department of Mathematics Yazd University, Yazd, Iran

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Contents

Preface

v

1.

1

A Brief Excursion into Group Theory 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

2.

. . . . . . examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

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Hypergroups 2.1 2.2 2.3 2.4 2.5 2.6 2.7

3.

Introduction . . . . . . . . . . . . . . . . . . The abstract definition of a group and some Subgroups . . . . . . . . . . . . . . . . . . . Normal subgroups and quotient groups . . . Group homomorphisms . . . . . . . . . . . Permutation groups . . . . . . . . . . . . . Direct product . . . . . . . . . . . . . . . . Solvable and nilpotent groups . . . . . . . .

33

Introduction and historical development of hypergroups Definition and examples of hypergroups . . . . . . . . . Some kinds of subhypergroups . . . . . . . . . . . . . . Homomorphisms of hypergroups . . . . . . . . . . . . . Regular and strongly regular relations . . . . . . . . . . Complete hypergroups . . . . . . . . . . . . . . . . . . . Join spaces . . . . . . . . . . . . . . . . . . . . . . . . .

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Polygroups 3.1 3.2 3.3 3.4

1 3 7 11 13 18 22 24

33 38 40 43 52 60 73 83

Definition and examples of polygroups . Extension of polygroups by polygroups . Subpolygroups and quotient polygroups Isomorphism theorems of polygroups . . vii

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83 86 89 95

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3.5 3.6 3.7 3.8 3.9 3.10 3.11 4.

γ ∗ relation on polygroups . . Generalized permutations . . Permutation polygroups . . . Representation of polygroups Polygroup hyperrings . . . . . Solvable polygroups . . . . . Nilpotent polygroups . . . . .

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Weak Polygroups 4.1 4.2 4.3 4.4

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Weak hyperstructures . . . . . . . . . . . . . . . . Weak polygroups as a generalization of polygroups Fundamental relations on weak polygroups . . . . Small weak polygroups . . . . . . . . . . . . . . . .

Combinatorial Aspects of Polygroups 5.1 5.2 5.3

98 105 107 112 119 123 131 139

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139 148 153 156 163

Chromatic polygroups . . . . . . . . . . . . . . . . . . . . 163 Polygroups derived from cogroups . . . . . . . . . . . . . 175 Conjugation lattice . . . . . . . . . . . . . . . . . . . . . . 180

Bibliography

189

Index

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Chapter 1

A Brief Excursion into Group Theory

1.1

Introduction

The concept of a group is one of the most fundamental in modern mathematics. Group theory can be considered the study of symmetry: the collection of symmetries of some object preserving some of its structure forms a group; in some sense all groups arise this way. Although permutations had been studied earlier, the theory of groups really began with Galois (1811-1832) who demonstrated that polynomials are best understood by examining certain groups of permutations of their roots. Since that time, groups have arisen in almost every branch of mathematics. There are three historical roots of group theory: (1) The theory of algebraic equations; (2) Number theory; (3) Geometry. Euler, Gauss, Lagrange, Abel and Galois were early researchers in the field of group theory. Galois is honored as the first mathematician linking group theory and field theory, with the theory that is now called Galois theory. Permutations were first studied by Lagrange (1770, 1771) on the theory of algebraic equations. Lagrange’s main object was to find out why cubic equations could be solved algebraically. In studying the cubic, for example, Lagrange assumes the roots of a given cubic equation are x0 , x00 and x000 . Then, taking 1, w, w2 as the cube roots of unity, he examines the expression R = x0 + wx00 + w2 x000 and notes that it takes just two different values under the six permutations of the roots x0 , x00 , x000 . But, he could not fully develop this insight because he viewed permutations only as rearrangements, and not as bijections that 1

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can be composed. Composition of permutations does appear in work of Ruffini and Abbati about 1800; in 1815 Cauchy established the calculus of permutations. Galois found that if r1 , r2 , . . . , rn are the n roots of an equation, there is always a group of permutations of the r’s such that every function of the roots invariable by the substitutions of the group is rationally known, and conversely, every rationally determinable function of the roots is invariant under the substitutions of the group. Galois also contributed to the theory of modular equations and to that of elliptic functions. His first publication on the group theory was made at the age of eighteen (1829), but his contributions attracted little attention until the publication of his collected papers in 1846. The number-theoretic strand was started by Euler and taken up by Gauss, who developed modular arithmetic and considered additive and multiplicative groups related to quadratic fields. Indeed, in 1761 Euler studied modular arithmetic. In particular he examined the remainders of powers of a number modulo n. Although Euler’s work is, of course, not stated in group theoretic terms he does provide an example of the decomposition of an abelian group into cosets of a subgroup. He also proves a special case of the order of a subgroup being a divisor of the order of the group. Gauss in 1801 was to take Euler’s work much further and gives a considerable amount of work on modular arithmetic which amounts to a fair amount of theory of abelian groups. He examines orders of elements and proves (although not in this notation) that there is a subgroup for every number dividing the order of a cyclic group. Gauss also examined other abelian groups. He looked at binary quadratic forms ax2 + 2bxy + cy 2 where a, b, c are integers. Gauss examined the behavior of forms under transformations and substitutions. He partitions forms into classes and then defines a composition on the classes. Gauss proves that the order of composition of three forms is immaterial so, in modern language, the associative law holds. In fact Gauss has a finite abelian group and later (in 1869) Schering, who edited Gauss’s works, found a basis for this abelian group. Geometry has been studied for a very long time so it is reasonable to ask what happened to geometry at the beginning of the 19th Century that was to contribute to the rise of the group concept. Geometry had began to lose its metric character with projective and non-euclidean geometries being studied. Also the movement to study geometry in n dimensions led

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to an abstraction in geometry itself. The difference between metric and incidence geometry comes from the work of Monge, his student Carnot and perhaps most importantly the work of Poncelet. Non-euclidean geometry was studied by Lambert, Gauss, Lobachevsky and J´anos Bolyai among others. M¨ obius in 1827, although he was completely unaware of the group concept, began to classify geometries using the fact that a particular geometry studies properties invariant under a particular group. Steiner in 1832 studied notions of synthetic geometry which were to eventually become part of the study of transformation groups. Arthur Cayley and Augustin Louis Cauchy were among the first to appreciate the importance of the theory, and to the latter especially are due a number of important theorems. The subject was popularized by Serret, Camille Jordan and Eugen Netto. Other group theorists of the nineteenth century were Bertrand, Charles Hermite, Frobenius, Leopold Kronecker, and Emile Mathieu. It was Walther von Dyck who, in 1882, gave the modern definition of a group. The study of what are now called Lie groups, and their discrete subgroups, as transformation groups, started systematically in 1884 with Sophus Lie; followed by work of Killing, Study, Schur, Maurer, and Cartan. The discontinuous (discrete group) theory was built up by Felix Klein, Lie, Poincar´e, and Charles Emile Picard, in connection in particular with modular forms. Other important mathematicians in this subject area include Emil Artin, Emmy Noether, Sylow, and many others.

1.2

The abstract definition of a group and some examples

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

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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,     a a b b A= 1 2 and B = 1 2 . a3 a4 b3 b4 Then, the product of A and B is also a 2×2 matrix. This product is defined as   a1 b1 + a2 b3 a1 b2 + a2 b4 AB = . a3 b1 + a4 b3 a3 b2 + a4 b4 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 1.2.1. Let G be a non-empty set together with a binary operation (usually called multiplication) that assigns to each ordered pair (a, b) of elements of G an element ab 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: (ab)c = a(bc). (2) For two arbitrary elements a and b, there exists x and y of G which satisfy the equations ax = b and ya = b. The following properties of a group are important. Theorem 1.2.2. (20 ) There is a unique element e in G such that for all g ∈ G, ge = eg = g.

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(200 ) For any element a ∈ G, there is a unique element a0 ∈ G such that aa0 = a0 a = e, where e is the element of G defined in (20 ). (3) The solutions x and y of the equations ax = b and ya = b are unique and we have x = a0 b and y = ba0 , 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 ax = a and ya = a. Also, if g is an arbitrary element of G, there are elements u and v of G such that au = g = va; so we have ge = (va)e = v(ae) = va = g. The second equality, (va)e = v(ae), follows from the associative law. Similarly, we obtain e0 g = g. Since the element g is arbitrary, we may take g = e to obtain e0 e = e. On the other hand, the element e satisfies ge = 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 ax = a is equal to a solution of ya = 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 aa0 = e = a00 a. Using (1) and (20 ), we have a00 = a00 e = a00 (aa0 ) = (a00 a)a0 = ea0 = 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 ax = b, then the left multiplication of a0 of (200 ) gives us a0 b = a0 (ax) = (a0 a)x = ex = x. Thus, a solution of ax = b is x = a0 b, and it is unique; similarly, the solution of ya = b is uniquely determined to be y = ba0 .  Corollary 1.2.3. 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 1.2.4. We have (a−1 )−1 = a and (ab)−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 (ab)(b−1 a−1 ) = ((ab)b−1 )a−1 = (a(bb−1 ))a−1 = (ae)a−1 = aa−1 = e and the uniqueness of the inverse. Notice the change in the order of the factors from ab to b−1 a−1 . 

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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, (xy)((z(uv))w) = x(((y(zu))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. We say that two elements a and b of a group G are commutative or commute if ab = ba. 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 1.2.5. (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) For all integers n ≥ 1, the set of complex roots of unity   2kπ 2kπ + isin | k = 0, 1, 2, . . . , n − 1 cos n n (i.e., complex zeros of xn − 1) is a group under multiplication.

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(5) 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. (6) 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 = ijk = −1. (7) If n is a positive integer, we can consider the set of all invertible n × n matrices over the real numbers. This is a group with matrix multiplication as operation. It is called the general linear group, GL(n). Geometrically, it contains all combinations of rotations, reflections, dilations and skew transformations of n-dimensional Euclidean space that fix a given point (the origin). If we restrict ourselves to matrices with determinant 1, then we obtain another group, the special linear group, SL(n). Geometrically, this consists of all the elements of GL(n) that preserve both orientation and volume of the various geometric solids in Euclidean space. If instead we restrict ourselves to orthogonal matrices, then we obtain the orthogonal group O(n). Geometrically, this consists of all combinations of rotations and reflections that fix the origin. These are precisely the transformations which preserve lengths and angles. Finally, if we impose both restrictions, then we obtain the special orthogonal group SO(n), which consists of rotations only. These groups are first examples of infinite non-abelian groups. 1.3

Subgroups

The concept of subgroups is one of the most basic ideas in group theory. Definition 1.3.1. 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 ab ∈ H;

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(2) a ∈ H implies a−1 ∈ H. Corollary 1.3.2. If H is a subgroup of G, then H is a group in its own right. Corollary 1.3.3. 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 ab−1 is in H, whenever a, b are in H. Corollary 1.3.4. 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. Example 1.3.5. (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 nonzero 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) Let G be the multiplicative group of all nonsingular 2 × 2 matrices over complex numbers. Let H be the set of the following eight matrices         10 i 0 0i 1 0 ± , ± , ± , ± . 01 0 −i i 0 0 −1 Then, H is a subgroup of G. (6) 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 | ax = xa, for all x ∈ G}. The center of a group G is a subgroup of G. (7) If H is a subgroup of G, then by the centralizer C(H) of H we mean the set {x ∈ G | xh = hx for all h ∈ H}. Then, C(H) is a subgroup of G. (8) Let a ∈ G, define N (a) = {x ∈ G | xa = ax}. Then, N (a) is a subgroup of G. N (a) is usually called the normalizer or centralizer of a in G.

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(9) If G is a group and a ∈ G, then the cyclic subgroup generated by a, denoted by < a >, 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 implies that G = H or H = M , then M is said to be a maximal subgroup of G. Proposition 1.3.6. 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 T is a family of subgroups of G, then Hi is a subgroup of G. i∈I

Definition 1.3.7. If X is a subset of a group G, then the smallest subgroup of G containing X, denoted by < X >, is called the subgroup generated by X. If X consists of a single element a, then < X >=< a >, the cyclic subgroup generated by a. Theorem 1.3.8. If X is a non-empty subset of a group G, then the subgroup < X > 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 and i y = b1 b2 . . . bm in H. Then, xy = a1 a2 . . . an b1 b2 . . . bm is a product of finite number of elements ai , bj such that either the factor or its inverse is in X, −1 −1 −1 consequently xy ∈ H. Further, x−1 = a−1 is n . . . a2 a1 . Since ai or ai −1 −1 −1 −1 −1 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 i any element of H, then x ∈ K, since ui ∈ K for all i. Hence, H ⊆ K. This proves that H is the subgroup of G generated by X.  Definition 1.3.9. 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 ab−1 ∈ H. Lemma 1.3.10. The relation a ≡ b mod H is an equivalence relation. Definition 1.3.11. If H is a subgroup of G and a ∈ G, then Ha = {ha | h ∈ H}. Ha is called a right coset of H in G. A left coset aH is defined similarly. The number of distinct right cosets of H is called the

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index of H in G and denoted by [G : H]. Lemma 1.3.12. For all a ∈ G, we have Ha = {x ∈ G | a ≡ x mod H}. The following corollary contains the basic properties of right cosets and it is useful in many applications. Corollary 1.3.13. Let H be a subgroup of G. (1) Every element a of G contained in exactly one coset of H. This coset is Ha. (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 1.3.14. 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 1.3.15. 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. We use the notation o(a) for the order of a. There are many useful corollaries of Lagrange theorem. Corollary 1.3.16. A finite cyclic group of prime order contains no nontrivial subgroup. Corollary 1.3.17. The order of an element of a finite group G divides the order |G|. Definition 1.3.18. Let A and B be two subsets of a group G. The set AB = {ab | 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 (AB)C = A(BC) for any three subsets A, B and C. The product of two subgroups is not necessarily a subgroup. We have the following theorem. Theorem 1.3.19. Let A and B be subgroups of a group G. Then, the following two conditions are equivalent:

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(1) The product AB is a subgroup of G; (2) We have AB = BA. Proof. Suppose that AB is a subgroup of G. Then, for any a ∈ A, b ∈ B, we have a−1 b−1 ∈ AB and so ba = (a−1 b−1 )−1 ∈ AB. Thus, BA ⊆ AB. Now, if x is any element of AB, then x−1 = ab ∈ AB and so x = (x−1 )−1 = (ab)−1 = b−1 a−1 ∈ BA, so AB ⊆ BA. Thus, AB = BA. On the other hand, suppose that AB = BA, i.e., if a ∈ A and b ∈ B, then ab = b1 a1 for some a1 ∈ A, b1 ∈ B. In order to prove that AB is a subgroup we must verify that it is closed and every element in AB has its inverse in AB. Suppose that x = ab ∈ AB and y = a0 b0 ∈ AB. Then, xy = aba0 b0 , but since ba0 ∈ BA = AB, ba0 = a2 b2 with a2 ∈ A, b2 ∈ B. Hence, xy = a(a2 b2 )b0 = (aa2 )(b2 b0 ) ∈ AB. Clearly, x−1 = b−1 a−1 ∈ BA = AB. Thus, AB is a subgroup of G.  1.4

Normal subgroups and quotient groups

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 aH = Ha 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 1.4.1. 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 1.4.2. 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. Example 1.4.3.

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(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    1b 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. (5) For every n ≥ 1, SL(n) is a normal subgroup of GL(n). Theorem 1.4.4. 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 XY as the subset of G obtained by taking the product of the two subsets X and Y of G. Then, XY 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 = xN for any x ∈ G. Whence XY = (N x)(N y) = N (xN )y = N (N x)y = N xy. This proves that XY is a coset of N . If Z ∈ G, then by the associative law for the product of subsets, we have (XY )Z = X(Y Z). Thus, the multiplication defined on G satisfies the associative law. By definition, we obtain (N x)(N y) = N xy; 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 1.4.5. (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 1.4.6. 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 it 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.  1.5

Group homomorphisms

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 a1 a1 ∗ a1 a2 a2 ∗ a1 ... ... an 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

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of these two groups: G 1 −1 1 1 −1 −1 −1 1

ˆ 0 1 G 0 0 1 1 1 0

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 1.5.1. 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 ˆ and G are isomorphic and write 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 . Examples 1.5.2. (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.

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Proposition 1.5.3. Let f be a homomorphism from a group G into a ˆ The following propositions hold: group G. (1) (2) (3) (4)

ˆ f (e) = e0 , the identity element of G. −1 −1 f (x ) = f (x) for all x ∈ G. The kernel of f is a normal subgroup of G. 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). 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 1.5.4. 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. G> >> >> > f >> 

/ G/kerf xx xx x xx g x| x

ϕ

ˆ G

ˆ by Proof. Suppose that K = kerf . Define a function g from G/K to G g(Kx) = f (x). The function g is well defined and does not depends on the choice of a representative from Kx. Since f is a homomorphism, we have g(KxKy) = g(Kxy) = f (xy) = f (x)f (y) = g(Kx)g(Ky). ˆ If the coset Kx lies in Hence, g is a homomorphism from G/K onto G. the kernel of g, then e = g(Kx) = 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 1.5.5. 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 into G such that f = gϕ if and only if N ⊆ kerf . In this case, we have 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. From Theorem 1.5.4, we get G/K = f (G) ∼ = g(G) ∼ = G/K = (G/N )/(K/N ).  Corollary 1.5.6. 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 1.5.7. 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) HK/H ∼ = K/H ∩ K. Proof. Since H is a normal subgroup of G, HK = KH and so according to Theorem 1.3.19, HK 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 HK/H. Theorem 1.5.4 proves that HK/H ∼ = K/kerf. But, the kernel of the canonical homomorphism is K, whence we get kerf = H ∩ K. This completes the proof.  Example 1.5.8. (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 1.5.9. 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 1.5.10. 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 1.5.11. 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 xg = 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 1.5.12. (1) Aut(Z) ∼ = Z2 ; (2) Aut(G) = 1 if and only if |G| ≤ 2.

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1.6

Polygroup Theory and Related Systems

Permutation groups

In this section, we study certain group of functions called permutation group, from a set X to itself. Permutation groups are an important class of groups. Definition 1.6.1. Let X be a set. A one to one function from X onto X is called a permutation on the set X. We denote the set of all permutations of X by SX . A function p defined on the set X is a permutation if and only if the following three conditions are satisfied: (1) p(x) ∈ X for all x ∈ X; (2) Given a ∈ X, there is an element x ∈ X such that p(x) = a; (3) p(u) = p(v) implies u = v. In the important special case when X = {x1 , x2 , . . . , xn }, we write Sn instead of SX . Note that |Sn | = n!. If f ∈ Sn , then f is a one to one mapping of X onto itself, and we could write f out by showing what it does to every element, e.g., f : x1 −→ x2 , x2 −→ x4 , x4 −→ x3 , x3 −→ x1 . But this is very cumbersome. be to write f out as  One short cut might  x1 x2 x3 . . . xn , x i1 x i2 x i3 . . . x in where xik is the image of xk under f . Return to our example just above, f might be represented by   x1 x2 x3 x4 . x2 x4 x1 x3 While this notation is a little handier there still is waste in it, for there seems to be no purpose served by the symbol x. We could equally well present the permutation as   1 2 3 ... n . i1 i2 i3 . . . in If p and q are permutations on a set X, the composition p ◦ q is also a permutation on X which is defined to be the product of p and q. We write the of p and   product   q as the juxtaposition pq. For example, p = 123 123 and q = are permutations of {1, 2, 3}. The product pq 321 231   123 1 is . 213 1 Notice

that some authors compute this product in the reverse order. This authors will write functions on the right: instead of f (x), they write (x)f .

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The set SX of permutations on X forms a group under the operation defined above. We call SX the symmetric group on X. Definition 1.6.2. Let X be a set and SX be the symmetric group on X. Any subgroup of SX is called a permutation group on X. If x ∈ X and f ∈ SX , then f fixes i if f (i) = i and f moves i if f (i) 6= i. We say σ1 , σ2 , . . . , σk are disjoint if no element of X is moved by more than one of σ1 , σ2 , . . . , σk . It is easy to see that τ σ = στ whenever σ and τ are disjoint. Definition 1.6.3. Let f be a permutation on the set {1, 2, . . . , n} and let i1 , i2 , . . . , ir (1 ≤ r ≤ n) be distinct elements of X. If f fixes the remaining n − r integers and if f (i1 ) = i2 , f (i2 ) = i3 , . . . , f (ir−1 ) = ir , f (ir ) = i1 , then f is called a cycle permutation or cycle and is denoted briefly (i1 i2 . . . ir ). r is called the length of the cycle. We multiply cycles by multiplying the permutations they represent. Lemma 1.6.4. Every permutation can be uniquely expressed as a disjoint cycles. A cycle of length 2 is called a transposition. Lemma 1.6.5. Every permutation is a product of transpositions. Let the permutation f on the set X = {1, 2, . . . , n} be expressible in some way as an even (respectively odd) number of transpositions. Then, every way of expressing f as a product of transpositions requires an even (respectively odd) number of transpositions. Definition 1.6.6. A permutation f ∈ Sn is said to be an even permutation if it can be represented as a product of an even number of transpositions. We call a permutation odd if it is not an even permutation. The rule for combining even and odd permutations is like that of combining even and odd numbers under addition. Let An be the subset of Sn consisting of all even permutations. Since the product of two even permutation is even, An must be a subgroup of Sn . An is called the alternative group of degree n. Lemma 1.6.7. An is a normal subgroup of Sn of order

n! 2 .

The alternative groups are among the most important examples of

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groups. Lemma 1.6.8. For n > 3, each element of An is a product of cycles of length 3. Proof. Suppose that σ ∈ An . Then, σ is an even permutation. Let (x y) and (a b) be two transpositions. If {a, b} ∩ {x, y} = ∅, then (a b)(x y) = (a b)(a x)(x a)(x y) = [(a b)(a x)][(x a)(x y)] = (x b a)(y a x). On the other hand, if {a, b} ∩ {x, y} 6= ∅, then we let b = x. So, (a b)(b y) = (y a b). . Lemma 1.6.9. Let N be a normal subgroup of An (n ≥ 5). If N contains a cycle of length 3, then N = An . Proof. By Lemma 1.6.8, it is enough to show that each cycle of length 3 is belong to N . Suppose that (a b c) is an arbitrary cycle of length 3 and (x y z) ∈ N . Consider σ ∈ Sn such that σ(a) = x, σ(b) = y and σ(c) = z. Then, σ −1 (x y z)σ = (a b c). If σ ∈ An , then (a b c) ∈ N . If σ 6∈ An , then we can choose u, v distinct from x, y, z. Since σ 6∈ An , σ is an odd permutation. So, σ(u v) ∈ An . Thus, ((u v)σ)−1 (x y z)((u v)σ) ∈ N . But ((u v)σ)−1 (x y z)((u v)σ) = σ −1 (u v)(x y z)(u v)σ = σ −1 (x y z)σ. Hence, σ −1 (x y z)σ ∈ N which implies that (a b c) ∈ N .  Lemma 1.6.10. Let N be a normal subgroup of An (n ≥ 5). If N contains a product of two disjoint transpositions, then N = An . Proof. Suppose that (x y)(a b) ∈ N such that {x, y} ∩ {a, b} = ∅. Since n ≥ 5, we choose z distinct from x, y, a, b. Now, we set σ = (a b z). Clearly, σ ∈ An . So, σ −1 (x y)(a b)σ = (z b a)(x y)(a b)(a b z) ∈ N . This implies that (a z)(x y) ∈ N . Since N is a subgroup, (x y)(a b)(a z)(x y) ∈ N . Thus, (a z b) ∈ N . Hence, N contains a cycle of length 3, and so by Lemma 1.6.9, N = An .  Theorem 1.6.11. An is simple for all n ≥ 5. Proof. Suppose that n ≥ 5 and N is a non-trivial normal subgroup of An . We show that N = An by considering the possible cases. (1) N contains a cycle of length 3; hence N = An by Lemma 1.6.9. (2) N contains an element σ, the product of disjoint cycles, at least

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one of which has length r ≥ 4. Then, σ = (a1 a2 . . . ar )τ (disjoint). Let δ = (a1 a2 a3 ) ∈ An . Then, σ −1 (δσδ −1 ) = τ −1 (ar ar−1 . . . a1 )(a1 a2 a3 )(a1 a2 . . . ar )τ (a3 a2 a1 ) = (a1 a3 ar ) ∈ N. Hence, N = An by Lemma 1.6.9. (3) N contains an element σ, the product of disjoint cycles, at least two of which have length 3, so that σ = (a1 a2 a3 )(a4 a5 a6 )τ (disjoint). Let δ = (a1 a2 a4 ) ∈ An . Then as above, N contains σ −1 (δσδ −1 ) = (a1 a4 a2 a6 a3 ). Hence, N = An by case (2). (4) N contains an element σ that is the product of one cycle of length 3 and some transpositions, say σ = (a1 a2 a3 )τ (disjoint), with τ a product of disjoint transpositions. Then, σ 2 = (a1 a3 a2 ) ∈ N , so N = An by Lemma 1.6.9. (5) Each element of N is the product of an even number of disjoint transpositions. Let σ ∈ N , with σ = (a1 a2 )(a3 a4 )τ (disjoint). Let δ = (a1 a2 a3 ) ∈ An . Then as above, N contains σ −1 (δσδ −1 ). Now, σ −1 (δσδ −1 ) = (a1 a3 )(a2 a4 ). Since n ≥ 5, there is an element b ∈ {1, . . . , n} distinct from a1 , a2 , a3 , a4 . Since η = (a1 a3 b) ∈ An and ζ = (a1 a3 )(a2 a4 ) ∈ N , we have ζ(ηζη −1 ) ∈ N . But ζ(ηζη −1 ) = (a1 a3 b) ∈ N . Hence, N = An by Lemma 1.6.9. Since the cases listed cover all the possibilities, An has no proper normal subgroups and hence is simple.  The English mathematician Cayley first noted that every group could be realized as a subgroup of SX for some X. Theorem 1.6.12. (Cayley’s theorem). Let G be a given group. Then, there exists a set X such that G is isomorphic to a permutation group on X. Proof. We choose the set X consisting of all the elements of G. For each element g ∈ G, let πg be a function on X defined by the formula πg (x) = xg (for x ∈ X). It follows that πg is a permutation on X. Furthermore, the associative law proves πgh = πg πh (for g, h ∈ G). Thus, the function π is a homomorphism from G into the symmetric group SX . Clearly, πg is the identity function on X if and only if g = e. This means that π is one to one. Hence, G is isomorphic to the image π(G) which is a permutation group on X. 

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Polygroup Theory and Related Systems

Direct product

There are many methods of constructing new groups from one or more given groups. One of the most important methods is the method of formation of direct product of groups. In this section, we discuss the concepts of direct and semidirect products of groups. Definition 1.7.1. If H and K are groups, then their direct product, denoted by H × K, is the group with elements all ordered pairs (h, k), where h ∈ H and k ∈ K, and with the operation (h, k)(h0 , k 0 ) = (hh0 , kk 0 ). It is easy to check that H × K is a group; the identity is (e, e0 ); the inverse (h, k)−1 is (h−1 , k −1 ). Notice that neither H nor K is a subgroup of H × K, but H × K does contain isomorphic replicas of each, namely, H × {e0 } = {(h, e0 ) | h ∈ H} and {e} × K = {(e, k) | k ∈ K}. Example 1.7.2. (1) Let V = {e, a, b, c} under a binary operation defined by the following table: ∗ e a b c e e a b c a a e c b b b c e a c c b a e Then, (G, ∗) is a group. This is known as Klein’s four group. We have V ∼ = Z2 × Z2 . (2) If (m, n) = 1, then Zm × Zn ∼ = Zmn . Theorem 1.7.3. Let G be a group with normal subgroups H and K. If HK = G and H ∩ K = {e}, then G ∼ = H × K. Proof. If a ∈ G, then a = hk for some h ∈ H and k ∈ K. We claim that h and k are uniquely determined by a. If a = h1 k1 for h1 ∈ H and k1 ∈ K, then hk = h1 k1 and h−1 h1 = kk1−1 ∈ H ∩ K = {e}. Hence, h = h1 and k = k1 . We define f : G −→ H × K by f (a) = (h, k), where a = hk. If a = hk and a0 = h0 k 0 , then aa0 = hkh0 k 0 which is not in the proper form for evaluating f . We consider h0 kh0−1 k −1 . Then, (h0 kh0−1 )k −1 ∈ K, since K is normal. Similarly, h0 (kh0−1 k −1 ) ∈ H, since H is normal. Therefore, h0 kh0−1 k −1 ∈ H ∩ K = {e} and h0 and k commute.

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Therefore, f (aa0 ) = f (hh0 kk 0 ) = (hh0 , kk 0 ) = (h, k)(h0 , k 0 ) = f (a)f (a0 ). Now, it is easy to see that f is an isomorphism.  Theorem 1.7.4. If A is a normal subgroup of H and B is normal subgroup of K, then A × B is a normal subgroup of H × K and (H × K)/(A × B) ∼ = (H/A) × (K/B). Proof. The homomorphism f : H × K −→ (H/A) × (K/B), defined by f (h, k) = (Ah, Bk), is surjective and kerf = A × B. The fundamental homomorphism theorem now gives the result.  It follows, in particular, that if N is normal subgroup of H, then N ×{e} is a normal subgroup of H × K. Corollary 1.7.5. If G = H × K, then G/(H × {e}) ∼ = K. Definition 1.7.6. Let G and H be two groups. If a homomorphism ϕ from G into AutH is given, we say that G acts on H via ϕ and G is an operator group on H; the homomorphism ϕ is called an action of G. The action ϕ of G is not necessarily an isomorphism. In particular, ϕ can be the trivial action, i.e., ϕ(g) = i for all g ∈ G. Via the trivial action, any group can act on H. For any g ∈ G, ϕ(g) is an automorphism of H. We denote the image ϕ(g)(h) of an element h ∈ H simply by hg . The action on H is given if and only if the function ϕ(g) : h 7→ hg are defined for each g ∈ G and satisfy the following formulas for all u, v ∈ H and x, y ∈ G: (uv)x = ux v x , uxy = (ux )y , ue = u, where e is the identity of G. Theorem 1.7.7. Let ϕ be an action of a group G on another group H. Let L be the cartesian product of G and H,i.e., the totally of pairs (g, h) of elements g ∈ G and h ∈ H. We define the product of two elements of L by the formula (g, h)(g 0 , h0 ) = (gg 0 , ϕ(g 0 )(h)h0 ) . Then, L forms a group with respect to this operation. Let eG denotes the identity of G; similarly, let eH denotes the identity of H. We define for g ∈ G and h ∈ H, γ(g) = (g, eH ),

η(h) = (eG , h).

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We set G = {γ(g) | g ∈ G}, H = {η(h) | h ∈ H}. Then, γ is an isomorphism from G onto G, η is an isomorphism from H onto H, and we have H is a normal subgroup of L = G H and G∩H = {e}, where e = (eG , eH ) is the identity of L. Furthermore, the formula γ(g)−1 η(h)γ(g) = η(hg ) holds for any g ∈ G and h ∈ H. Proof. First, we verify the associative law. Let g, u, x ∈ G and h, v, y ∈ H. Then, [(g, h)(u, v)](x, y) = (gu, hu v)(x, y) = ((gu)x, (hu v)x y) = (g(ux), ((hu )x v x )y) = (g(ux), hux (v x y)) = (g, h)(ux, v x y) = (g, h)[(u, v)(x, y)]. By definition, e = (eG , eH ) is the identity; the inverse of (g, h) is given by (g −1 , ϕ(g −1 )(h)−1 ). So, L forms a group with respect to the operation defined above. The definition of the operation also shows that γ(gg 0 ) = γ(g)γ(g 0 ), η(hh0 ) = η(h)η(h0 ); so both γ and η are homomorphisms and G = γ(G) and H = η(H) are subgroups. Furthermore, by definition we have (g, h) = γ(g)η(h), γ(g)−1 η(h)γ(g) = η(hg ). Hence, H is a normal subgroup of L = G H. Clearly, we have G ∩ H = {e}.  The group constructed in the above theorem is called the semidirect product of G and H with respect to the action ϕ. 1.8

Solvable and nilpotent groups

The goal of this section is to introduce the concept of solvable and nilpotent groups. Definition 1.8.1. A sequence of subgroups {e} = G0 ⊆ G1 ⊆ . . . ⊆ Gn = G

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of a group G is called a normal series of G, if Gi is a normal subgroup of Gi+1 , for i = 0, . . . , n − 1. Definition 1.8.2. A group G is said to be solvable if it has a normal series {e} = G0 ⊆ G1 ⊆ . . . ⊆ Gn = G such that each of its factor group Gi+1 /Gi is abelian, for every i = 0, . . . , n − 1. The above series is referred to as a solvable series of G. Example 1.8.3. (1) (2) (3) (4)

Abelian groups are solvable. S3 and S4 are solvable. The dihedral groups are solvable. Any group whose order has the form pm , where p is prime, is solvable.

Theorem 1.8.4. Any subgroup H of a solvable group G is solvable. Proof. Suppose that {e} = G0 ⊆ G1 ⊆ . . . ⊆ Gn = G is a solvable series for G. We show that {e} = H ∩ G0 ⊆ H ∩ G1 ⊆ . . . ⊆ H ∩ Gn = H is a solvable series for H. Since Gi is normal in Gi+1 , where i = 0, . . . , n−1, we conclude that Ni = H ∩ Gi is normal in Ni+1 = H ∩ Gi+1 . Now, we define a mapping f : Ni+1 −→ Gi+1 /Gi by f (x) = xGi , for all x ∈ Ni+1 . Clearly, f is a homomorphism. Moreover, if x ∈ Ni+1 , then x ∈ kerf ⇔ xGi = Gi ⇔ x ∈ Gi ⇔ x ∈ H ∩ Gi . This yields that kerf = H ∩ Gi = Ni . Hence, by the fundamental theorem of group homomorphism, Ni+1 /Ni ∼ = f (Ni+1 ). As f (Ni+1 ) is a subgroup of Gi+1 /Gi and Gi+1 /Gi is abelian, f (Ni+1 ) is also abelian. Consequently, Ni+1 /Ni is abelian. This proves that H is a solvable group.  Theorem 1.8.5. If N is a normal subgroup of a solvable group G, then G/N is also solvable. Proof. Suppose that {e} = G0 ⊆ G1 ⊆ . . . ⊆ Gn = G is a solvable series for G. We consider the following series {N } = G0 N/N ⊆ G1 N/N ⊆ . . . ⊆ Gn N/N = G/N. Let 0 ≤ i ≤ n−1 be an arbitrary element. Suppose that x ∈ Gi+1 N . Then, x = gy for some g ∈ Gi+1 and y ∈ N . Hence, xGi N = gyGi N = gyN Gi = gN Gi = Gi gN = Gi gyN = Gi N gy = Gi N x.

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This proves that Gi N is a normal subgroup of Gi+1 N . Thus, Gi+1 N/Gi N ∼ = (Gi+1 N/N )/(Gi N/N ). Now, we define f : Gi+1 −→ Gi+1 N/Gi N by f (x) = Gi N x, for all x ∈ Gi+1 . Then, f is a homomorphism. As Gi+1 N = N Gi+1 , for y ∈ Gi+1 N , we can write y = zg for some z ∈ N and g ∈ Gi+1 . Then, Gi N y = Gi N zg = Gi N g = f (g). This shows that f is onto. Since Gi ⊆ kerf , the following function is a homomorphism f : Gi+1 /Gi −→ Gi+1 N/Gi N f (Gi x) = Gi N x, for all x ∈ Gi+1 . Clearly, f is also onto. Thus, Gi+1 N/Gi N is a homomorphic image of the abelian group Gi+1 /Gi . So that it must be itself abelian. Consequently, each factor group is abelian. This prove that G/N is solvable.  Theorem 1.8.6. Let N be a normal subgroup of a group G. If both N and G/N are solvable, then G is solvable. Proof. Since N and G/N are solvable, there exist the following solvable series for N and G/N , respectively, {e} = N0 ⊆ N1 ⊆ . . . ⊆ Nm = N, {N } = G0 /N ⊆ G1 /N ⊆ . . . ⊆ Gn /N = G/N. Here each Gi is a subgroup of G containing N . Since Gi /N is normal in Gi+1 /N , each Gi is normal in Gi+1 . Moreover, G0 = N . Now, {e} = N0 ⊆ N1 ⊆ . . . ⊆ Nm = N = G0 ⊆ G1 ⊆ . . . ⊆ Gn = G is a solvable series for G. Therefore, we conclude that G is solvable.  Corollary 1.8.7. If two groups H and K are solvable, then H × K is solvable. Proof. Let G = H × K. Alternatively, we can consider H ∼ = (H × {e}) and K∼ = ({e} × H) and clearly, G = (H × {e})({e} × K). Hence, G/H ∼ = K. Since K is solvable, we have G/H is solvable. Hence, G is solvable by Theorem 1.8.6.  Lemma 1.8.8. Let N be a normal subgroup of G and G/N be abelian. If x, y ∈ G, then xyx−1 y −1 ∈ N . Proof. It is straightforward.  Proposition 1.8.9. Sn is not solvable for n ≥ 5.

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Proof. Suppose that Sk is solvable for some k ≥ 5 and {e} = G0 ⊆ G1 ⊆ . . . ⊆ Gm−1 ⊆ Gm = Sk is a solvable series for Sk . By induction, we show that Gm−i contains all of the cycles of length 3, where 0 ≤ i ≤ m. Indeed, since G0 = {e}, we obtain a contradiction and the proof completes. Let (x y z) be a cycle of length 3 in Sk . Since k ≥ 5, there exist u, v distinct from x, y, z. Now, suppose that σ = (z u y) and δ = (y x v). Since Gm−1 is a normal subgroup of Sk and Sk /Gm−1 is abelian, by Lemma 1.8.8, we obtain σδσ −1 δ −1 ∈ Gm−1 . Therefore, (x y z) = (z u y)(y x v)(z y u)(y v x) = σδσ −1 δ −1 is an element of Gm−1 . Now, let Gm−j contains all of the cycles of length 3. Since (x y z) ∈ Gm−j and Gm−j−1 is a normal subgroup of Gm−j , similar to the above discussion, we obtain (x y z) ∈ Gm−j−1 .  The commutator subgroup G0 of a group G is the subgroup generated by the set {x−1 y −1 xy | x, y ∈ G}. That is, every element of G0 has the form ai11 ai22 . . . aikk , where each aj has the form x−1 y −1 xy, each ij = ±1 and k is any positive integer. x−1 y −1 xy is called the commutator of x and y. The commutator subgroup is also called a derived subgroup. Lemma 1.8.10. Let G be a group and G0 be its commutator subgroup. Then, (1) G0 is a normal subgroup of G; (2) For any normal subgroup N of G, G/N is an abelian group if and if N contains G0 . Proof. (1) Assume that a, b ∈ G. Since (a−1 b−1 ab)−1 = b−1 a−1 ba is again a commutator, we conclude that each element of G0 is a product of finite number of commutators. Now, suppose that a ∈ G and x ∈ G0 . Then, x = g1 g2 . . . gk , where for each i = 1, . . . , k, gi is −1 a commutator, so gi = a−1 i bi ai bi for some ai , bi ∈ G. So, we have a−1 xa = (a−1 g1 a)(a−1 g2 a) . . . (a−1 gk a). Further, −1 a−1 gi a = a−1 a−1 i bi ai bi a = (a−1 ai a)−1 (a−1 bi a)−1 (a−1 ai a)(a−1 bi a).

Hence, a−1 gi a is again a commutator. Therefore, a−1 xa is a product of commutators which implies that a−1 xa ∈ G0 .

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(2) Suppose that G/N is abelian and a, b are two elements of G. Then, (aN )(bN ) = (bN )(aN ) or abN = baN . So, a−1 b−1 ab ∈ N . We conclude that N contains every commutator a−1 b−1 ab. Since G0 is generated by all the commutators, G0 ⊆ N . Conversely, suppose that G0 ⊆ N and a, b are two elements of G. Then, a−1 b−1 ab ∈ G0 gives a−1 b−1 ab ∈ N which implies that abN = baN or (aN )(bN ) = (bN )(aN ). This shows that G/N is abelian.  Corollary 1.8.11. G/G0 is abelian. Corollary 1.8.12. A group G is abelian if and only if G0 = {e}. Note that by using Corollary 1.8.12 and Theorem 1.6.11, one can give a short proof for Proposition 1.8.9. Definition 1.8.13. Let G be a group. We define the sequence of subgroups G(i) of G inductively by (1) G(0) = G; (2) G(1) = G0 , commutator subgroup of G; (3) G(i) = (G(i−1) )0 , commutator subgroup of G(i−1) , if i > 1. Theorem 1.8.14. A group G is solvable if and only if G(n) = {e} for some n ≥ 1. Proof. Suppose that G is solvable and {e} = G0 ⊆ G1 ⊆ . . . ⊆ Gn = G is a solvable series for G. We prove inductively that G(k) ⊆ Gn−k (∗) (0) for all 0 ≤ k ≤ n. If k = 0, then G = G = G0 . Let G(k) ⊆ Gn−k (k+1) for some k. This implies that G = (G(k) )0 ⊆ (Gn−k )0 . Since Gn−k /Gn−k−1 is abelian, by Lemma 1.8.10, (Gn−k )0 ⊆ Gn−k−1 . Consequently, G(k+1) ⊆ Gn−k−1 . Thus, by induction, (∗) holds. In particular, G(n) ⊆ G0 . Therefore, G(n) = {e}. Conversely, assume that G(n) = {e} for some n. Then {e} = G(n) ⊆ G(n−1) ⊆ . . . ⊆ G(1) ⊆ G(0) = G is a normal series for G such that G(j) /G(j+1) = G(j) /(G(j) )0 is abelian. Thus, G is solvable.  We now introduce the class of nilpotent groups. Definition 1.8.15. A group G is said to be nilpotent if it has a series of subgroups {e} = G0 ⊆ G1 ⊆ . . . ⊆ Gn = G such that for every 1 ≤ i ≤ n,

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(1) Gi is a normal subgroup of G; (2) Gi /Gi−1 ≤ Z (G/Gi−1 ). Such series is called a central series of G. For a nilpotent group, the smallest n such that G has a central series of length n is called the nilpotency class of G; and G is said to be nilpotent of class n. Let H and K be subgroups of G. We define [H, K] =< h−1 k −1 hk | h ∈ H, k ∈ K > . Lemma 1.8.16. A series of G, say {e} = G0 ⊆ G1 ⊆ . . . ⊆ Gn = G is a central series if and only if for each i = 1, . . . , n, [Gi , G] ≤ Gi−1 . Proof. If the given series is a central series, then for each i = 1, . . . , n, Gi−1 is a normal subgroup of G and Gi /Gi−1 ≤ Z (G/Gi−1 ). Then, for any x ∈ Gi and y ∈ G, (xGi−1 )(yGi−1 ) = (yGi−1 )(xGi−1 ), that is xyGi−1 = yxGi−1 . Hence, x−1 y −1 xy ∈ Gi−1 . This implies that [Gi , G] ≤ Gi−1 . Conversely, suppose that for each i = 1, . . . , n, [Gi , G] ≤ Gi−1 . Let x ∈ Gi and y ∈ G. Then, x−1 y −1 xy ∈ Gi−1 . In particular, since Gi−1 ≤ Gi , if x ∈ Gi−1 then y −1 xy ∈ Gi−1 . Thus, Gi−1 is a normal subgroup of G. Moreover, xyGi−1 = yxGi−1 for every x ∈ Gi and y ∈ G, and so Gi /Gi−1 ≤ Z (G/Gi−1 ). Thus, the series is a central series.  Example 1.8.17. (1) Abelian groups are nilpotent. (2) The quaternion group Q8 is nilpotent. Theorem 1.8.18. If G is nilpotent, then all subgroups and all quotient groups of G are nilpotent. Proof. Suppose that G is nilpotent and {e} = G0 ⊆ G1 ⊆ . . . ⊆ Gn = G is a central series for G. Let H be a subgroup of G and N be a normal subgroup of G. Then, we have {e} = H ∩ G0 ⊆ H ∩ G1 ⊆ . . . ⊆ H ∩ Gn = H, {N } = G0 N/N ⊆ G1 N/N ⊆ . . . ⊆ Gn N/N = G/N. For each i = 1, . . . , n, [Gi , G] ≤ Gi−1 . Hence, [Gi ∩ H, H] ≤ H ∩ [Gi , G] ≤ H ∩ Gi−1

(∗) (∗∗)

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and [Gi N/N, G/N ] = [Gi , G]N/N ≤ Gi−1 N/N. Therefore by Lemma 1.8.15, (∗) and (∗∗) are central series, so that H and G/N are nilpotent.  In the following lemma, we present a non nilpotent group. Lemma 1.8.19. S3 is not nilpotent. Proof. Assume that S3 is nilpotent. Then, there exist a central series {e} = G0 ⊆ G1 ⊆ . . . ⊆ Gn = S3 . We have G1 /G0 ≤ Z(S3 /G0 ). So, G1 ≤ Z(S3 ) = {e} which implies that G1 = {e}. Similarly, we obtain G2 = {e}, . . . , Gn = S3 = {e}, that is a contradiction.  By the above lemma, we see that Theorem 1.8.6 is not true for nilpotent groups, i.e., if N is a normal subgroup of a group G and both N and G/N are nilpotent, then G is not nilpotent, in general. Lemma 1.8.20. If two groups H and K are nilpotent, then H × K is nilpotent. Proof. If H and K are both nilpotent, then there are central series {e} = H0 ⊆ H1 ⊆ . . . ⊆ Hm = H, {e} = K0 ⊆ K1 ⊆ . . . ⊆ Kn = K. By inserting repetition of terms if necessary, we may assume without loss generality that m = n. Then, we have ({e} × {e}) = (H0 × K0 ) ⊆ (H1 × K1 ) ⊆ . . . ⊆ (Hn × Kn ) = G. For each i = 1, . . . , n, [Hi × Ki , G] = ([Hi , H] × [Ki , K]) ≤ (Hi−1 × Ki−1 ). Now, by Lemma 1.8.16, we conclude that G is nilpotent.  Definition 1.8.21. We define subgroups Γn (G) and Zn (G) of G respectively as follows. Let Γ1 (G) = G and Z0 (G) = {e}. Then, for each integer n > 1, Γn (G) = [Γn−1 (G), G] and for each integer n > 0, Zn (G)/Zn−1 (G) = Z(G/Zn−1 (G)).

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Then, G = Γ1 (G) ≥ Γ2 (G) ≥ Γ3 (G) ≥ . . . , {e} = Z0 (G) ≤ Z1 (G) ≤ Z2 (G) ≤ . . . · The first sequence is called the lower central series of G and the second sequence is called the upper central series of G. It is not difficult to see that the terms of the lower and upper central series are normal subgroups of G. Theorem 1.8.22. The following three statements are equivalent: (1) G is nilpotent. (2) Γn (G) = {e} for some integer n. (3) Zn (G) = G for some integer n. Proof. If Γn (G) = {e} for some n, then G = Γ1 (G) ≥ Γ2 (G) ≥ . . . ≥ Γn (G) = {e} is a central series of G, and so G is nilpotent. Similarly, if Zn (G) = G for some n, then {e} = Z0 (G) ≤ Z1 (G) ≤ . . . ≤ Zn (G) = G is a central series of G, and so G is nilpotent. Conversely, assume that G is nilpotent and {e} = G0 ⊆ G1 ⊆ . . . ⊆ Gn = G is a central series of G. We prove first by induction on i that Gi ≤ Zi (G), for all 0 ≤ i ≤ n. This is trivial for i = 0. Suppose that i > 0 and, inductively, that Gi−1 ≤ Zi−1 (G). Then, Gi−1 Zi−1 (G) = Zi−1 (G). By hypothesis, Gi /Gi−1 ≤ Z(G/Gi−1 ). Therefore, Gi Zi−1 (G)/Zi−1 (G) ≤ Z(G/Zi−1 (G)) = Zi (G)/Zi−1 (G). Hence, Gi ≤ Zi (G). Thus, the induction argument goes through. In particular, since G = Gn ≤ Zn (G), we obtain Zn (G) = G. Now, we prove by induction on j that Γj+1 (G) ≤ Gn−j , for all 0 ≤ j ≤ n. This is trivial for j = 0. Assume that j > 0 and, inductively, that Γj (G) ≤ Gn−j+1 . Then, by Lemma 1.8.16, [Gn−j+1 , G] ≤ Gn−j . Hence, Γj+1 (G) = [Γj (G), G] ≤ [Gn−j+1 , G] ≤ Gn−j .

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Again, the induction argument goes through. Γn+1 (G) ≤ G0 = {e}, we obtain Γn+1 (G) = {e}. 

In particular, since

Corollary 1.8.23. Let G be a nilpotent group. Then, for any central series of G, say {e} = G0 ⊆ G1 ⊆ . . . ⊆ Gn = G, we have Γn−i+1 (G) ≤ Gi ≤ Zi (G), for all 0 ≤ i ≤ n. Moreover, the least integer r such that Γr+1 (G) = {e} is equal to the least integer r such that Zr (G) = G. Proof. Suppose that r is the least integer such that Zr (G) = G. We show that r is also the least integer such that Γr+1 (G) = {e}. Let Γk+1 (G) = {e} for some k < r. We set Gi = Γk−i+1 (G). Then, Gk = Γ1 (G), . . . , G0 = Γk+1 (G) = {e}. Hence, {e} = G0 ⊆ G1 ⊆ . . . ⊆ Gk = G is a central series. Now, by Theorem 1.8.22, it follows that Zk (G) = G; this is contrary to the definition of r.  We conclude that if G is a nilpotent group, then the lower central series of G is its most rapidly descending central series and the upper central series of G is its most rapidly ascending central series. Lemma 1.8.24. For each non-negative integer i, G(i) ≤ Γi+1 (G). Proof. We prove by induction on i. If i = 0, then G(0) = G = Γ1 (G). Suppose that i > 0 and, inductively, that G(i−1) ≤ Γi (G). Then, G(i) = [G(i−1) , G(i−1) ] ≤ [Γi (G), G] = Γi+1 (G). This completes the proof.  Theorem 1.8.25. Each nilpotent group is solvable. Proof. Suppose that G is a nilpotent group. By Corollary 1.8.23, there exist a non-negative integer r such that Γr+1 = {e}. By Lemma 1.8.24, we get G(i) ≤ Γi+1 (G) = {e}. Now, by Theorem 1.8.14, we conclude that G is solvable.  Lemma 1.8.19 shows that the converse of Theorem 1.8.25 is not true in general.

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Hypergroups

2.1

Introduction and historical development of hypergroups

The hypergroup notion was introduced in 1934 by F. Marty [103], at the 8th Congress of Scandinavian Mathematicians. He published some notes on hypergroups, using them in different contexts: algebraic functions, rational fractions, non-commutative groups. Hypergroups are a suitable generalization of groups. We know in a group, the composition of two elements is an element, while in a hypergroup, the composition of two elements is a set. The motivating example was the following: Let G be a group and K be a subgroup of G. Then, G/K = {xK | x ∈ G} becomes a hypergroup, where the composition is defined in a usual manner. In [119], Prenowitz represented several kinds of geometries (projective, descriptive and spherical) as hypergroups, and later, with Jantosciak [124], founded geometries on join spaces, a special kind of hypergroups, which in the last decades were shown to be useful instrument in the study of several matters: graphs, hypergraphs and binary relations. Several kinds of hypergroups have been intensively studied, such as regular hypergroups, reversible regular hypergroups, canonical hypergroups, cogroups and cyclic hypergroups. The situations that occur in hypergroup theory, are often extremely diversified and complex with respect to group theory. For instance, there are homomorphisms of various types between hypergroups and there are several kinds of subhypergroups, such as: closed, invertible, ultraclosed, conjugable. Around the 1940’s, the general aspects of the theory, the connections with groups and various applications in geometry were studied in France by F. Marty, M. Krasner, M. Kuntzmann, R. Croisot, in U.S.A. by M. Dresher,

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O. Ore, W. Prenowitz, H.S. Wall, J.E. Eaton, H. Campaigne, L. Griffiths, in Russia by A. Dietzman, A. Vikhrov, in Italy by G. Zappa, in Japan by Y. Utumi. Over the following two decades, other interesting results on hyperstructures were obtained, for instance, in Italy, A. Orsatti studied semiregular hypergroups, in Czechoslovakia, K. Drbohlav studied hypergroups of two sided classes, in Romania, M. Benado studied hyperlattices. The theory knew an important progress starting with the 1970’s, when its research area enlarged. In France, M. Krasner, M. Koskas and Y. Sureau investigated the theory of subhypergroups and the relations defined on hyperstructures; in Greece, J. Mittas, and his students M. Konstantinidou, K. Serafimidis, S. Ioulidis and C.N. Yatras studied the canonical hypergroups, the hyperrings, the hyperlattices, Ch. Massouros obtained important results about hyperfields and other hyperstructures. G. Massouros, together with J. Mittas studied applications of hyperstructures to Automata. D. Stratigopoulos continued some of Krasner ideas, studying in depth noncommutative hyperrings and hypermodules. T. Vougiouklis, L. Konguetsof and later S. Spartalis, A. Dramalidis analyzed especially the cyclic hypergroups, the P -hyperstructures. Significant contributions to the study of regular hypergroups, complete hypergroups, of the heart and of the hypergroup homomorphisms in general or with applications in Combinatorics and Geometry were brought by the Italian mathematician P. Corsini and his group of research, among whom we mention M.de Salvo, R. Migliorato, F. de Maria, G. Romeo, P. Bonansinga. Also around 1970’s, some connections between hyperstructures and ordered systems, particularly lattices, were established by T. Nakano and J.C. Varlet. Around the 1980’s and 90’s, associativity semyhypergroups were analyzed in the context of semigroup theory by T. Kepka and then by J. Jezec, P. Nemec and K. Drbohlav, and in Finland by M. Niemenmaa. In U.S.A., R. Roth used canonical hypergroups in solving some problems of character theory of finite groups, while S. Comer studied the connections among hypergroups, combinatorics and the relation theory. J. Jantosciak continued the study of join spaces, introduced by W. Prenowitz, he considered a generalization of them for the noncommutative case and studied correspondences between homomorphisms and the associated relations. In America, hyperstructures have been studied both in U.S.A. (at Charleston, South Carolina – The Citadel, New York-Brooklyn College, CUNY, Cleveland, Ohio – John Carroll University) and in Canada (at Universit´e

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de Montr´eal). A big role in spreading this theory is played by the Congresses on Algebraic Hyperstructures and their Applications. The first three Congresses were organized by P. Corsini in Italy. The contribution of P. Corsini in the development of Hyperstructure Theory has been decisive. He has delivered lectures about hyperstructures and their applications in several countries, several times, for instance in Romania, Thailand, Iran, China, Montenegro, making known this theory. After his visits in these countries, hyperstructures have had a substantial development. Coming back to the Congresses on Algebraic Hyperstructures, the first two were organized in Taormina, Sicily, in 1978 and 1983, with the names: “Sistemi Binari e loro Applicazioni” and “Ipergruppi, Strutture Multivoche e Algebrizzazione di Strutture d’Incidenza”. The third Congress, called “Ipergruppi, altre Strutture Multivoche e loro Applicazioni” was organized in Udine in 1985. The fourth congress, organized by T. Vougiouklis in Xanthi in 1990, used already the name of Algebraic Congress on Hyperstructures and their Applications, also known as AHA Congress. After 1990, AHA Congresses have been organized every three years. Beginning with the 90’s Hyperstructure Theory represents a constant concern also for the Romanian mathematicians, the decisive moment being the fifth AHA Congress, organized in 1993 at the University “Al.I.Cuza” of Iasi by M. Stefanescu. This domain of the modern algebra is a topic of a great interest also for the Romanian researches, who have published a lot of papers on hyperstructures in national or international journals, have given communications in conferences and congresses, have written Ph.D. theses in this field. The sixth AHA Congress was organized in 1996 at the Agriculture University of Prague by T. Kepka and P. Nemec, the seventh was organized in 1999 by R. Migliorato in Taormina, Sicily, then the eighth was organized in 2002 by T. Vougiouklis in Samothraki, Greece. All these congresses were organized in Europe. Nowadays, one works successfully on Hyperstructures in the following countries of Europe: • in Greece, at Thessaloniki (Aristotle University), at Alexandropoulis (Democritus University of Thrace), at Patras (Patras University), Orestiada (Democritus University of Thrace), at Athens; • in Italy at Udine University, at Messina University, at Rome (Uni-

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versita’ “La Sapienza”), at Pescara (D’Annunzio University), at Teramo (Universita’ di Teramo), Palermo University; • in Romania, at Iasi (“Al.I. Cuza” University), Cluj (“BabesBolyai” University), Constanta (“Ovidius” University); • in Czech Republic, at Praha (Charles University, Agriculture University), at Brno (Brno University of Technology, Military Academy of Brno, Masaryk University), Olomouc (Palacky University); • in Montenegro, at Podgorica University. Let us continue with the following AHA Congresses. The ninth congress on hyperstructures, organized in 2005 by R. Ameri in Babolsar, Iran, was the first of this kind in Asia. In the past millenniums, Iran gave fundamental contributions to Mathematics and in particular, to Algebra (for instance Khwarizmi, Kashi, Khayyam and recently Zadeh), many scientists have well understood the importance of hyperstructures, on the theoretical point of view and for the applications to a wide variety of scientific sectors. Nowadays, hyperstructures are cultivated in many universities and research centers in Iran, among which we mention Yazd University, Shahid Bahonar University of Kerman, Mazandaran University, Kashan University, Ferdowsi University of Mashhad, Tehran University, Tarbiat Modarres University, Zahedan (Sistan and Baluchestan University), Semnan University, Islasmic Azad University of Kerman, Shahid Beheshti University of Tehran, Center for Theoretical Physics and Mathematics of Tehran, Zanjan (Institute for Advanced Studies in Basic Sciences). Another Asian country where hyperstructures have had success is Thailand. In Chulalornkorn University of Bangkok, important results have been obtained by Y. Kemprasit and her students Y. Punkla, S. Chaoprakhoi, N. Triphop, C. Namnak on the connections among hyperstructures, semigroups and rings. There are other Asia centers for researches in hyperstructures. We mention here India (University of Calcutta, Aditanar College of Arts and Sciences, Tiruchendur, Tamil Nadu), Korea (Chiungju National University, Chiungju National University of Education, Gyeongsang National University, Jinju), Japan (Hitotsubashi University of Tokyo), Sultanate of Oman (Education College for Teachers), China (Northwest University of Xian, Yunnan University of Kunming). Hypertructures have been also cultivated in Germany, Netherlands, Belgium, Macedonia, Serbia, Slovakia, Spain, Uzbekistan, Australia. The

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tenth AHA Congress was held in Brno, Czech Republic in the autumn ˇarka. Hoˇskov´a, at the Military Academy of 2008. It was organized by S´ of Brno. The eleventh AHA Congress was held in Pescara, Italy in the autumn of 2011. It was organized by A. Maturo, at the Universit`a degli Studi “G. d’Annunzio” Chieti-Pescara. More than 800 papers and some books have been written till now on hyperstructures. Many of them are dedicated to the applications of hyperstructures in other topics. We shall mention here some of the fields connected with hyperstructures and only some names of mathematicians who have worked in each topic: • Geometry (W. Prenowitz, J. Jantosciak, and later G. Tallini), • Codes (G. Tallini), • Cryptography and Probability (L. Berardi, F. Eugeni, S. Innamorati, A. Maturo), • Automata (G. Massouros, J. Chvalina, L. Chvalinova), • Artificial Intelligence (G. Ligozat), • Median Algebras, Relation Algebras, C-algebras (S. Comer), • Boolean Algebras (A.R. Ashrafi, M. Konstantinidou), • Categories (M. Scafati, M.M. Zahedi, C. Pelea, R. Bayon, N. Ligeros, S.N. Hosseini, B. Davvaz, M.R. Khosharadi-Zadeh), • Topology (J. Mittas, M. Konstantinidou, M.M. Zahedi, R. Ameri, S. Hoˇskov´ a), • Binary Relations (J. Chvalina, I.G. Rosenberg, P. Corsini, V. Leoreanu, B. Davvaz, S. Spartalis, I. Chajda, S. Hoˇskov´a, I. Cristea, M. De Salvo, G. Lo Faro), • Graphs and Hypergraphs (P. Corsini, I.G. Rosenberg, V. Leoreanu, M. Gionfriddo, A. Iranmanesh, M.R. Khosharadi-Zadeh), • Lattices and Hyperlattices ( J.C. Varlet, T. Nakano, J. Mittas, A. Kehagias, M. Konstantinidou, K. Serafimidis, V. Leoreanu, I.G. Rosenberg, B. Davvaz, G. Calugareanu, G. Radu, A.R. Ashrafi), • Fuzzy Sets and Rough Sets (P. Corsini, M.M. Zahedi, B. Davvaz, R. Ameri, R.A. Borzooei, V. Leoreanu, I. Cristea, A. Kehagias, A. Hasankhani, I. Tofan, C. Volf, G.A. Moghani, H. Hedayati), • Intuitionistic Fuzzy Hyperalgebras (B. Davvaz, R.A. Borzooei, Y.B. Jun, W.A. Dudek, L. Torkzadeh), • Generalized Dynamical Systems (M.R. Molaei) and so on. Another topic which has aroused the interest of several mathematicians, is that one of Hv -structures, introduced by T. Vougiouklis and studied

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then also by B. Davvaz, M.R. Darafsheh, M. Ghadiri, R. Migliorato, S. Spartalis, A. Dramalidis, A. Iranmanesh, M.N. Iradmusa, A. Madanshekaf. Hv -structures are a special kind of hyperstructures, for which the weak associativity holds. Recently, n-ary hyperstructures, introduced by B. Davvaz and T. Vougiouklis, represent an intensively studied field of research. Therefore, there are good reasons to hope that Hyperstructure Theory will be one of the more successful fields of research in algebra. 2.2

Definition and examples of hypergroups

Let H be a non-empty set and ◦ : H × H −→ P ∗ (H) be a hyperoperation, where P ∗ (H) is the family of non-empty subsets of H. The couple (H, ◦) is called a hypergroupoid. For any two non-empty subsets A and B of H and x ∈ H, we define S A◦B = a ◦ b, A ◦ x = A ◦ {x} and x ◦ B = {x} ◦ B. a∈A,b∈B

Definition 2.2.1. A hypergroupoid (H, ◦) is called a semihypergroup if for all a, b, c of H we have (a ◦ b) ◦ c = a ◦ (b ◦ c), which means that S S u◦c= a ◦ v. u∈a◦b

v∈b◦c

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 2.2.2. A hypergroupoid (H, ◦) which is both a semihypergroup and a quasihypergroup is called a hypergroup. 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 2.2.3. (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.

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(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 =< x, y >. 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}. (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 R be an equivalence relation in H, such that for all x of H, the equivalence class R(x) of x has at least three elements. For any subset A of H, R(A) denotes the S S set R(x), while R(A) denotes the set R(x). The R(x)∩A6=∅

R(x)⊆A

couple (R(A), R(A)) is called a rough set. If for all x, y of H, we define x ◦ y = R({x, y})\R({x, y}), then (H, ◦) is a hypergroup. (13) 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.

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(14) 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.

2.3

Some kinds of subhypergroups

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 2.3.1. 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. Among the mathematicians who studied this topic, we mention F. Marty, M. Dresher, O. Ore, M. Krasner who analyzed closed and invertible subhypergroups. M. Koskas considered another type of subhypergroups are complete parts. Later Y. Sureau has studied ultraclosed, invertible and conjugable subhypergroups. Corsini has obtained important results about ultraclosed and complete parts. Also, Leoreanu has studied and obtained other interesting results on subhypergroups. Let us present now the definition of these types of subhypergroups. Let (H, ◦) be a hypergroup and (K, ◦) be a subhypergroup of it. Definition 2.3.2. 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.

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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 2.3.3. (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}; 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}. Other examples can be found in [22]. Lemma 2.3.4. 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

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all x of H we have x ∈ x ◦ K. From here, we obtain that y ∈ y ◦ K = x ◦ K.  Similar to Lemma 2.3.4, 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 2.3.5. 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 2.3.6. 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.  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 2.3.7. 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

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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. Conversely, 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 2.3.8. 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.  2.4

Homomorphisms of hypergroups

Homomorphisms of hypergroups are studied by Dresher, Ore, Krasner, Kuntzmann, Koskas, Jantosciak, Corsini, Freni, Davvaz and many others. In this section, we study several kinds of homomorphisms. The main references are [22; 86]. Definition 2.4.1. Let (H1 , ◦) and (H2 , ∗) be two hypergroups. A map f : H1 −→ H2 , is called (1) a homomorphism or inclusion homomorphism if for all x, y of H1 , we have f (x ◦ y) ⊆ f (x) ∗ f (y); (2) a good homomorphism if for all x, y of H1 , we have f (x ◦ y) = f (x) ∗ f (y); (3) an isomorphism if it is a one to one and onto good homomorphism. If f is an isomorphism, then H1 and H2 are said to be isomorphic, which is denoted by H1 ∼ = H2 .

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Example 2.4.2. Let H1 = {a, b, c} and H2 = {0, 1, 2} be two hypergroups with the following hyperoperations: ◦ a b c 2 ∗ 0 1 a a H1 H1 0 0 H2 H2 1 b H1 b b 1 H2 1 c H1 b c 2 H2 1 {1, 2} and let f : H1 −→ H2 is defined by f (a) = 0, f (b) = 1 and f (c) = 2. Clearly, f is an inclusion homomorphism. Proposition 2.4.3. Let (H1 , ◦) and (H2 , ∗) be two hypergroups and f : H1 −→ H2 be a good homomorphism. Then, Imf is a subhypergroup of H2 . Proof. For every a, b ∈ H1 , we have f (a)∗f (b) = f (a◦b) ⊆ Imf . Moreover, there exist x, y ∈ H1 such that a ∈ x ◦ b and a ∈ b ◦ y and consequently f (a) ∈ f (x) ∗ f (b) and f (a) ∈ f (b) ∗ f (y).  We employ for simplicity of notation xf = f −1 (f (x)) and for a subset A of H1 , Af = f −1 (f (A)) = ∪{xf | x ∈ A}. Notice that the defining condition for an inclusion homomorphism is equivalent to x ◦ y ⊆ f −1 (f (x) ∗ f (y)). It is also clear for an inclusion homomorphism that (x ◦ y)f ⊆ f −1 (f (x) ∗ f (y)). The defining condition for an inclusion homomorphism is also valid for sets. That is, if A, B are non-empty subsets of H1 , then it follows that f (A ◦ B) ⊆ f (A) ∗ f (B). Applying the above relation for A = xf and B = yf , we obtain xf ◦ yf ⊆ f −1 (f (x) ∗ f (y)) and (xf ◦ yf )f ⊆ f −1 (f (x) ∗ f (y)). Homomorphisms having various types of properties are defined and studied in the literature. Each of these properties can be viewed as a condition on f −1 (f (x) ∗ f (y)). We consider four types of homomorphisms in the following definition. Definition 2.4.4. Let (H1 , ◦) and (H2 , ∗) be two hypergroups and f : H1 −→ H2 be a mapping. Then, given x, y ∈ H1 , f is called a homomorphism of

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1, 2, 3, 4,

if if if if

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f −1 (f (x) ∗ f (y)) = (xf ◦ yf )f ; f −1 (f (x) ∗ f (y)) = (x ◦ y)f ; f −1 (f (x) ∗ f (y)) = xf ◦ yf ; f −1 (f (x) ∗ f (y)) = (x ◦ y)f = xf ◦ yf .

Note that x ◦ y ⊆ (x ◦ y)f , x ◦ y ⊆ xf ◦ yf and that (x ◦ y)f ⊆ (xf ◦ yf )f , xf ◦ yf ⊆ (xf ◦ yf )f . Hence, a homomorphism of any type 1 through 4 is indeed an inclusion homomorphism. Observe that a one to one homomorphism of H1 onto H2 of any type 1 through 4 is an isomorphism. Proposition 2.4.5. Let (H1 , ◦) and (H2 , ∗) be two hypergroups, A, B are non-empty subsets of H1 and f : H1 −→ H2 be a mapping. Then, f is a homomorphism of (1) (2) (3) (4)

type type type type

1 2 3 4

implies implies implies implies

f −1 (f (A) ∗ f (B)) = (Af ◦ Bf )f ; f −1 (f (A) ∗ f (B)) = (A ◦ B)f ; f −1 (f (A) ∗ f (B)) = Af ◦ Bf ; f −1 (f (A) ∗ f (B)) = (A ◦ B)f = Af ◦ Bf .

Proof. Each part is established by a straightforward set theoretic argument.  Proposition 2.4.6. Let (H1 , ◦) and (H2 , ∗) be two hypergroups and f : H1 −→ H2 be a mapping. Then, f is a homomorphism of (1) type 4 if and only if f is a homomorphism of type 2 and type 3; (2) type 1 if f is a homomorphism of type 2 or type 3. Proof. (1) It is trivial. (2) Suppose that x, y ∈ H1 and f is a homomorphism of type 2. Then, (x ◦ y)f ⊆ (xf ◦ yf )f ⊆ f −1 (f (x) ∗ f (y)) = (x ◦ y)f . Similarly, if f is a homomorphism of type 3, then xf ◦ yf ⊆ (xf ◦ yf )f ⊆ f −1 (f (x) ∗ f (y)) = xf ◦ yf . Hence, in either case, f is a homomorphism of type 1. Thus, (2) holds.  2 %

&

&

%

4

1

3 Homomorphism Types

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The defining condition for a homomorphism of type 1 or type 2 can easily be simplified if the homomorphism is onto. Proposition 2.4.7. Let (H1 , ◦) and (H2 , ∗) be two hypergroups and f : H1 −→ H2 be an onto mapping. Then, given x, y ∈ H1 , f is a homomorphism of (1) type 1 if and only if f (xf ◦ yf ) = f (x) ∗ f (y); (2) type 2 if and only if f (x ◦ y) = f (x) ∗ f (y). Proof. It is straightforward.  Corollary 2.4.8. Let (H1 , ◦) and (H2 , ∗) be two hypergroups, A, B be nonempty subsets of H1 and f : H1 −→ H2 be an onto mapping. Then, f is a homomorphism of (1) type 1 implies f (Af ◦ Bf ) = f (A) ∗ f (B); (2) type 2 implies f (A ◦ B) = f (A) ∗ f (B). Example 2.4.9. Consider H1 = {0, 1, 2} and H2 = {a, b} together with the following hyperoperations: ◦ 0 1 2 0 0 {0, 1} {0, 2} 1 {0, 1} 1 {1, 2} 2 {0, 2} {1, 2} 2

∗ a b a a {a, b} b {a, b} b

and suppose that f : H1 −→ H2 is defined by f (0) = f (1) = a and f (2) = b. Then, f is a good homomorphism of type 4. On a hypergroup H, we are concerened with equivalence relations for which the family of equivalence classes forms a hypergroup under the hyperopertation induced by that on H. For an equivalence relation ρ on H, we may use xρ , x or ρ(x) to denote the equivalence class of x ∈ H. Moreover, generally, if A is a non-empty subset of H, then Aρ = ∪{xρ | x ∈ A}. We let H/ρ (read H modulo ρ) denote the family {xρ | x ∈ H} of classes of ρ. The hyperoperation on H induces a hyperoperation ⊗ on H/ρ defined by xρ ⊗ yρ = {zρ | z ∈ xρ ◦ yρ }, where x, y ∈ H. The structure (H/ρ, ⊗) is known as a factor or quotient structure. Note that in the definition of ⊗, the condition z ∈ xρ ◦ yρ maybe replaced by z ∈ (xρ ◦ yρ )ρ or zρ ⊆ (xρ ◦ yρ )ρ . Obviously, ∪(xρ ⊗ yρ ) = (xρ ◦ yρ )ρ .

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Proposition 2.4.10. Let (H, ◦) be a hypergroup. Then, (H/ρ, ⊗) is a hypergroup if and only if for all x, y, z ∈ H, ((xρ ◦ yρ )ρ ◦ zρ )ρ = (xρ ◦ (yρ ◦ zρ )ρ )ρ . Proof. In H/ρ, we have (xρ ⊗ yρ ) ⊗ zρ = {uρ | u ∈ xρ ◦ yρ } ⊗ zρ = {tρ | t ∈ uρ ◦ zρ , u ∈ xρ ◦ yρ } = {tρ | t ∈ (xρ ◦ yρ )ρ ◦ zρ }. Similarly, xρ ⊗ (yρ ⊗ zρ ) = {tρ | t ∈ xρ ◦ (yρ ◦ zρ )ρ }. Therefore, ⊗ is associative. Reproducibility in (H/ρ, ⊗) is a consequence of reproducibility in H. Suppose that xρ , yρ ∈ H/ρ. Let u, v ∈ H such that y ∈ x ◦ u, v ◦ x. Then, obviousely, yρ ∈ xρ ⊗ uρ , vρ ⊗ xρ . Hence, the proposition holds.  Definition 2.4.11. Let ρ be an equivalence relation on a hypergroup (H, ◦). Then, given x, y ∈ H, ρ is said to be of type type type type

1, 2, 3, 4,

if if if if

H/ρ is a hypergroup; xρ ◦ yρ ⊆ (x ◦ y)ρ ; (x ◦ y)ρ ⊆ xρ ◦ yρ ; xρ ◦ yρ = (x ◦ y)ρ .

Observe that being of type 2 is equivalent to (xρ ◦ yρ )ρ = (x ◦ y)ρ , and of type 3 to (xρ ◦ yρ )ρ = xρ ◦ yρ . Note that for an equivalence of type 2, xρ ⊗ yρ = {zρ | z ∈ x ◦ y}. Proposition 2.4.12. Let ρ be an equivalence relation on a hypergroup (H, ◦). Then, ρ is of (1) type 4 if and only if ρ is of type 2 and type 3; (2) type 1 if ρ is of type 2 or type 3. Proof. (1) It is straightforward. (2) Let x, y, z ∈ H. Suppose that ρ is of type 2. Then, we obtain ((xρ ◦ yρ )ρ ◦ zρ )ρ = ((x ◦ y) ◦ z)ρ and (xρ ◦ (yρ ◦ zρ )ρ )ρ = (x ◦ (y ◦ z))ρ . Suppose that ρ is of type 3. Then, we have ((xρ ◦ yρ )ρ ◦ zρ )ρ = (xρ ◦ yρ ) ◦ zρ and (xρ ◦ (yρ ◦ zρ )ρ )ρ = xρ ◦ (yρ ◦ zρ ). Hence, in either case, Proposition 2.4.10 applies and yields that H/ρ is a hypergroup. Therefore, ρ is of type 1 and (2) holds.  Homomorphisms between hypergroups and equivalence relations on hypergroups are closely related. The following results are fundamental.

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Theorem 2.4.13. Let (H1 , ◦) and (H2 , ∗) be two hypergroups and f : H1 −→ H2 be an onto mapping.We denote also by f the equivalence relation by f on H1 whose classes comprise the family {xf | x ∈ H1 }. Then, for n = 1, 2, 3 or 4, f is an equivalence relation of type n on H1 for which H1 /f is canonically isomorphic to H2 if and only if f is a homomorphism of type n. Proof. Let n = 1. Suppose that f is a homomorphism of type 1. In order to show f is an equivalence relation of type 1 on H1 , Proposition 2.4.10 is employed. Let x, y, z ∈ H1 . By Corollary 2.4.8 (1), ((xf ◦ yf )f ◦ zf )f = f −1 (f ((xf ◦ yf )f ◦ zf )) = f −1 (f (xf ◦ yf ) ∗ f (z)) = f −1 (f (x) ∗ f (y) ∗ f (z)). Similarly, we obtain (xf ◦ (yf ◦ zf )f )f = f −1 (f (x) ∗ f (y) ∗ f (z)). Thus, f is an equivalence relation of type 1. The canonical mapping θ of H1 /f onto H2 is given for x ∈ H1 by θ(xf ) = f (x). It is clearly well defined and one to one. Moreover, for x, y ∈ H1 , θ(xf ⊗ yf ) = θ({zf | z ∈ xf ◦ yf }) = {f (z) | z ∈ xf ◦ yf } = f (xf ◦ yf ) and θ(xf ) ∗ θ(yf ) = f (x) ∗ f (y). Therefore, Proposition 2.4.7 (1) yields that H1 /f is canonically isomorphic to H2 if and only if f is a homomorphism of type 1. The theorem is then established for n = 1. Now, let n > 1. If f is a homomorphism of type n, then Proposition 2.4.6 and the theorem for n = 1 imply that H1 /f is canonically isomorphic to H2 . On the other hand, if H1 /f is canonically isomorphic to H2 , then the above relations yields for x, y ∈ H1 that f −1 (f (x) ∗ f (y)) = (xf ◦ yf )f . Therefore, for n = 2, 3 or 4, f is an equivalence relation of type n if and only if f is a homomorphism of type n.  Corollary 2.4.14. Let (H, ◦) be a hypergroup. Let θ be an equivalence relation on H. We denote also by θ the canonical mapping of H onto H/θ. Then, for n = 1, 2, 3 or 4, θ is an equivalence relation of type n on H if and only if H/θ is a hypergroup and θ is a homomorphism of type n of H onto H/θ. Proof. By Proposition 2.4.13, if θ is an equivalence relation of type 1, 2, 3 or 4, then H/θ is a hypergroup. Note that for x ∈ H, θ(x) = xθ , an element

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of H/θ, and that θ−1 (θ(x)) = xθ , a subset of H. This compatibility allows the theorem to be applied and yields the corollary.  Three equivalence relations on a hypergroup are of such importance that they will be referred to as the fundamental equivalences. They arise as one tries to discriminate between pairs of elements by means of the hyperoperation. Definition 2.4.15. Let (H, ◦) be a hypergroup. Let x, y ∈ H. Then, x and y are said to be operationally equivalent or 0-equivalent if x ◦ a = y ◦ a and a ◦ x = a ◦ y for every a ∈ H. The elements x and y are said to be inseparable or i-equivalent if x ∈ a ◦ b when and only when y ∈ a ◦ b for every a, b ∈ H. Also, x and y are said to be essentially indistinguishable or e-equivalent if they are both operationally equivalent and inseparable. Obviously, the relations o-, i- and e-equivalence, denoted respectively by o, i, e, are equivalence relations on a hypergroup H. For x ∈ H, the o-, i- and e-equivalence classes of x are hence denoted by xo , xi and xe , respectively. Proposition 2.4.16. Let (H, ◦) be a hypergroup and x, y ∈ H. (1) xo ◦ yo = x ◦ y; o-equivalence is of type 2. (2) (x ◦ y)i = x ◦ y; i-equivalence is of type 3. (3) xe = xo ∩ xi ; xe ◦ ye = (x ◦ y)e = x ◦ y; e-equivalence is of type 4. Proof. It is an immediate consequence of definition.  Corollary 2.4.17. Given that H is a hypergroup, H/o, H/i and H/e are hypergroups. The canonical mappings of H onto H/o, H/i and H/e are homomorphisms of type 2, 3 and 4, respectively. Definition 2.4.18. A hypergroup H in which xo = x for each x ∈ H, xi = x for each x ∈ H or xe = x for each x ∈ H is said respectively to be o-reduced, i-reduced or e-reduced. An e-reduced hypergroup is simply said to be reduced. Example 2.4.19. Let H = {a, b, c, d}. Let the hyperoperation ◦ on H be given by the following table: ◦ a a {a, b} b {a, b} c {c, d} d {c, d}

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

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One easily checks that (H, ◦) is a hypergroup. The o-equivalence classes are seen to be {a, b}, c and d, whereas the i-equivalence classes are a, b and {c, d}. Therefore, the e-equivalence classes are all singletons, that is to say, H is e-reduced. Example 2.4.20. Let H = Z×Z∗ , where Z∗ is the set of non-zero integers. Let ρ be the equivalence relation that puts equivalent “fraction” into classes, that is, for (x, y) ∈ H, (x, y)ρ = {(u, v) | xv = yu}. The hyperoperation ◦ on H is given for (w, x), (y, z) ∈ H by (w, x) ◦ (y, z) = (wz + xy, xz)ρ . Then, (H, ◦) is a hypergroup in which (x, y)o = (x, y)i = (x, y)e = (x, y)ρ , for (x, y) ∈ H. Furthermore, (H/ρ, ⊗) ∼ = (Q, +). Proposition 2.4.21. Let (H1 , ◦), (H2 , ∗) and (H3 , •) be hypergroups. For n = 1, 2, 3 or 4, let f be a homomorphism of type n of H1 onto H2 and g be a homomorphism of type n of H2 onto H3 . Then, gf is a homomorphism of type n of H1 onto H3 . Proof. Suppose that x, y ∈ H1 . We first observe that for z ∈ H1 , zgf = f −1 g −1 gf (z) = f −1 (f (z)g ). Let n = 1. By the above relation, we obtain
gf (xgf ◦ ygf ) = gf f −1 (f (x)g ) ◦ f −1 (f (y)g ) . Since f is onto, Corollary 2.4.8 (1) applies and yields
gf f −1 (f (x)g ) ◦ f −1 (f (y)g ) = g(f (x)g ∗ f (y)g ). Then, Proposition 2.4.7 (1) gives g (f (x)g ∗ f (y)g ) = gf (x) • gf (y). Hence, gf is a homomorphism of type 1 by the above relations and Proposition 2.4.7 (1). Let n = 2. Similar to the previous case, but simpler. Let n = 3. Since g is of type 3, f −1 g −1 (gf (x) • gf (y)) = f −1 (f (x)g ∗ f (y)g ).

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Since f is onto, Proposition 2.4.5 (3) applies and gives f −1 (f (x)g ∗ f (y)g ) = f −1 (f (x)g ) ◦ f −1 (f (y)g . Now, we obtain f −1 (f (x)g ) ◦ f −1 (f (y)g ) = xgf ◦ ygf . So, by the above relations, we conclude that gf is a homomorphism of type 3. Let n = 4. Then, gf is a homomorphism of type 4 by Proposition 2.4.6 (1).  In the following definition, we introduce more types of homomorphisms of hypergroups that appeared in the literature under various names. For a, b, we denote a/b = {x | a ∈ x ◦ b} and b\a = {y | a ∈ b ◦ y}. Definition 2.4.22. Let (H1 , ◦) and (H2 , ∗) be two hypergroups and f : H1 −→ H2 be a mapping. We say that f is a homomorphism of type 5, if for all x, y ∈ H1 , f is a good homomorphism and furthermore (1) f (x/y) = f (x/f −1 (f (y))), (2) f (y\x) = f (f −1 (f (y)\x)); type 6, if for all x, y ∈ H1 , f is a good homomorphism and furthermore (3) f (x/f −1 (f (y))) = f (x)/f (y), (4) f (f −1 (f (y))\x) = f (y)\f (x); type 7, if for all x, y ∈ H1 , f is a good homomorphism and furthermore (5) f (x/y) = f (x)/f (y), (6) f (y\x) = f (y)\f (x). Theorem 2.4.23. If f : H1 −→ H2 is a homomorphism of type 7, then f is a homomorphism of type 4. Proof. In general, if f is an inclusion homomorphism, then for every x, y ∈ H1 , f −1 (f (x)) ◦ f −1 (f (y)) ⊆ f −1 (f (x) ∗ f (y)). Now, let f be a homomorphism of type 7. Suppose that z ∈ f −1 (f (x) ∗ f (y)). Then, f (z) ∈ f (x) ∗ f (y), which implies that f (y) ∈ f (x)\f (z) and

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so f (y) ∈ f (x\z). Thus, there exists y 0 ∈ x\z such that f (y) = f (y 0 ), consequently z ∈ x ◦ y 0 ⊆ f −1 (f (x)) ◦ f −1 (f (y)). Therefore, f −1 (f (x) ∗ f (y)) ⊆ f −1 (f (x)) ◦ f −1 (f (y)).  Theorem 2.4.24. If f : H1 −→ H2 is an onto homomorphism of type 4, then f is a homomorphism of type 6. Proof. We know that an onto homomorphism of type 4 is a good homomorphism. Suppose that u ∈ f (z)/f (x). Then, there exists y such that f (y) = u and so f (z) ∈ f (y) ∗ f (x). Consequently, z ∈ f −1 (f (y) ∗ f (x)) = f −1 (f (y)) ◦ f −1 (f (x)), it follows that there exist a, b such that z ∈ a ◦ b, where a ∈ f −1 (f (y)) and b ∈ f −1 (f (x)). Hence, f (y) = f (a), f (x) = f (b) and so u = f (a) ∈ f (z/b) ⊆ f (z/f −1 (f (x))). Therefore, f (z)/f (x) ⊆ f (z/f −1 (f (x))). Note that the inverse inclusion is always true. Similarly, we can prove f (f −1 (f (y))\x) = f (y)\f (x).  Notice that, in general, a homomorphism of type 4 is not good. Example 2.4.25. Let H(Z) be the hypergroup (N, ◦), where for all x, y, x ◦ y = {x + y, |x − y|}. Let f : H(Z) −→ H(Z) be the function defined by  0 if x ∈ 2N f (x) = 1 if x ∈ 2N + 1. Then, f is a homomorphism of type 4 but is not a good homomorphism.

2.5

Regular and strongly regular relations

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.

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Definition 2.5.1. 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 2.5.2. 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) 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 2.5.3. 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. 

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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 2.5.4. 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. Let h1 ρf h2 and a be an arbitrary element of H. If u ∈ h1 ◦ a, then 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.  Theorem 2.5.5. 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. 

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Corollary 2.5.6. 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. Proof. It is obvious.  Theorem 2.5.7. 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 2.5.8. 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 2.5.9. β ∗ is the smallest strongly regular relation on H. 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.

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(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. Definition 2.5.10. 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 ∗ [74]. 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 [94] and studied mainly by Corsini, Davvaz, Freni, Leoreanu, Vougiouklis and many others. Freni in [73] introduced the relation γ as a generalization of the relation β. Definition 2.5.11. Let H be a semihypergroup. Then, we set γ1 = {(x, x) | x ∈ H} and for every integer n > 1, γn is the relation defined as follows: n n Q Q x γn y ⇐⇒ ∃(z1 , . . . , zn ) ∈ H n , ∃σ ∈ Sn : x ∈ zi , y ∈ zσ(i) . i=1

i=1

Obviously, for n ≥ 1, the relations γn are symmetric, and the relation S γ= γn is reflexive and symmetric. n≥1

Let γ ∗ be the transitive closure of γ. If H is a hypergroup, then γ = γ ∗ [73]. Theorem 2.5.12. The relation γ ∗ is a strongly regular relation. Proof. Clearly, γ ∗ is an equivalence relation. In order to prove that it is strongly regular, we have to show first that xγy =⇒ (x ◦ a) γ (y ◦ a) and (a ◦ x) γ (a ◦ y), for every a ∈ H. If xγy, then there is n ∈ N such that xγn y. Hence, there n n Q Q exist (z1 , . . . , zn ) ∈ H n and σ ∈ Sn such that x ∈ zi and y ∈ zσ(i) . i=1

i=1

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For every a ∈ H, set a = zn+1 and let τ be a permutation of Sn+1 such that τ (i) = σ(i), ∀i ∈ {1, 2, . . . , n}; τ (n + 1) = n + 1. n Q

For all v ∈ x ◦ a and for all w ∈ y ◦ a, we have v ∈ x ◦ a ⊆

zi ◦ a =

i=1

and w ∈ y ◦ a ⊆

n Q i=1

zσ(i) ◦ a =

n Q

zσ(i) ◦ zn+1 =

i=1

n+1 Q

n+1 Q

zi

i=1

zτ (i) . So, v γn+1 w and

i=1

hence v γ w. Thus, (x ◦ a) γ (y ◦ a). In the same way, we can show that (a ◦ x) γ (a ◦ y). Moreover, if x γ ∗ y, then there exist m ∈ N and (w0 = x, w1 , . . . , wm−1 , wm = y) ∈ H m+1 such that x = w0 γ w1 γ . . . γ wm−1 γ wm = y. Now, we obtain x ◦ a = w0 ◦ a γ w1 ◦ a γ w2 ◦ a γ . . . γ wm−1 ◦ a γ wm ◦ a = y ◦ a. Finally, for all v ∈ x ◦ a = w0 ◦ a and for all w ∈ wm ◦ a = y ◦ a, taking z1 ∈ w1 ◦ a, z2 ∈ w2 ◦ a, . . . , zm−1 ∈ wm−1 ◦ a, we have v γ z1 γ z2 γ . . . γ zm−1 γ w, and so v γ ∗ w. Therefore, x ◦ aγ ∗ y ◦ a. Similarly, we obtain a ◦ xγ ∗ a ◦ y. Hence, γ ∗ is strongly regular.  Corollary 2.5.13. The quotient H/γ ∗ is a commutative semigroup. Furthermore, if H is a hypergroup, then H/γ ∗ is a commutative group. Proof. Since γ ∗ is a strongly regular relation, the quotient H/γ ∗ is a semigroup under the following operation: γ ∗ (x1 ) ⊗ γ ∗ (x2 ) = γ ∗ (z), for all z ∈ x1 ◦ x2 . Moreover, if H is a hypergroup, then H/γ ∗ is a group. Finally, if σ is the cycle of S2 such that σ(1) = 2, for all z ∈ x1 ◦ x2 and w ∈ xσ(1) ◦ xσ(2) , we have zγ2 w, so zγ ∗ w and γ ∗ (x1 ) ⊗ γ ∗ (x2 ) = γ ∗ (z) = γ ∗ (x2 ) ⊗ γ ∗ (x1 ).  Theorem 2.5.14. The relation γ ∗ is the smallest strongly regular relation on a semihypergroup H such that the quotient H/γ ∗ is commutative semigroup. Proof. Suppose that ρ is a strongly regular relation such that H/ρ is a commutative semigroup and ϕ : H −→ H/ρ is the canonical projection. Then, ϕ is a good homomorphism. Moreover, if x γn y, then there exist n n Q Q (z1 , . . . , zn ) ∈ H n and σ ∈ Sn such that x ∈ zi and y ∈ zσ(i) , whence i=1

i=1

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ϕ(x) = ϕ(z1 ) ⊗ . . . ⊗ ϕ(zn ) and ϕ(y) = ϕ(zσ(1) ) ⊗ . . . ⊗ ϕ(zσ(n) ). By the commutativity of H/ρ, it follows that ϕ(x) = ϕ(y) and x ρ y. Thus, x γn y implies x ρ y, and obviously, x γ y implies that x ρ y. Finally, if x γ ∗ y, then there exist m ∈ N and (w0 = x, w1 , . . . , wm−1 , wm = y) ∈ H m+1 such that x = w0 γ w1 γ . . . γ wm−1 γ wm = y. Therefore, x = w0 ρ w1 ρ . . . ρ wm−1 ρ wm = y, and transitivity of ρ implies that x ρ y. Therefore, γ ∗ ⊆ ρ.  Complete parts were introduced and studied for the first time by M. Koskas [94]. Later, this topic was analyzed by P. Corsini [29], Y. Sureau [138] and B. Davvaz et al [54; 55] mostly in the general theory of hypergroups. M. De Salvo studied complete parts from a combinatorial point of view. A generalization of them, called n-complete parts, was introduced by R. Migliorato. Other authors gave a contribution to the study of complete parts and of the heart of a hypergroup. Among them, V. Leoreanu analyzed the structure of the heart of a hypergroup in her Ph.D. Thesis. We present now the definitions. Definition 2.5.15. Let (H, ◦) be a semihypergroup and A be a non-empty subset of H. We say that A is a complete part of H if for any non-zero natural number n and for all a1 , . . . , an of H, the following implication holds: A∩

n Q

ai 6= ∅ =⇒

i=1

n Q

ai ⊆ A.

i=1

Theorem 2.5.16. If (H, ◦) is a semihypergroup and R is a strongly regular relation on H, then for all z of H, the equivalence class of z is a complete part of H. Proof. Let a1 , . . . , an be elements of H, such that z∩

n Q

ai 6= ∅.

i=1

Then, there exists y ∈

n Q

ai , such that y R z. The homomorphism π :

i=1

H −→ H/R  n is good nand H/R is a semigroup.n It follows that π(z) = Q Q Q π(y) = π ai = π(ai ). This means that ai ⊆ z.  i=1

i=1

i=1

Now, we want to determine some necessary and sufficient conditions so

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that the relation γ is transitive. Definition 2.5.17. Let M be a non-empty subset of H. We say that M is a γ-part of H if for any non-zero natural number n, for all (z1 , . . . , zn ) ∈ H n and for all σ ∈ Sn , we have M∩

n Q

zi 6= ∅ =⇒

i=1

n Q

zσ(i) ⊆ M.

i=1

Lemma 2.5.18. Let M be a non-empty subsets of H. Then, the following conditions are equivalent: (1) M is a γ-part of H; (2) x ∈ M, xγy =⇒ y ∈ M ; (3) x ∈ M, xγ ∗ y =⇒ y ∈ M . Proof. (1=⇒2) If (x, y) ∈ H 2 is a pair such that x ∈ M and x γ y, then n Q there exist n ∈ N, (z1 , . . . , zn ) ∈ H n and σ ∈ Sn such that x ∈ M ∩ zi and y ∈

n Q

zσ(i) . Since M is a γ-part of H, we have

i=1

n Q

i=1

zσ(i) ⊆ M and

i=1

y ∈ M. (2=⇒3) Assume that (x, y) ∈ H 2 such that x ∈ M and x γ ∗ y. Obviously, there exist m ∈ N and (w0 = x, w1 , . . . , wm−1 , wm = y) ∈ H m+1 such that x = w0 γ w1 γ . . . γ wm−1 γ wm = y. Since x ∈ M , applying (2) m times, we obtain y ∈ M . n n Q Q (3=⇒1) Suppose that M ∩ xi 6= ∅ and x ∈ M ∩ xi . For every σ ∈ Sn and for every y ∈

n Q

i=1

i=1

xσ(i) , we have x γ y. Thus, x ∈ M and x γ ∗ y.

i=1

Finally, by (3), we obtain y ∈ M , whence

n Q

xσ(i) ⊆ M . 

i=1

Before proving the next theorem, we introduce the following notations: Let H be a semihypergroup. For all x ∈ H, we set   n Q n • Tn (x) = (x1 , . . . , xn ) ∈ H | x ∈ xi ; i=1  n  S Q • Pn (x) = xσ(i) | σ ∈ Sn , (x1 , . . . , xn ) ∈ Tn (x) ; S i=1 • Pσ (x) = Pn (x). n≥1

From the preceding notations and definitions, it follows at once the following:

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Lemma 2.5.19. For every x ∈ H, Pσ (x) = {y ∈ H | xγy}. Proof. For all x, y ∈ H, we have n Q

xγy ⇐⇒ ∃n ∈ N, ∃(x1 , . . . , xn ) ∈ H n , ∃σ ∈ Sn : x ∈

xi , y ∈

i=1

n Q

xσ(i)

i=1

⇐⇒ ∃n ∈ N : y ∈ Pn (x) ⇐⇒ y ∈ Pσ (x).  Theorem 2.5.20. Let H be a semihypergroup. Then, the following conditions are equivalent: (1) γ is transitive; (2) γ ∗ (x) = Pσ (x), for all x ∈ H; (3) Pσ (x) is a γ-part of H, for all x ∈ H. Proof. (1=⇒2) By Lemma 2.5.19, for all x, y ∈ H, we have y ∈ γ ∗ (x) ⇐⇒ xγ ∗ y ⇐⇒ xγy ⇐⇒ y ∈ Pσ (x). (2=⇒3) By Lemma 2.5.18, if M is a non-empty subset of H, then M is a γ-part of H if and only if it is union of equivalence classes modulo γ ∗ . In particular, every equivalence class modulo γ ∗ is a γ-part of H. (3=⇒1) If x γ y and y γ z, then there exist m, n ∈ N, (x1 , . . . , xn ) ∈ n Q Tn (x), (y1 , . . . , ym ) ∈ Tm (y), σ ∈ Sn and τ ∈ Sm such that y ∈ xσ(i) and z ∈

m Q

i=1

yτ (i) . Since Pσ (x) is a γ-part of H, we have

i=1

x∈

n Q

xi ∩ Pσ (x) =⇒

i=1

=⇒

n Q i=1 m Q

xσ(i) ⊆ Pσ (x) =⇒ y ∈

m Q

yi ∩ Pσ (x)

i=1

yτ (i) ⊆ Pσ (x) =⇒ z ∈ Pσ (x)

i=1

=⇒ ∃k ∈ N : z ∈ Pk (x) =⇒ z γ x. Therefore, γ is transitive.  2.6

Complete hypergroups

In this section, we study the concept of complete hypergroups. Definition 2.6.1. Let A be a non-empty subset of H. The intersection of the parts of H which are complete and contain A is called the complete closure of A in H; it will be denoted by C(A). A semihypergroup H is complete, if it satisfies one of the following conditions:

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(1) ∀(x, y) ∈ H 2 , ∀a ∈ x ◦ y, C(a) = x ◦ y. (2) ∀(x, y) ∈ H 2 , C(x ◦ y) = x ◦ y. (3) ∀(m, n) ∈ N2 , 2 ≤ m, n, ∀(x1 , . . . , xn ) ∈ H n , ∀(y1 , . . . , ym ) ∈ H m , n Q i=1

xi ∩

m Q

yj 6= ∅ =⇒

j=1

n Q

xi =

i=1

m Q

yj .

j=1

A hypergroup is complete if it is a complete semihypergroup. If (H, ◦) is a complete semihypergroup, then either there exist a, b ∈ H such that β ∗ (x) = a ◦ b or β ∗ (x) = {x}. An element e ∈ H is called an identity if a ∈ e ◦ a ∩ a ◦ e for all a ∈ H. An element x0 is called an inverse of x if an identity e exists such that e ∈ x ◦ x0 ∩ x0 ◦ x. Definition 2.6.2. A regular hypergroup H is a hypergroup which it has at least one identity and every element has at least one inverse. A regular hypergroup H is said to be reversible, if it satisfies the following conditions: ∀(a, b, x) ∈ H 3 : a ∈ b ◦ x =⇒ ∃x0 ∈ i(x) : b ∈ a ◦ x0 , a ∈ x ◦ b =⇒ ∃x00 ∈ i(x) : b ∈ x00 ◦ a. If H is regular, for every x ∈ H, we denote i(x) the set of the inverses of x. Theorem 2.6.3. If (H, ◦) is a complete hypergroup, then (1) ωH = {e ∈ H : ∀x ∈ H, x ∈ x ◦ e ∩ e ◦ x}, which means that ωH is the set of two-sided identities of H. (2) H is regular (i.e. H has at least one identity and any element has an inverse) and reversible. Proof. (1) If u ∈ ωH , then for all a ∈ H, we have a ∈ C(a) = a ◦ ωH = a ◦ u. Similarly we have a ∈ u ◦ a, which means that u is a two-sided identity of H. Conversely, any two-sided identity u of H is an element of ωH , since ϕ(u) = 1. (2) Let a, a0 , a00 be elements of H and e be a two-sided identity, such that e ∈ a0 ◦ a ∩ a ◦ a00 . Then, a0 ◦ a = ωH = a ◦ a00 and a ◦ a0 ⊆ a ◦ a0 ◦ a ◦ a00 ⊆ a ◦ ωH ◦ a00 = ωH ◦ a ◦ a00 = ωH . Hence, a ◦ a0 = ωH , so a0 is an inverse of a. Moreover, if a ∈ b ◦ c, then ωH = a0 ◦ a ⊆ a0 ◦ b ◦ c, so for any inverse c0 of c, we have c0 ∈ ωH ◦ c0 ⊆ a0 ◦ b ◦ c ◦ c0 = a0 ◦ b ◦ ωH = a0 ◦ b.

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Similarly, from here we obtain b0 ∈ c ◦ a0 , and so b0 ◦ a ⊆ c ◦ a0 ◦ a = C(c), whence c ∈ C(c) = b0 ◦ a. In a similar way, we obtain b ∈ a ◦ c0 .  Definition 2.6.4. A hypergroup (H, ◦) is called flat if for all subhypergroup K of H, we have ωK = ωH ∩ K. Theorem 2.6.5. Any complete hypergroup is flat. Proof. Let H be a complete hypergroup and let K be a subhypergroup H. We have ωH ∩ K = {e ∈ K : ∀a ∈ H, x ∈ e ◦ x ∩ x ◦ e} ⊆ ωK . Moreover, y ∈ CK (x) ⇒ yβK x ⇒ yβH x ⇒ y ∈ CH (x), which means that CK (x) ⊆ CH (x). Clearly, ωH ∩ K 6= ∅. If x ∈ ωH ∩ K ⊆ ωK , then CK (x) = ωK , CH (x) = ωH . Hence, ωK ⊆ ωH whence ωK ⊆ ωH ∩ K. Hence, ωK =ωH ∩ K.  Corollary 2.6.6. If K is a subhypergroup of a complete hypergroup (H, ◦), then ωK = ωH . Proof. Set x ∈ ωH ∩ K. We have ωH = C(x ◦ x) = x ◦ x ⊆ ωH ∩ K, whence ωH ⊆ ωH ∩ K, then we apply the above theorem. Hence, ωK =ωH .  Theorem 2.6.7. Let H, H 0 be complete hypergroups and f : H → H 0 be a 0 . good homomorphism. Then, we have f (ωH ) = ωH Proof. Let x ∈ ωH . Then, x ◦ x = ωH , whence f (x) ◦ f (x) = f (ωH ). On the other hand, f (x) is an identity of H 0 , since x is an identity of H, which 0 = f (x) ◦ f (x) = f (ωH ).  means that f (x) ∈ ωH 0 . Hence, ωH Now, we let n ≥ 2. Definition 2.6.8. A hypergroup H issaid tobe an n-complete hypergroup n n Q Q if for all (z1 , . . . , zn ) ∈ H n , zi = β zi . i=1

i=1

If H is n-complete, then β = βn . A hypergroup H is said to be n∗ -complete if there exists n ∈ N such ∗ 6= βn∗ . that βn∗ = β and βn−1 Definition 2.6.9. A hypergroup H is said be γn -complete if for all  to n n Q Q (z1 , z2 , . . . , zn ) ∈ H n and for all σ ∈ Sn : γ zi = zσ(i) . i=1

i=1

Corollary 2.6.10. If H is a commutative hypergroup, then H is a γn complete hypergroup if and only if H is an n-complete hypergroup. We begin with some properties which are valid in every hypergroup. They concern the relation γn and will be prove only for n > 1. The case

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n = 1 is always trivially true. We suppose that H = (H, ◦) is a hypergroup. Proposition 2.6.11. For any positive integer n, we have (1) γn ⊆ γn+1 ; ∗ (2) γn∗ ⊆ γn+1 . Proof. (1) If x γn y, then there exist (z1 , z2 , . . . , zn ) ∈ H n and σ ∈ Sn such n n Q Q that x ∈ zi , y ∈ zσ(i) . Since H is a hypergroup, so there exists i=1

i=1

(t1 , t2 ) ∈ H 2 such that zn ∈ t1 ◦ t2 . 0 = t2 , so x ∈ Now let zi0 = zi for 1 ≤ i ≤ n − 1 and zn0 = t1 , zn+1 0 0 0 0 0 z1 ◦ z2 . . . ◦ zn ◦ zn+1 . Now, let σ(j) = n and we define σ ∈ Sn+1 as follows:  if 1 ≤ i ≤ j  σ(i) σ 0 (i) = n + 1 if i = j + 1  σ(i − 1) if j + 2 ≤ i. Then, σ 0 ∈ Sn+1 and y ∈

n+1 Q i=1

zσ0 0 (i) . Therefore, x γn+1 y.

(2) It follows from (1).  Proposition 2.6.12. A hypergroup H is γn -complete if and only if for all n n Q Q σ ∈ Sn and for all x ∈ zi , we have γ(x) = zσ(i) . i=1

i=1

Proof. Suppose that H is γn -complete, for σ ∈ Sn . If x ∈

n Q

zi , then we

i=1

have S

γ(x) ⊆ x∈

n Q

γ(x) =

n Q

zσ(i) =⇒ γ(x) ⊆

i=1

zi

n Q

zσ(i) .

i=1

i=1

Now, if y ∈

n Q

zσ(i) , then

i=1

x γn y =⇒ x γ y =⇒ y ∈ γ(x) =⇒

n Q

zσ(i) ⊆ γ(x).

i=1

Qn Conversely, for every σ ∈ Sn , (z1 , z2 , . . . , zn ) ∈ H n and x ∈ i=1 zi we have n n n Q Q S Q γ(x) = zσ(i) =⇒ γ( zi ) = Q γ(x) = zσ(i) .  i=1

i=1

x∈

n i=1

zi

i=1

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Proposition 2.6.13. If H is a γn -complete hypergroup, then γ ∗ = γn . Proof. We know that in every hypergroup γ ∗ = γ, so it is suffices to prove that: γ ⊆ γn . Suppose that x γ y. Then, there exists m ∈ N such that x γm y. If m ≤ n, then γm ⊆ γn . If m > n, then there exist m m Q Q (z1 , z2 , . . . , zm ) ∈ H m and σ ∈ Sm such that x ∈ zi , y ∈ zσ(i) . There i=1 n−1 Q

exists t1 ∈ H such that t1 ∈ zn ◦zn+1 ◦. . . zm , x ∈ 1 ≤ i ≤ n − 1 and un = t1 . Thus, x ∈

i=1

zi ◦t1 . Let ui = zi for

i=1

n Q

ui . We have y = γ(x) =

i=1

n Q

uσ(i)

i=1

which implies that x γn y.  Example 2.6.14. If H is a semihypergroup, then Proposition 2.6.13 maybe not correct: let H be the following semihypergroup: H 1 2 3 4

1 1, 2, 3 1, 2 1, 3 1, 3

2 1, 2 1, 2, 3 2, 3 1, 3

3 1, 3 2, 3 1, 2, 3 1, 2, 3

4 1, 3 1, 3 1, 2, 3 1, 2, 3

Then, H is γ2 -complete, 4γ ∗ 4 but not 4γ2 4. Lemma 2.6.15. For all (a, b, x) ∈ H 3 , a γn b =⇒ [(a ◦ x) γn+1 (b ◦ x) and (x ◦ a) γn+1 (x ◦ b)]. Proof. If a γn b, then there exist (z1 , z2 , . . . , zn ) ∈ H n and σ ∈ Sn such n n n Q Q Q that a ∈ zi , b ∈ zσ(i) . But, we have a ◦ x ⊆ ( zi ) ◦ x and i=1

b◦x ⊆ (

n Q

i=1

i=1

zσ(i) ) ◦ x. Now let ui := zi , for all 1 ≤ i ≤ n, un+1 := x and

i=1

σ 0 ∈ Sn+1 with σ 0 (i) = σ(i), σ 0 (n + 1) = n + 1. Then, we have a◦x⊆

n+1 Q

ui and b ◦ x ⊆

i=1

n+1 Q

uσ0 (i) ,

i=1

which implies that (a ◦ x) γn+1 (b ◦ x). The rest can be proved in an analogous way.  Lemma 2.6.16. For all (a, b, x) ∈ H 3 , ∗ ∗ a γn∗ b =⇒ [(a ◦ x) γn+1 (b ◦ x) and (x ◦ a) γn+1 (x ◦ b)].

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Proof. If a γn∗ b, then there exists (z1 , z2 , . . . , zm ) ∈ H m such that a = z1 γn z2 γn . . . zm−1 γn zm = b. Now, we have a ◦ x = z1 ◦ x γn+1 z2 ◦ x . . . zm−1 ◦ x γn+1 zm ◦ x = b ◦ x. ∗ It means that (a ◦ x) γn+1 (b ◦ x). The rest can be proven similarly. 

Let γ0∗ = ∅. We define the γn∗ -complete hypergroups. Definition 2.6.17. A hypergroup H is said to be γn∗ -complete if n is the ∗ )H . smallest natural number such that (γn∗ )H = γH and (γn∗ )H 6= (γn−1 Example 2.6.18. Let H be the following hypergroup: H a b c d

a a, b c, d a, b c, d

b c, d a, b c, d a, b

c a, b c, d a, b c, d

d c, d a, b c, d a, b

Then, H is a γ3∗ -complete hypergroup. Corollary 2.6.19. Let H be a commutative hypergroup. Then, H is a γn∗ -complete hypergroup if and only if H is an n∗ -complete hypergroup. Proposition 2.6.20. A hypergroup H is γ1∗ -complete if and only if H is an abelian group. Proof. Suppose that H is a γ1∗ -complete hypergroup, so γ1∗ = γ and γ2 ⊆ γ1 . Now, for each x ∈ z1 ◦z2 and y ∈ z2 ◦z1 , we have x γ2 y, so x = y. Therefore, z1 ◦ z2 = z2 ◦ z1 is singleton, and so H is an abelian group. n n Q Q Conversely, if H is an abelian group, then ∀(x, y) ∈ H 2 ; zσ(i) = zi and |

n Q

zi |= 1. By definition, x γn y if and only if x =

i=1

n Q i=1

i=1

zi , y =

n Q

i=1

zσ(i) .

i=1

Thus, x = y and x γ1 y.  Corollary 2.6.21. H is a γn∗ -complete hypergroup if and only if n is the minimum integer such that H/γn∗ is an abelian group. Proposition 2.6.22. In every hypergroups (H, ◦ ) the following conditions are equivalent: (1) H is γ2∗ -complete hypergroup;

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(2) (H/γ2∗ , ⊗ ) is an abelian group; (3) (H/γ2∗ , ⊗ ) is a commutative hypergroup. Proposition 2.6.23. Every finite hypergroup is γn∗ -complete. Proof. Since H is finite, the succession γ1∗ ⊆ γ2∗ ⊆ . . . is stationary. Thus, ∗ ∃n ∈ N : γn∗ = γ and γn∗ 6= γn−1 .  Proposition 2.6.24. If H is a γn -complete hypergroup, then there exists ∗ m ≤ n such that H is γm -complete. Proof. One proves that, if H is γn -complete, then γn = γ, so γn∗ = γ and ∗ ∗ ∗ there exists a m ≤ n such that γm = γ and γm−1 6= γm .  Let ϕ be the canonical projection. Then, we define DH = ϕ−1 (1H/γ ∗ ). Theorem 2.6.25. We have 2 (1) If (v, w) ∈ DH and v γn w, then γ = γn+1 ; ∗ 2 . (2) If (v, w) ∈ DH and v γn∗ w, then γ = γn+1 2 such that y ∈ x◦v and y ∈ Proof. (1) If x γ y, then there exists (v, w) ∈ DH x ◦ w, by hypothesis v γn w. Now, using Lemma 2.6.15, (x ◦ v) γn+1 (x ◦ w), whence x γn+1 y, so γ ⊆ γn+1 . (2) It follows from (a) and Lemma 2.6.16.  2 2 , v γn∗ w and there exists (u0 , w0 ) ∈ DH Theorem 2.6.26. If (v, w) ∈ DH 0 ∗ 0 ∗ ∗ such that u 6∈ γn−1 w , then H is γn -complete or γn+1 -complete.

Example 2.6.27. Both of the two possibilities of Theorem 2.6.26 are verifiable, as the following examples shows: H a b c d H0 a b c

a a b c d

b b a d c a a b c

c c d a, b a, b b b a, b c

d d c a, b a, b c c c a, b

Proposition 2.6.28. If H is a γn -complete hypergroup, then for all (z1 , z2 , n Q . . . , zn ) ∈ H n and for all σ ∈ Sn , zσ(i) is a γ-part of H. i=1

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Proof. By using Proposition 2.6.18, the proof is straightforward.  Corollary 2.6.29. If H is γ2 -complete, then H is a complete hypergroup. Corollary 2.6.30. If H is a γ2 -complete hypergroup, then (1) DH is the set of identity elements of H. (2) H is regular and reversible. Definition 2.6.31. A KH hypergroup is a hypergroup constructed from a hypergroup H = (H, ◦ ) and a family {A(x)}x∈H of non-empty subsets of H such that

Setting KH

∀(x, y) ∈ H 2 : x 6= y =⇒ A(x) ∩ A(y) = ∅. S = A(x) and defining the following hyperoperation ∗: x∈H

2 ; a ∈ A(x), b ∈ A(y), a ∗ b := ∀(a, b) ∈ KH

S

A(z).

z∈x◦y

(H, ◦ ) is a hypergroup if and only if (KH , ∗ ) is a hypergroup. In this case, KH is said to be a KH hypergroup generated by H. For all P ∈ P ∗ (H), let S K(P ) := A(x). x∈P

Proposition 2.6.32. (H, ◦) is a hypergroup if and only if (KH , ∗) is a hypergroup. Proof. It is straightforward. Theorem 2.6.33. If P is a γ-part of (H, ◦), then K(P ) is γ-part of (KH , ∗). m , and ∗ Proof.1 Suppose that (z1 , z2 , . . . , zm ) ∈ KH

u∈∗

m Q

m Q

zi ∩ K(P ) 6= ∅. Let

i=1

zi ∩ K(P ), so there exists (x1 , x2 , . . . , xm ) ∈ H m such that for all

i=1

1 ≤ i ≤ m, zi ∈ A(xi ), u∈

S m Q

y∈◦

A(y) =⇒ ∃ y1 ∈ ◦

m Q

xi : u ∈ A(y1 ).

i=1

xi

i=1

1 If

(◦ ) is a hyperoperation, then ◦

m Q i=1

by hyperoperation ( ◦ ).

xi denotes the hyperproduct of the elements xi

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Since u ∈ K(P ), there exists y2 ∈ P such that u ∈ A(y2 ). Therefore, m Q xi ∩ P. Since P is a A(y1 ) ∩ A(y2 ) 6= ∅, which implies that y1 = y2 ∈ ◦ γ-part of (H, ◦), for all σ ∈ Sm , ◦

m Q

i=1

xσ(i) ⊆ P. Now, for σ ∈ Sm , assume

i=1

that v∈∗

m Q i=1

m Q

Then, there exists w ∈ ◦

S

zσ(i) = y∈◦

A(y).

Qm

xσ(i)

i=1

xσ(i) such that v ∈ A(w). Since ◦

i=1

m Q

xσ(i) ⊆ P ,

i=1

we have A(w) ⊆ K(P ). Therefore, v ∈ K(P ).  Theorem 2.6.34. For every (x, y) ∈ H 2 and (u, v) ∈ A(x) × A(y), the following conditions are pairwise equivalent: (1) u (γn )KH v; (2) x (γn )H y; (3) A(x) (γn )KH A(y). n and Proof. (1=⇒2) Let u (γn )KH v. Then, there exist (z1 , z2 , . . . , zn ) ∈ KH n n Q Q σ ∈ Sn such that u ∈ ∗ zi and v ∈ ∗ zσ(i) . Now, for every 1 ≤ i ≤ n i=1

i=1

there exists yi with zi ∈ A(yi ) such that S u∈ A(w) and v ∈ w∈◦

n Q i=1

Thus, there exist w1 ∈ ◦

n Q

w0 ∈◦

yi

A(w0 ).

S yσ(i)

i=1

n Q

n Q

yi and w2 ∈ ◦

i=1

yσ(i) such that u ∈ A(w1 ),

i=1

v ∈ A(w2 ). So, w1 = x, w2 = y and we obtain x (γn )H y. (2=⇒3) Suppose that x(γn )H y. Then, there exist (z1 , z2 , . . . , zn ) ∈ H n n n Q Q and σ ∈ Sn such that x ∈ ◦ zi , y ∈ ◦ zσ(i) . Now, let yi ∈ A(zi ) for i=1

1 ≤ i ≤ n. Then, n Q ∗ yi = i=1

S w∈◦

n Q

A(w) and ∗ zi

i=1 n Q i=1

w0 ∈◦

i=1

So, A(x) ⊆ ∗

n Q

yi and A(y) ⊆ ∗

i=1

A(w0 ).

S

yσ(i) =

n Q

zσ(i)

i=1

n Q

yσ(i) .

Therefore, we have

i=1

A(x) (γn )KH A(y). (3=⇒1) It is clear.  Theorem 2.6.35. For every (x, y) ∈ H 2 and (u, v) ∈ A(x) × A(y), the following conditions are pairwise equivalent.

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(1) u(γn∗ )KH v; (2) x(γn∗ )H y; (3) A(x) (γn∗ )KH A(y). Proof. (1=⇒2) Let u (γn∗ )KH v. Then, there exist m ∈ N, (x0 , x1 , . . . , xm ) ∈ m+1 KH such that u = x0 (γn )KH x1 . . . xm−1 (γn )KH xm = v. For all 0 ≤ j < n Q n zij and σ j ∈ Sn such that xj ∈ ∗ m − 1, there exist (z1j , z2j , . . . , znj ) ∈ KH and xj+1 ∈ ∗

n Q i=1

i=1

zσj j (i) . Now, for every 1 ≤ i ≤ n, 0 ≤ j < m − 1, there

exists yij such that zij ∈ A(yij ), S xj ∈ A(w) and xj+1 ∈ w∈◦

n Q i=1

yij

Thus, there exist wj ∈ ◦

n Q

w0 ∈◦

i=1

n Q i=1

A(w0 ).

S

yij and wj+1 ∈ ◦

n Q i=1

yj j

σ (i)

yσj j (i) , where xj ∈ A(wj )

and xj+1 ∈ A(wj+1 ) such that w0 (γn )H w1 . . . wm−1 (γn )H wm . So, w0 = x, wm = y. Therefore, we have x (γn∗ )H y. (2=⇒3) Suppose that x (γn∗ )H y. Then, there exist m ∈ N and (x0 , x1 , . . . , xm ) ∈ H m+1 such that x = x0 (γn )H x1 . . . xm−1 (γn )H xm = y. For all 0 ≤ j < m − 1, there exist (z1j , z2j , . . . , znj ) ∈ H n and σ j ∈ Sn such n n Q Q that xj ∈ ◦ zij and xj+1 ∈ ◦ zσj j (i) . For all yij ∈ A(zij ), we have i=1

∗

n Q i=1

yij =

i=1

S w∈◦

n Q i=1

A(w) and ∗

i=1

zij

For all 0 ≤ j < m − 1, A(xj ) ⊆ ∗

n Q

n Q i=1

yσj j (i) =

A(w0 ).

S w0 ∈◦

n Q

zj j i=1 σ (i)

yij and A(xj+1 ) ⊆ ∗

n Q i=1

j yσ(i) . Now, we

have A(x0 ) (γn )KH A(x1 ) . . . A(xm−1 ) (γn )KH A(xm ). Therefore, A(x) (γn∗ )KH A(y). (3=⇒1) It is clear.  Theorem 2.6.36. If H is hypergroup and n ≥ 2, then (1) (γn )H = γH ⇐⇒ (γn )KH = γKH ; ∗ ∗ (2) (γn∗ )H = γH ⇐⇒ (γn∗ )KH = γK . H

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Proof. (1) Suppose that (γn )H = γH . It is enough to show γKH ⊆ (γn )KH . If u γKH v, then there exists (x, y) ∈ H 2 such that u ∈ A(x) and v ∈ A(y). So, x γH y and x (γn )H y. Thus, we have A(x) (γn )KH A(y), and so u (γn )KH v. Conversely, if (γn )KH = γKH , then it is enough to show γH ⊆ (γn )H . We have A(x) (γ)KH A(y). Hence, A(x) (γn )KH A(y), so we have x (γn )H y. (2) The proof is similar to the proof of part (1).  Corollary 2.6.37. For n ≥ 2, H is γn∗ -complete if and only if KH is γn∗ complete. Definition 2.6.38. Let A be a non-empty subset of H. The intersection of the γ-parts of H which contain A is called γ-closure of A in H. It will be denoted Cγ (A). Theorem 2.6.39. Let A be a non-empty subset of H. We pose • G1 (A) := A, • Gn+1 (A) := {x | ∃p ∈ N, ∃(h1 , . . . , hp ) ∈ H p , ∃σ ∈ Sp : p p Q Q x∈ hσ(i) and hi ∩ Gn (A) 6= ∅}, i=1 i=1 S • G(A) := Gn (A). 1≤n

Then, G(A) = Cγ (A). Proof. It is necessary to prove (1) G(A) is a γ-part of H; (2) If A ⊆ B and B is a γ-part of H, then G(A) ⊆ B. Therefore, (1) Let

p Q

xi ∩ G(A) 6= ∅. Then, there exists n ∈ N such that

i=1

Gn (A) 6= ∅. For every σ ∈ Sp and y ∈ Gn+1 (A) and

p Q

p Q

p Q

xi ∩

i=1

xσ(i) we have y ∈

i=1

xσ(i) ⊆ G(A), and so G(A) is a γ-part of H.

i=1

(2) We have A = G1 (A) ⊆ B. Suppose that B is a γ-part of H and Gn (A) ⊆ B. We prove that this implies Gn+1 (A) ⊆ B. For every z ∈ Gn+1 (A) there exist p ∈ N, (x1 , . . . , xp ) ∈ H p and σ ∈ Sp p p Q Q such that z ∈ xσ(i) , xi ∩ Gn (A) 6= ∅. Since Gn (A) ⊆ B, i=1

i=1

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xi ∩ B 6= ∅. Hence, z ∈

i=1

p Q

71

xσ(i) ⊆ B and so Gn+1 (A) ⊆ B. 

i=1

Lemma 2.6.40. We have (1) ∀n ≥ 2, ∀x ∈ H, Gn (G2 (x)) = Gn+1 (x). (2) x ∈ Gn (y) ⇔ y ∈ Gn (x). Proof. (1) We have  p Q G2 (G2 (x)) = z | ∃p ∈ N, ∃(h1 , . . . , hp ) ∈ H p , ∃σ ∈ Sp : z ∈ hσ(i) , p Q

i=1



hi ∩ G2 (x) 6= ∅

i=1

= G3 (x). Now, we proceed by induction. Suppose that Gn−1 (G2 (x)) = Gn (x). Then,  p Q Gn (G2 (x)) = z | ∃p ∈ N, ∃(h1 , . . . , hp ) ∈ H p , ∃σ ∈ Sp : z ∈ hσ(i) , i=1  p Q hi ∩ Gn−1 (G2 (x)) 6= ∅ i=1  p Q = z | ∃p ∈ N, ∃(h1 , . . . , hp ) ∈ H p , ∃σ ∈ Sp : z ∈ hσ(i) , i=1  p Q hi ∩ Gn (x) 6= ∅ i=1

= Gn+1 (x). (2) We prove by induction. It is clear that x ∈ G2 (y) ⇔ y ∈ G2 (x). Suppose that x ∈ Gn−1 (y) ⇔ y ∈ Gn−1 (x). If x ∈ Gn (y), then there exist q ∈ N, (a1 , . . . , aq ) ∈ H q and σ ∈ Sq such that x∈

q Q i=1

aσ(i) and

q Q

ai ∩ Gn−1 (y) 6= ∅,

i=1

by this it follows that there exists v ∈

q Q

ai ∩Gn−1 (y). Therefore, v ∈ G2 (x)

i=1

is obtained. From v ∈ Gn−1 (y) we have y ∈ Gn−1 (G2 (x)) = Gn (x).  Theorem 2.6.41. The relation x G y ⇔ x ∈ G({y}) is an equivalence relation. Proof. We write Cγ (x) instead of Cγ ({x}). Clearly, G is reflexive. Now, suppose that xGy and yGz. If P is a γ-part of H and z ∈ P , then Cγ (z) ⊆ P, y ∈ P and consequently x ∈ Cγ (y) ⊆ P . For this reason x ∈ Cγ (z) that is xGz. The symmetrically of G follows in a direct way from the preceding

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lemma.  Theorem 2.6.42. For all x, y ∈ H, one gets x G y ⇔ x γ ∗ y. Proof. Let xγ y. Then, there exists n ∈ N such that x γn y. So, there exist n n Q Q zi and y ∈ zσ(i) . We (z1 , . . . , zn ) ∈ H n and σ ∈ Sn such that x ∈ have

n Q

i=1

i=1

zi ∩ {x} 6= ∅, so y ∈ G2 (x). Hence,

i=1

x ∈ G2 (y) =⇒ xGy =⇒ γ ⊆ G. Since G is an equivalence relation, γ ∗ ⊆ G. Conversely, if xGy, then there exists n ∈ N such that x ∈ Gn+1 (y), from 1 ) ∈ H m , ∃σ 1 ∈ Sm : this it follows that ∃m ∈ N, ∃(z11 , . . . , zm x∈

m Q i=1

x1 ∈

r Q i=1

m Q i=1

m Q

zi1 ∩ Gn (y). Therefore, i=1 N, (z12 , . . . , zr2 ) ∈ H r , σ 2 ∈ Sr

Thus, x1 ∈ exist r ∈

zσ1 1 (i) and

zσ2 2 (i) ,

r Q i=1

zi1 ∩ Gn (y) 6= ∅.

x γ x1 and x1 ∈ Gn (y) and so there such that

zi2 ∩ Gn−1 (y) 6= ∅ =⇒ ∃x2 ∈

r Q i=1

zi2 ∩ Gn−1 (y) =⇒ x1 γx2 .

So, as a consequence one obtains ∃xn ∈

s Q i=1

zin ∩ Gn−(n−1) (y) =⇒ xn ∈ G1 (y) = {y} =⇒ xn = y,

and so x γ x1 . . . γ xn = y. Therefore, G ⊆ γ ∗ .  Theorem 2.6.43. If B is a non-empty subset of H, then S Cγ (B) = Cγ (b). b∈B

Proof. It is clear for every b ∈ B, Cγ (b) ⊆ Cγ (B), because every γ-part S containing B contains {b}. Therefore, Cγ (b) ⊆ Cγ (B). In order to b∈B S prove the converse remember that Cγ (B) = Gn (B). One clearly has 1≤n

G1 (B) = B =

S b∈B

{b} =

S

G1 (b).

b∈B

We demonstrate the theorem by induction. Suppose that it is true for n, S S that is, Gn (B) ⊆ Gn (b) and we prove that Gn+1 (B) ⊆ Gn+1 (b). If b∈B

b∈B

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z ∈ Gn+1 (B), then there exist q ∈ N, (x1 , . . . , xq ) ∈ H q and σ ∈ Sq such that q q Q Q z∈ xσ(i) and xi ∩ Gn (B) 6= ∅, i=1

by the hypothesis induction

i=1 q Q

 xi ∩

i=1

 Gn (b) 6= ∅. Hence, there ex-

b∈B

q Q xi ∩ Gn (b ) 6= ∅. Since z ∈ xσ(i) , one gets i=1 i=1 S z ∈ Gn+1 (b0 ) and so one has prove Gn+1 (B) ⊆ Gn+1 (b). Therefore, b∈B S Cγ (B) ⊆ Cγ (b).  0

ists b ∈ B such that

q Q

S

0

b∈B

Theorem 2.6.44. If A ∈ P ∗ (H), one has DH ◦ A = A ◦ DH = Cγ (A). Proof. It is straightforward.  Corollary 2.6.45. Let A ∈ P ∗ (H). Then, A is a γ-part of H if and only if A ◦ DH = A. Proof. We have Cγ (A) = A ◦ DH = A.  Corollary 2.6.46. If A is a γ-part of H, then for every B ∈ P ∗ (H), A ◦ B and B ◦ A are γ-parts of H. Proof. We have: Cγ (A◦B) = A◦B ◦DH = A◦DH ◦B = Cγ (A)◦B = A◦B. 

2.7

Join spaces

Join spaces were introduced by W. Prenowitz [118; 120; 121; 122] to provide a common algebraic framework in which classical geometries could be axiomatized and studied. The underlying algebraic structure used was a hypergroup. Then this concept applied by him and J. Jantosciak both in Euclidian and in non Euclidian geometry [124]. Using this notion, several branches of non Euclidian geometry were rebuilt: descriptive geometry, projective geometry and spherical geometry. Then, several important examples of join spaces have been constructed in connection with binary relations, graphs, lattices, fuzzy sets and rough sets. In this section, we study the concept of join space. The main references for this section are [56; 86].

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In order to define a join space, we recall the following notation: If a, b are elements of a hypergroupoid (H, ◦), then we denote a/b = {x ∈ H | S a ∈ x ◦ b}. Moreover, by A/B we intend the set a/b. a∈A,b∈B

Definition 2.7.1. A commutative hypergroup (H, ◦) is called a join space if the following condition holds for all elements a, b, c, d of H: a/b ∩ c/d 6= ∅ =⇒ a ◦ d ∩ b ◦ c 6= ∅ (transposition axiom). Elements of H are called points and are denoted by a, b, c, . . .. Sets of points are denoted by A, B, C, · · · . The elementary algebra of join space theory for sets of points include the result (1) (2) (3) (4)

(A/B)/C = A/(B ◦ C); A 6= ∅ implies B ⊆ A/(A/B); A ◦ (B/C) ⊆ (A ◦ B)/C; A/(B/C) ⊆ (A ◦ C)/B.

A set of points M is said to be linear if it is closed under join and extension (a, b ∈ M imply a ◦ b ⊆ M and a/b ⊆ M ). If M is linear, then M = M ◦ M = M/M . The linear sets are the closed subhypergroups of H. For a set of points A, the intersection of all linear sets containing A (the least linear set containing A) is denoted by < A > and is called the linear space spanned or generated by A. We use < A, B > for < A ∪ B >. Important results concerning linear sets are a formula for the linear span of two intersecting linear sets and a weak modularity property (1) M and N linear and M ∩ N 6= ∅ imply < M, N >= M/N ; (2) L, M and N linear, L∩M 6= ∅ and L ⊆ N imply < L, M > ∩N =< L, M ∩ N >. Both of these results depend on the transposition axiom. A point e is said to be a (scalar) identity if e ◦ a = a for every point a. If H has an identity, it is unique. In a join space H with identity e the following hold: (1) (2) (3) (4)

For each a there exists a unique a−1 such that e ∈ a ◦ a−1 ; a−1 = e/a; a/b = a ◦ b−1 ; e ∈ M for any non-empty linear set M .

The transposition axiom is needed here to prove the uniqueness of the inverse point and that an extension reduce to a join. Join spaces with identity

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have been studied also in [112] by the name of canonical hypergroups. Some important examples of join spaces were presented in Section 2.2 (see Examples 2.2.3. (9), (10), (11), (12), (14)). We give here some other examples. Example 2.7.2. (1) Let (L, ∧, ∨) be a distributive lattice. If for all a, b of L we define a ◦ b = {x ∈ L | x = (a ∧ b) ∨ (a ∧ x) ∨ (b ∧ x)}, then (L, ◦) is a non geometrical join space in which every element is an identity. The above hyperoperation can be considered in a more general context, that one of a median semilattice. A median semi-lattice is a meet semilattice (S, ∧), such that the following conditions hold: • Every principal ideal is a distributive lattice; • Any three elements of S have an upper bound whenever each pair of them has an upper bound. (2) Let V be a vector space over an ordered field F . If for all a, b of V we define a ◦ b = {λa + µb | λ > 0, µ > 0, λ + µ = 1}, then (V, ◦) is a join space, called an affine join space over F . (3) Let G = (V, E) be a connected simple graph. We say that a subset A of V is convex if for all different elements a, b of A, we have that A contains all points on all geodetics from a to b. Denote by (a, b) the least convex set containing {a, b}. A convex set P is called prime if V \ P is convex. Finally, G is called a strong prime convex intersection graph if: • For any convex set A and any point x, which does not belong to A, there exists a prime convex set P , such that A ⊆ P , x ∈ V \P ; • For any (a, b), (c, d) such that (a, b) ∩ (c, d) = ∅, there exists a convex prime set P such that (a, b) ⊆ P and (c, d) ⊆ V \ P . If G satisfies the above two conditions and for all different elements a, b of V we define a ◦ b = (a, b) and a ◦ a = a, then (V, ◦) is a join space. (4) Denote by )a, b( an open real interval. We define the following hyperoperation on the Cartesian plane R2 : for all different elements (x1 , x2 ), (y1 , y2 ) of R2 , we have (x1 , x2 )◦(y1 , y2 )={(z1 , z2 ) | z1 ∈)x1 , x2 ( and z2 ∈)x2 , y2 (} and for all element (x1 , x2 ) of R2 , we

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have (x1 , x2 ) ◦ (x1 , x2 ) = (x1 , x2 ). Then, (R2 , ◦) is a geometric join space, not provided with identity elements. (5) Let G = (V, E) be a connected simple graph. We define the following hyperoperation on V : for all different elements x, y of V , we have x ◦ x = x and x ◦ y is the set of all points z ∈ V , which belong to some paths γ : x − y. Then, (V, ◦) is a non-geometric join space in which every element is an identity. Three classical types of geometry are readily formulated as join spaces. Definition 2.7.3. (1) A descriptive join space or ordered join geometry is a join space that satisfies the axioms a ◦ a = a/a = a; a, b, c distinct and c ∈ {a, b} imply c ∈ a ◦ b, b ∈ a ◦ c or a ∈ b ◦ c. (2) A spherical join space is a join space with identity e that satisfies the axioms a ◦ a = a; a/a = {e, a, a−1 }. (3) A projective join space is a join space with identity e that satisfies the axiom a ◦ a = a/a ⊆ {e, a}. Proposition 2.7.4. If H is one of the classical join spaces, then the “line” spanned by points a and b, i.e., < a, b > is given by (1) a ∪ b ∪ a ◦ b ∪ a/b ∪ b/a, for H descriptive; (2) e ∪ a ∪ b ∪ a−1 ∪ b−1 ∪ a ◦ b ∪ a ◦ b−1 ∪ a−1 ◦ b ∪ a−1 ◦ b−1 , for H spherical; (3) e ∪ a ∪ b ∪ a ◦ b, for H projective. Proof. It is straightforward.  Let M be a non-empty linear subset of the join space H. For any point a the coset of M in H containing a is denoted by (a)M and is given by (a)M = (a ◦ M )/M . The family (H : M ) = {(a)M | a ∈ H} of all cosets of M in H is a partition of H. For the set of points A let (A)M = (A ◦ M )/M = ∪{(a)M | a ∈ A}.

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Then, the join cosets (a)M and (b)M satisfies (a)M ◦ (b)M ⊆ (a ◦ b)M , so that the equivalence relation on H corresponding to the partition of H into cosets of M is a regular equivalence relation. Proposition 2.7.5. The family (H : M ) of cosets M in H under the join operation (a)M ? (b)M = {(x)M | x ∈ (a)M ◦ (b)M } is a join space with identity M = (m)M , where m ∈ M . Proof. Note that (a)−1 M = M/a and that a ◦ b ∩ M 6= ∅ if and only if (a)−1 = (b) .  M M (H : M ) under ? is known as the factor join space H modulo M . Analogues of the three classical isomorphism theorems of group theory hold for join spaces. Let (H, ◦) and (K, ·) be join spaces. A mapping ϕ : H −→ K is said to be a good homomorphism if ϕ(a ◦ b) = ϕ(a) · ϕ(b). If ϕ is also one to one and onto, then ϕ is said to be an isomorphism and the notation H ∼ = K is used. If ϕ is a good homomorphism and K has identity e, then {x ∈ H | ϕ(x) = e} is called the kernel of ϕ and is denoted by kerϕ. Theorem 2.7.6. (First Isomorphism Theorem) Let H and K be join spaces. Let K has identity. Let ϕ be a good homomorphism of H onto K. Then, kerϕ is a linear subset of H and (H : kerϕ) ∼ = K. Theorem 2.7.7. (Second Isomorphism Theorem) Let H be a join space. Let M and N be linear subsets such that M ∩ N 6= ∅. Then, (< M, N >: M ) ∼ = (N : M ∩ N ). Theorem 2.7.8. (Third Isomorphism Theorem) Let H be a join space. Let M and N be linear subsets such that ∅ 6= N ⊆ M . Then, (M : N ) is a linear subset of (H : N ) and (H : M ) ∼ = ((H : N ) : (M : N )). A version of Jordan-H¨ older theorem also holds for join spaces. Theorem 2.7.9. (Jordan-H¨older Theorem) Let H be a join space. Let M and N be linear subsets such that ∅ 6= N ⊆ M . Suppose that N = A0 ⊆ · · · ⊆ Am = M and N = B0 ⊆ · · · ⊆ Bn = M where each Ai and each Bj are linear and for i = 1, · · · , m and for j = 1, · · · , n that Ai−1 and Bj−1 are maximal proper linear subsets of Ai and

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Bj respectively. Then, m = n and there exists a one to one corresponding between the families of factor spaces {(Ai : Ai−1 ) | i = 1, · · · , m} and {(Bj : Bj−1 ) |j = 1, · · · , n} such that the correspondents are isomorphic. Proof. The chains A0 ⊆ · · · ⊆ Am and B0 ⊆ · · · ⊆ Bn are refined respectively by Ai,j =< Ai−1 , Ai ∩ Bj > for i = 1, · · · , m and j = 0, · · · , n and Bj,i =< Bj−1 , Bj ∩ Ai > for i = 0, · · · , m and j = 1, · · · , n. Thus, A0 = A1,0 ⊆ · · · ⊆ A1,n = A1 = A2,0 ⊆ · · · ⊆ Am,n = Am and B0 = B1,0 ⊆ · · · ⊆ B1,m = B1 = B2,0 ⊆ · · · ⊆ Bn,m = Bn . Next, by the second isomorphism theorem followed by the weak modularity property (Ai,j : Ai,j−1 ) = (< Ai−1 , Ai ∩ Bj >:< Ai−1 , Ai ∩ Bj−1 >) = (, Ai ∩ Bj >:< Ai−1 , Ai ∩ Bj−1 >) ∼ = (Ai ∩ Bj :< Ai−1 , Ai ∩ Bj−1 > ∩(Ai ∩ Bj )) = (Ai ∩ Bj :< Ai−1 ∩ (Ai ∩ Bj ), Ai ∩ Bj−1 >) = (Ai ∩ Bj :< Ai−1 ∩ Bj , Ai ∩ Bj−1 >). Similarly, we have (Bj,i : Bj,i−1 ) ∼ = (Bj ∩ Ai :< Bj−1 ∩ Ai , Bj ∩ Ai−1 >). Hence, (Ai,j : Ai,j−1 ) ∼ = (Bj,i : Bj,i−1 ) for i = 1, · · · , m and j = 1, · · · , n. Therefore, given the maximality condition on Ai−1 and Bj−1 in Ai and Bj respectively, the conclusion of the theorem readily follows.  If N is a closed subhypergroup of a join space H and {x, y} ⊆ H, then we define the following binary relation: xJN y if x ◦ N ∩ y ◦ N 6= ∅. Theorem 2.7.10. JN is an equivalence relation on H and the equivalence class of an element a is JN (a) = (a ◦ N )/N . In particular, JN (a) = N for all a ∈ N .

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Proof. Clearly, JN is reflexive and symmetric. Now, suppose that a◦N ∩b◦N 6= ∅ and b◦N ∩c◦N 6= ∅. It follows that b ∈ (a◦N )/N ∩(c◦N )/N and since (H, ◦) is a join space, we obtain a ◦ N ∩ c ◦ N 6= ∅, which means that aJN c. Hence, JN is also transitive, and so it is an equivalence relation on H. We check now that for all a ∈ H we have JN (a) = (a ◦ N )/N . If d ∈ JN (a), then d ◦ N ∩ a ◦ N 6= ∅. Hence, there exist v ∈ a ◦ N and m ∈ N such that v ∈ d ◦ m, whence it follows that d ∈ v/m ⊆ (a ◦ N )/N . We obtain JN (a) ⊆ (a ◦ N )/N . Now, let y ∈ (a ◦ N )/N . Then, there exist u ∈ a ◦ N and m ∈ N , such that u ∈ y ◦ m, whence y ◦ N ∩ a ◦ N 6= ∅, which means that y ∈ JN (a) and so, (a ◦ N )/N ⊆ JN (a). Clearly, if a ∈ N , then JN (a) = N , since N is closed.  Canonical hypergroups are a particular case of join spaces. The structure of canonical hypergroups was individualized for the first time by M. Krasner as the additive structure of hyperfields. In 1970, J. Mittas was the first who studied them independently from the other operations. In 1973, P. Corsini analyzed the sd-hypergroups, which are a particular type of canonical hypergroups and in 1975 Roth used canonical hypergroups in the character theory of finite groups. W. Prenowitz and J. Jantosciak emphasized the role of canonical hypergroups in geometry, while J.R. McMullen and J.F. Price underlined the role of a generalization of canonical hypergroups in harmonic analysis and particle physics. Some connected hyperstructures with canonical hypergroups were introduced and analyzed by P. Corsini, P. Bonansinga, K. Serafimidis, M. Kostantinidou, J. Mittas, De Salvo. We mention here some of them: strongly canonical, i.p.s. hypergroups, quasi-canonical hypergroups (also called polygroups), feebly (quasi)canonical hypergroups. Let us see now what a canonical hypergroup is. Definition 2.7.11. We say that a hypergroup (H, ◦) is canonical if (1) it is commutative, (2) it has a scalar identity (also called scalar unit), which means that ∃e ∈ H, ∀x ∈ H, x ◦ e = e ◦ x = x, (3) every element has a unique inverse, which means that for all x ∈ H, there exists a unique x−1 ∈ H, such that e ∈ x ◦ x−1 ∩ x−1 ◦ x, (4) it is reversible, which means that if x ∈ y ◦ z, then there exist the inverses y −1 of y and z −1 of z, such that z ∈ y −1 ◦x and y ∈ x◦z −1 . Clearly, the identity of a canonical hypergroup is unique. Indeed, if e is a scalar identity and e0 is an identity of a canonical hypergroup (H, ◦),

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then we have e ∈ e ◦ e0 = {e0 }. Some interesting examples of a canonical hypergroup is the following ones (see [9]). Example 2.7.12. (1) Let C(n) = {e0 , e1 , · · · , ek(n) }, where k(n) = n/2 if n is an even natural number and k(n) = (n−1)/2 if n is an odd natural number. For all es , et of C(n), define es ◦ et = {ep , ev },where p = min{s + t, n−(s+t)}, v = |s−t|. Then (C(n), ◦) is a canonical hypergroup. (2) Let (S, T ) be a projective geometry, i.e., a system involving a set S of elements called points and a set T of sets of points called lines, which satisfies the following postulates: • Any lines contains at least three points; • Two distinct points a, b are contained in a unique line, that we shall denote by L(a, b); • If a, b, c, d are distinct points and L(a, b) ∩ L(c, d) 6= ∅, then L(a, c) ∩ L(b, d) 6= ∅. Let e be an element which does not belong to S and let S 0 = S∪{e}. We define the following hyperoperation on S 0 : • For all different points a, b of S, we consider a◦b=L(a, b)\{a, b}; • If a ∈ S and any line contains exactly three points, let a◦a={e}, otherwise a ◦ a={a, e}; • For all a ∈ S 0 , we have e ◦ a = a ◦ e = a. Then (S 0 , ◦) is a canonical hypergroup. In what follows, we present some basic results of canonical hypergroups. Theorem 2.7.13. If (H, ◦) is a canonical hypergroup, then the following implication holds for all x, y, z, t of H: x ◦ y ∩ z ◦ t 6= ∅ =⇒ x ◦ z −1 ∩ t ◦ y −1 6= ∅. Proof. Let u ∈ x ◦ y ∩ z ◦ t. Since H is reversible, we obtain u−1 ∈ z −1 ◦ t−1 , whence u ◦ u−1 ⊆ x ◦ y ◦ z −1 ◦ t−1 . If e is an identity of H, we obtain e∈(x◦z −1 )◦(t◦y −1 )−1 . Hence there exists an element v∈x◦z −1 ∩t◦y −1 .  Theorem 2.7.14. A commutative hypergroup is canonical if and only if it is a join space with a scalar identity. Proof. Suppose that (H, ◦) is a canonical hypergroup. For all a, b of H we

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have a/b = a◦b−1 . Then the implication =⇒ follows by the above theorem. Conversely, let us check that the inverse of an element is unique. Let e be the scalar identity. If e ∈ a ◦ b ∩ a ◦ c, then a ∈ e/b ∩ e/c, whence it follows that e ◦ c ∩ e ◦ b 6= ∅, hence b = c = a−1 . Let us check now the reversibility of H. We have a ∈ b ◦ c if and only if b ∈ a/c. From e ∈ b ◦ b−1 we obtain b ∈ e/b−1 , hence a ◦ b−1 ∩ e ◦ c 6= ∅, which means that c ∈ a ◦ b−1 . Therefore, H is canonical.  Theorem 2.7.15. If (H, ◦) is a join space and N is a closed subhypergroup of H, then the quotient (H/JN , ⊗) is a canonical hypergroup, where for all a, b of H/JN , we have a ⊗ b = {c | c ∈ a ◦ b}. Proof. First, we check that the hyperoperation ⊗ is well defined. In other words, we have to check that if a1 JN a2 and x ∈ H, then for all z ∈ a1 ◦ x, there exists w ∈ a2 ◦ x, such that zJN w. Indeed, from a1 JN a2 , it follows there exist m, n of N and v of H, such that v ∈ a1 ◦ m ∩ a2 ◦ n. If z ∈ a1 ◦ x, then we have a1 ∈ z/x ∩ v/m, hence z ◦ N ∩ v ◦ x 6= ∅, whence z ◦N ∩N ◦a2 ◦x 6= ∅. It follows that there exists w ∈ a2 ◦x, such that zJN w. Therefore the hyperoperation ⊗ is well defined. Since (H, ◦) is a join space, it follows that (H/JN , ⊗) is a join space, too. Moreover, notice that N is a scalar identity for (H/JN , ⊗), and according to the above theorem, we obtain that (H/JN , ⊗) is a canonical hypergroup. 

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Chapter 3

Polygroups

3.1

Definition and examples of polygroups

Quasicanonical hypergroups were introduced by P. Corsini and later, they were studied by P. Bonansinga and Ch.G. Massouros. They satisfy all the conditions of canonical hypergroups, except the commutativity. Later, S.D. Comer introduced this class of hypergroups independently, using the name of polygroups. He emphasized the importance of polygroups, by analyzing them in connections to graphs, relations, Boolean and cylindric algebras. Another connection between polygroups and artificial intelligence was considered and analyzed by G. Ligozat. Some of these results are exposed in [31]. The double cosets hypergroups are particular quasicanonical hypergroups and they were analyzed by K. Drbohlav, D.K. Harrison and S.D. Comer. Definition 3.1.1. A polygroup is a system ℘ =< P, ., e,−1 >, where e ∈ P, −1 is a unitary operation on P , · maps P × P into the non-empty subsets of P , and the following axioms hold for all x, y, z ∈ P : (P1 ) (x · y) · z = x · (y · z), (P2 ) e · x = x · e = x, (P3 ) x ∈ y · z implies y ∈ x · z −1 and z ∈ y −1 · x. The following elementary facts about polygroups follow easily from the axioms: e ∈ x·x−1 ∩x−1 ·x, e−1 = e, (x−1 )−1 = x, and (x·y)−1 = y −1 ·x−1 , where A−1 = {a−1 | a ∈ A}. A polygroup in which every element has order 2 (i.e., x−1 = x for all x) is called symmetric. As in group theory it can be shown that a symmetric polygroup is commutative. 83

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Example 3.1.2. (1) Double coset algebra. Suppose that H is a subgroup of a group G. Define a system G//H =< {HgH | g ∈ G}, ∗, H,−I >, where (HgH)−I = Hg −1 H and (Hg1 H) ∗ (Hg2 H) = {Hg1 hg2 H |h ∈ H}. The algebra of double cosets G//H is a polygroup introduced in (Dresher and Ore [69]). (2) Prenowitz algebras. Suppose G is a projective geometry with a set P of points and suppose, for p 6= q, pq denoted the set of all points on the unique line through p and q. Choose an object I 6∈ P and form the system PG =< P ∪ {I}, ·, I,−1 > where x−1 = x and I · x = x · I = x for all x ∈ P ∪ {I} and for p, q ∈ P ,  pq \ {p, q} if p 6= q p·q = {p, I} if p = q. PG is a polygroup (Prenowitz [119]). (3) Conjugacy class polygroups. In dealing with a symmetry group two symmetric operations belong to the same class if they present the same map with respect to (possibly) different coordinate systems where one coordinate system is converted into the other by a member of the group. In the language of group theory this means the elements a, b in a symmetric group G belong to the same class if there exists a g ∈ G such that a = gbg −1 , i.e., a and b are conjugate. The collection of all conjugacy classes of a group G is denoted by G and the system < G, ∗, {e},−1 > is a polygroup where e is the identity of G and the product A ∗ B of conjugacy classes A and B consists of all conjugacy classes contained in the elementwise product AB. This hypergroup was recognized by Campaigne [8] and by Diatzman [68]. Now, we illustrate constructions using the dihedral group D4 . This group is generated by a counter-clockwise rotation r of 90o and a horizontal reflection h. The group consists of the following 8 symmetries: {1 = r0 , r, r2 = s, r3 = t, h, hr = d, hr2 = v, hr3 = f }.

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The dihedral groups occur frequently in art and nature. Many of the decorative designs used on floor coverings, pottery, and buildings have one of the dihedral groups as a group of symmetry. In the case of D4 there are five conjugacy classes: {1}, {s}, {r, t}, {d, f } and {h, v}. Let us denote these classes by C1 , . . . , C5 respectively. Then, the polygroup D4 is ∗ C1 C2 C3 C4 C5

C1 C1 C2 C3 C4 C5

C2 C2 C1 C3 C4 C5

C3 C3 C3 C1 , C2 C5 C4

C4 C4 C4 C5 C1 , C2 C3

C5 C5 C5 C4 C3 C1 , C2

As a sample of how to calculate the table entries consider C3 · C3 . To determine this product compute the elementwise product of the conjugacy classes{r, t}{r, t} = {s, 1} = C1 ∪C2 . Thus, C3 ·C3 consists of the two conjugacy classes C1 , C2 . (4) Character polygroups. Closely related to the conjugacy classes of ˆ = {χ1 , χ2 , . . . , χk } be the a finite group are its characters. Let G collection of irreducible characters of a finite group G where χ1 is ˆ of G is the system trivial character. The character polygroup G −1 ˆ ∗, χ1 , > where the product χi ∗ χj is the set of irreducible < G, ˆ was components in the elementwise product χi χj . The system G ˆ investigated by R. Roth [129] who consider a duality between G and G. Before calculating Dˆ4 we need to know the five irreducible characters of the dihedral group D4 . These are given by the following character table. (since characters are constant on conjugacy classes it is usual to list only the conjugacy classes across the top of the table.) χ1 χ2 χ3 χ4 χ5

: : : : :

C1 C2 C3 C4 C5 1 1 1 1 1 1 1 −1 1 −1 1 1 −1 −1 1 1 1 1 −1 −1 2 −2 0 0 0

We illustrate the calculation of the polygroup product of two characters by considering χ5 ∗ χ5 . The pointwise product of χ5 with

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itself yields the following (non-irreducible) character: C1 C2 C3 C4 C5 χ5 χ5 : 4 4 0 0 0 This character can be written as a sum of irreducible characters in exactly one way: χ5 χ5 = χ1 + χ2 + χ3 + χ4 . This is indicated by the entry in the lower right hand corner of the polygroup table ˆ 4 . In general the polygroup product of two characters χi ∗ χj for D tells which irreducible characters are in the product χi χj , but not the multiplicity. Using i in place of the character χi the polygroup ˆ 4 is D 1 2 3 4 5 3.2

1 1 2 3 4 5

2 2 1 4 3 5

3 3 4 1 2 5

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

Extension of polygroups by polygroups

Let < P1 , ·, e1 ,−1 > and < P2 , ∗, e2 ,−I > be two polygroups. Then, on P1 × P2 we can define a hyperproduct as follows: (x1 , y1 ) ◦ (x2 , y2 ) = {(x, y) | x ∈ x1 x2 , y ∈ y1 ∗ y2 }. We recall this the direct hyperproduct of P1 and P2 . Clearly, P1 × P2 equipped with the usual direct hyperproduct becomes a polygroup. In [20], extensions of polygroups by polygroups were introduced in the following way. Suppose that A =< A, ·, e,−1 > and B =< B, ·, e,−1 > are two polygroups whose elements have been renamed so that A ∩ B = {e}. A new system A[B] =< M, ∗, e,I > called the extension of A by B is formed in the following way: Set M = A ∪ B and let eI = e, xI = x−1 , e ∗ x = x ∗ e = x for all x ∈ M , and for all x, y ∈ M \ {e}  x·y if x, y ∈ A    x if x ∈ B, y ∈ A  x∗y = y if x ∈ A, y ∈ B    x · y if x, y ∈ B, y 6= x−1   x · y ∪ A if x, y ∈ B, y = x−1 .

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In this case A[B] is a polygroup which is called the extension of A by B. In the last case, e occurs in both x · y and A. If A = {e, a1 , a2 , . . .} and B = {e, b1 , b2 , . . .}, the table for ∗ in A[B] has the form e a1 a2 .. . b1 b2 .. .

e a1 e a1 a1 a1 a1 a2 a2 a1 .. .. . . b1 b1 b2 b2 .. .. . .

a2 a2 a1 a2 a2 a2 .. . b1 b2 .. .

... ... ... ... .. .

b1 b1 b1 b1 .. .

. . . b1 ∗ b1 . . . b2 ∗ b1 .. .. . .

b2 b2 b2 b2 .. .

... ... ... ... .. .

b1 ∗ b2 b2 ∗ b2 .. .

... ... .. .

Several special cases of the algebra A[B] are useful. Before describing them we need to assign names to the two 2-elements polygroups. Let 2 denotes the group Z2 and let 3 denotes the polygroup S3 // < (1 2) >∼ = Z3 /θ, where θ is the special conjugation with blocks {0}, {1, 2}. The multiplication table for 3 is 0 1 0 0 1 1 1 {0, 1} The names 2 and 3 are suggested by the color schemes that represent the algebras (see Chapter 5). Example 3.2.1. (Adjoining a new identity element). The system 3[M] is the result of adding a “new” identity to the polygroup M. The system 2[M] is almost as good. For example, suppose that R is the system with table 0 1 2 0 0 1 2 1 1 {0, 2} {1, 2} 2 2 {1, 2} {0, 1} Then, 0 a 1 2

0 a 1 2 0 a 1 2 a {0, a} 1 2 1 1 {0, a, 2} {1, 2} 2 2 {1, 2} {0, a, 1} 3[R]

0 a 1 2

0 0 a 1 2

a 1 2 a 1 2 0 1 2 1 {0, a, 2} {1, 2 2 {1, 2} {0, a, 1} 2[R]

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The element “a” acts like the “old” identity on R. Example 3.2.2. The tables for R[2] and R[3] are given below: 0 1 2 a

0 1 2 a 0 1 2 a 1 {0, 2} {1, 2} a 2 {1, 2} {0, 1} a a a a {0, 1, 2} R[2]

0 1 2 a

0 1 2 a 0 1 2 a 1 {0, 2} {1, 2} a 2 {1, 2} {0, 1} a a a a {0, 1, 2, a} R[3]

Example 3.2.3. As an example of A[B], where neither A nor B are minimal, we consider R[R] whose table is given below: 0 1 2 a b

0 1 2 a b 0 1 2 a b 1 {0, 2} {1, 2} a b 2 {1, 2} {0, 1} a b a a a {0, 1, 2, b} {a, b} b b b a, b {0, 1, 2, a}

We finish this section by showing that the extension construction will always yiels a polygroup. Theorem 3.2.4. A[B] is a polygroup. Proof. The second condition of Definition 3.1.1 is clear. It is enough to check the conditions (P1 ) and (P3 ) of Definition 3.1.1. Without loss of generality we may assume x, y, z 6= e and not all elements belong to A. Note that (1) If u ∈ B and v ∈ A, then u ∗ v = v ∗ u = u. If exactly one of x, y, z belong to B, then (1) implies that both sides of (P1 ) equal the element in {x, y, z} ∩ B. If exactly two of x, y, z belong to B, say u and v, then (1) implies that both sides of (P1 ) equal u ∗ v. We assume that x, y, z ∈ B \ {e} and show that (2) u ∈ (x ∗ y) ∗ z implies u ∈ x ∗ (y ∗ z). If u 6∈ A, then u ∈ w ∗ z for some w ∈ x ∗ y. Now, if w 6∈ A, w ∈ x · y and u ∈ w · z so u ∈ (xy)z = x(yz) (in B) ⊆ x ∗ (y ∗ z). Also, if w ∈ A, u ∈ w ∗ z = z (so u = z) and e ∈ xy. Thus, u = z ∈ (xy)z = x(yz) ⊆ x ∗ (y ∗ z). Now, suppose that u ∈ A. Then, z −1 ∈ x ∗ y. Since z −1 6∈ A so z −1 ∈ xy (in B), so e ∈ (xy)z = x(yz). Thus, x−1 ∈ y · z ⊆ y ∗ z and hence u ∈ A ⊆ x ∗ (y ∗ z). The proof of the opposite inclusion x ∗ (y ∗ z) ⊆ (x ∗ y) ∗ z. is similar to

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(2). The condition (P3 ) is clear if x, y, z ∈ A. Since x ∈ B \ {e} implies y or z belongs to B \ {e} and x ∈ A implies z ∈ B \ {e}, we may assume at least two of x, y, z belong to B \ {e}. On the other hand, if x, y, z ∈ B \ {e}, then x ∈ y ∗ z implies x ∈ y · z (in B) from which (P3 ) follows. Therefore, we may assume exactly two of x, y, z belong to B \ {e}. This reduces to two cases: (3) x ∈ y ∗ z where x, y ∈ B \ {e} and z ∈ A. By (1), y ∗ z = y so x = y. Thus, y = x = x ∗ z −1 using (1) again and z ∈ A ⊆ x−1 ∗ x = y −1 ∗ x. (4) x ∈ y ∗ z where x ∈ A and y, z ∈ B \ {e}. In this case y = z −1 so the desired conclusion follows using (1). This completes the proof of (P3 ) and hence the theorem. 

3.3

Subpolygroups and quotient polygroups

In this section, we study the concepts of subpolygroups and normal subpolygroups. In particular, by using the notion of the right coset of a subpolygroup, we show that there is a polygroup structure on the set of all right cosets of a given normal subpolygroup. Definition 3.3.1. A non-empty subset K of a polygroup P is a subpolygroup of P if (1) a, b ∈ K implies a · b ⊆ K, (2) a ∈ K implies a−1 ∈ K. For simplicity of notations, sometimes we may write xy instead of x · y. Definition 3.3.2. A subpolygroup N of a polygroup P is normal in P if a−1 N a ⊆ N , for all a ∈ P. The following corollaries are direct consequences of Definitions 3.3.1 and 3.3.2. Corollary 3.3.3. Let N be a normal subpolygroup of a polygroup P . Then, (1) N a = aN , for all a ∈ P ; (2) (N a)(N b) = N ab, for all a, b ∈ P ; (3) N a = N b, for all b ∈ N a.

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Corollary 3.3.4. Let K and N be subpolygroups of a polygroup P with N normal in P . Then, (1) N ∩ K is a normal subpolygroup of K; (2) N K = KN is a subpolygroup of P ; (3) N is a normal subpolygroup of N K. Definition 3.3.5. If N is a normal subpolygroup of P , then we define the relation x ≡ y(modN ) if and only if xy −1 ∩ N 6= ∅. This relation is denoted by xNP y. Lemma 3.3.6. The relation NP is an equivalence relation. Proof. (1) Since e ∈ xx−1 ∩N for all x ∈ P ; then xNP x, i.e., NP is reflexive. (2) Suppose that xNP y. Then, there exists z ∈ xy −1 ∩ N which implies z −1 ∈ yx−1 and z −1 ∈ N , this means that yNP x, and so NP is symmetric. (3) Let xNP y and yNP z where x, y, z ∈ P . Then, there exist a ∈ xy −1 ∩ N and b ∈ yz −1 ∩ N . So x ∈ ay and z −1 ∈ y −1 b, then z −1 x ⊆ y −1 bay. Since ba ⊆ N and N is a normal subpolygroup, then y −1 bay ⊆ N . Therefore, z −1 x ∩ N 6= ∅, which satisfies the condition for xNP z, and so NP is transitive.  Let NP (x) be the equivalence class of the element x ∈ P . Suppose that [P : N ] = {NP (x) | x ∈ P }. On [P : N ] we consider the hyperoperation defined as follows: NP (x) NP (y) = {NP (z) | z ∈ NP (x)NP (y)}. For a subpolygroup K of P and x ∈ P , denote the right coset of K by Kx and let P/K is the set of all right cosets of K in P . Lemma 3.3.7. If N is a normal subpolygroup of P , then N x = NP (x). Proof. Suppose that y ∈ N x. Then, there exists n ∈ N such that y ∈ nx, which implies that n ∈ yx−1 , and so yx−1 ∩ N 6= ∅. Thus, N x ⊆ NP (x). Similarly, we have NP (x) ⊆ N x.  Therefore, we conclude that [P : N ] = P/N . Lemma 3.3.8. Let N be a normal subpolygroup of P . Then, for all x, y ∈ P , N xy = N z for all z ∈ xy. Proof. Suppose that z ∈ xy. Then, it is clear that N z ⊆ N xy. Now, let a ∈ N xy. Then, we obtain y ∈ (N x)−1 a or y ∈ x−1 N a, and so xy ⊆ xx−1 N a. Since N is a normal subpolygroup, we obtain xy ⊆ xN x−1 a ⊆ N a. Therefore, for every z ∈ xy, we have z ∈ N a which implies that a ∈ N z. This completes the proof. 

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Corollary 3.3.9. NP (x)NP (y).

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For all x, y ∈ P , we have NP (NP (x)NP (y)) =

Definition 3.3.10. An equivalence relation ρ on a polygroup P is called a (full) conjugation on P if (1) xρy implies x−1 ρy −1 ; (2) z ∈ xy and z 0 ρz implies z 0 ∈ x0 y 0 for some x0 ρx and y 0 ρy. A conjugation ρ on P is a special conjugation if for all x ∈ P , (3) xρe implies x = e. Using the notation of equivalence classes, a conjugation relation on P can be described, alternatively, as an equivalence relation ρ such that for all x, y ∈ P , (10 ) (ρ(x))−1 = ρ(x−1 ); (20 ) ρ(xy) ⊆ ρ(x)ρ(y). A conjugation is special if ρ(e) = {e}. Lemma 3.3.11. ρ is a conjugation of P if and only if (1) ρ(x)−1 = {y −1 | y ∈ ρ(x)} = ρ(x−1 ); (2) ρ(ρ(x)y) = ρ(x)ρ(y). Proof. Assume that ρ is a conjugation. Condition (1) of Definition 3.3.10 easily implies (1) of lemma. Condition (2) of definition of conjugation implies that ρ(x)ρ(y) is a union of ρ-classes so ρ((ρ(x))y) ⊆ ρ(x)ρ(y). Now, assume that z ∈ x0 · y 0 , x0 ρx, and y 0 ρy. By (P3 ), y 0 ∈ x0−1 · z so y ∈ x00 · z 0 , where x00 ρ x0−1 and z 0 ρ z by (2) of conjugation definition. By (P3 ) again and (1), z ρ z 0 ∈ (ρ(x)) · y which shows equality in (2). A similar argument establish that a ρ with properties (1) and (2) is a conjugation.  Corollary 3.3.12. The equivalence relation NP is a conjugation on P . Proposition 3.3.13. < [P : N ], , NP (e),−I > is a polygroup, where NP (a)−I = NP (a−1 ).

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Proof. For all a, b, c ∈ P , we have (NP (a) NP (b)) NP (c) = {NP (x) | x ∈ NP (a)NP (b)} NP (c) = {NP (y) | y ∈ NP (x)NP (c), x ∈ NP (a)NP (b)} = {NP (y) | y ∈ NP (NP (a)NP (b))NP (c)} = {NP (y) | y ∈ (NP (a)NP (b))NP (c)}, NP (a) (NP (b) NP (c)) = NP (a) {NP (x) | x ∈ NP (b)NP (c)} = {NP (y) | y ∈ NP (a)NP (x), x ∈ NP (b)NP (c)} = {NP (y) | y ∈ NP (a)NP (NP (b)NP (c))} = {NP (y) | y ∈ NP (a)(NP (b)NP (c))}. Since (NP (a)NP (b))NP (c) = NP (a)(NP (b)NP (c)), we get (NP (a) NP (b)) NP (c) = NP (a) (NP (b) NP (c)). Therefore, is associative. It is easy to see that NP (e) is the unit element in [P : N ], and NP (x−1 ) is the inverse of the element NP (x). Now, we show that NP (c) ∈ NP (a) NP (b) implies NP (a) ∈ NP (c) NP (b−1 ) and NP (b) ∈ NP (a−1 ) NP (c). We have NP (c) ∈ NP (a) NP (b), and hence NP (c) = NP (x) for some x ∈ NP (a)NP (b). Therefore, there exist y ∈ NP (a) and z ∈ NP (b) such that x ∈ yz, so y ∈ xz −1 . This implies that NP (y) ∈ NP (x) NP (z −1 ), and so NP (a) ∈ NP (c) NP (b−1 ). Similarly, we obtain NP (b) ∈ NP (a−1 ) NP (c). Therefore, [P : N ] is a polygroup.  Corollary 3.3.14. If N is a normal subpolygroup of P , then < P/N, , N,−I > is a polygroup, where N x N y = {N z | z ∈ xy} and (N x)−I = N x−1 . Definition 3.3.15. Let < P1 , ·, e1 ,−1 > and < P2 , ∗, e2 ,−I > be polygroups. Let f be a mapping from P1 into P2 such that f (e1 ) = e2 . Then, f is called (1) an inclusion homomorphism if f (x · y) ⊆ f (x) ∗ f (y), for all x, y ∈ P1 ; (2) a strong homomorphism or a good homomorphism if f (x · y) = f (x) ∗ f (y), for all x, y ∈ P1 ; (3) a homomorphism of type 2 if f −1 (f (x) ∗ f (y)) = f −1 f (x · y), for all x, y ∈ P1 ; (4) a homomorphism of type 3 if f −1 (f (x) ∗ f (y)) = (f −1 f (x)) · (f −1 f (y)), for all x, y ∈ P1 ;

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(5) a homomorphism of type 4 if f −1 (f (x)∗f (y)) = f −1 f (x·y) = (f −1 f (x))·(f −1 f (y)), for all x, y ∈ P1 ; where for non-empty subset A of P1 , by f −1 f (A) we mean S −1 f −1 f (A) = f f (x). x∈A

Clearly, a strong homomorphism f is called an isomorphism if f is one to one and onto. The defining condition for any types of homomorphism is also valid for sets. For instance, if f is an inclusion homomorphism of P1 into P2 and A, B are non-empty subsets of P1 , then it follows that f (A · B) ⊆ f (A) ∗ f (B). Notice that a homomorphism of the types 2 through 4 is indeed an inclusion homomorphism. Observe that a one to one homomorphism of P1 onto P2 of any type 2 through 4 is an isomorphism. The defining condition for a homomorphism of type 2 can easily be simplified if the homomorphism is onto. Corollary 3.3.16. Let f be a mapping of P1 onto P2 . Then, f is a homomorphism of type 2 if and only if f is a strong homomorphism. The types of homomorphism introduced above have appeared in [86] and in the literature under various names. For example, homomorphisms of type 2 have been dealt with by Corsini [22], Koskas [94], Mittas [112], Prenowitz and Jantociak [123] and Sureau [138]. Also, Corsini [22] and Koskas [94] have also studied homomorphisms of type 3. Why are factor polygroups important? Well, when P is finite and N 6= {e}, P/N is smaller than P , and its structure is usually less complicated than that of P . At the same time, P/N simulates P in many ways. In fact, we may think of a factor polygroup of P as a less complicated approximation of P . What makes factor polygroups important is that one can often deduce properties of P by examine the less complicated polygroup P/N instead. An excellent illustration of this is given in the next example. Example 3.3.17. We illustrate constructions using the dihedral group D4 . This group is generated by a counter-clockwise rotation r of 90o and a horizontal reflection h. The group consists of the following 8 symmetries: {1 = r0 , r, r2 = s, r3 = t, h, hr = d, hr2 = v, hr3 = f }.

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The dihedral groups occur frequently in art and nature. Many of the decorative designs used on floor coverings, pottery, and buildings have one of the dihedral groups as a group of symmetry. Let S3 be the symmetric group of degree 3, i.e., S3 = {i, (1 2), (1 3), (2 3), (1 2 3), (1 3 2)} Let A denote the polygroup S3 //(1 2). The multiplication table for A is A B A A B B B {A, B} Now, the polygroup D4 × A is isomorphic to ℘, where ℘ has the following multiplication table: ◦ A0 A1 A2 A3 A4 A5 A6 A7 B0 B1 B2 B3 B4 B5 B6 B7

A0 A0 A1 A2 A3 A4 A5 A6 A7 B0 B1 B2 B3 B4 B5 B6 B7

A1 A1 A0 A3 A2 A5 A4 A7 A6 B1 B0 B3 B2 B5 B4 B7 B6

A2 A2 A3 A1 A0 A7 A6 A4 A5 B2 B3 B1 B0 B7 B6 B4 B5

A3 A3 A2 A0 A1 A6 A7 A5 A4 B3 B2 B0 B1 B6 B7 B5 B4

A4 A4 A5 A6 A7 A0 A1 A2 A3 B4 B5 B6 B7 B0 B1 B2 B3

A5 A5 A4 A7 A6 A1 A0 A3 A2 B5 B4 B7 B6 B1 B0 B3 B2

A6 A6 A7 A5 A4 A3 A2 A0 A1 B6 B7 B5 B4 B3 B2 B0 B1

A7 A7 A6 A4 A5 A2 A3 A1 A0 B7 B6 B4 B5 B2 B3 B1 B0

B0 B0 B1 B2 B3 B4 B5 B6 B7 A0 , B 0 A1 , B 1 A2 , B 2 A3 , B 3 A4 , B 4 A5 , B 5 A6 , B 6 A7 , B 7

B1 B1 B0 B3 B2 B5 B4 B7 B6 A1 , B 1 A0 , B 0 A3 , B 3 A2 , B 2 A5 , B 5 A4 , B 4 A7 , B 7 A6 , B 6

B2 B2 B3 B1 B0 B7 B6 B4 B5 A2 , B 2 A3 , B 3 A1 , B 1 A0 , B 0 A7 , B 7 A6 , B 6 A4 , B 4 A5 , B 5

B3 B3 B2 B0 B1 B6 B7 B5 B4 A3 , B 3 A2 , B 2 A0 , B 0 A1 , B 1 A6 , B 6 A7 , B 7 A5 , B 5 A4 , B 4

B4 B4 B5 B6 B7 B0 B1 B2 B3 A4 , B 4 A5 , B 5 A6 , B 6 A7 , B 7 A0 , B 0 A1 , B 1 A2 , B 2 A3 , B 3

B5 B5 B4 B7 B6 B1 B0 B3 B2 A5 , B 5 A4 , B 4 A7 , B 7 A6 , B 6 A1 , B 1 A0 , B 0 A3 , B 3 A2 , B 2

B6 B6 B7 B5 B4 B3 B2 B0 B1 A6 , B 6 A7 , B 7 A5 , B 5 A4 , B 4 A3 , B 3 A2 , B 2 A0 , B 0 A1 , B 1

B7 B7 B6 B4 B5 B2 B3 B1 B0 A7 , B 7 A6 , B 6 A4 , B 4 A5 , B 5 A2 , B 2 A3 , B 3 A1 , B 1 A0 , B 0

Let N = {A0 , A1 , B0 , B1 }, and consider the factor polygroup ℘ by N , ℘/N = {N, A2 N, A4 N, A6 N } The multiplication table for ℘/N is given in the following table: N N N A2 N A2 N A4 N A4 N A6 N A6 N

A2 N A2 N N A6 N A4 N

A4 N A4 N A6 N N A2 N

A6 N A6 N A4 N A2 N N

℘/N provides a good opportunity to demonstrate how a factor polygroup of P is related to P itself. Suppose that we arrange the heading of the Cayley table for ℘ in such a way that elements from the same coset of N are in adjacent columns. Then, the multiplication table for ℘ can be blocked off into boxes which are cosets of N , and the substitution that replaces a box containing the element X with the coset XN yields the Cayley table for ℘/N .

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◦ A0 A1 B0 B1 A2 A3 B2 B3 A4 A5 B4 B5 A6 A7 B6 B7

A0 A0 A1 B0 B1 A2 A3 B2 B3 A4 A5 B4 B5 A6 A7 B6 B7

A1 A1 A0 B1 B0 A3 A2 B3 B2 A5 A4 B5 B4 A7 A6 B7 B6

B0 B0 B1 A0 , B 0 A1 , B 1 B2 B3 A2 , B 2 A3 , B 3 B4 B5 A4 , B 4 A5 , B 5 B6 B7 A6 , B 6 A7 , B 7

B1 B1 B0 A1 , B 1 A0 , B 0 B3 B2 A3 , B 3 A2 , B 2 B5 B4 A5 , B 5 A4 , B 4 B7 B6 A7 , B 7 A6 , B 6

A2 A2 A3 B2 B3 A1 A0 B1 B0 A7 A6 B7 B6 A4 A5 B4 B5

A3 A3 A2 B3 B2 A0 A1 B0 B1 A6 A7 B6 B7 A5 A4 B5 B4

B2 B2 B3 A2 , B 2 A3 , B 3 B1 B0 A1 , B 1 A0 , B 0 B7 B6 A7 , B 7 A6 , B 6 B4 B5 A4 , B 4 A5 , B 5

B3 B3 B2 A3 , B 3 A2 , B 2 B0 B1 A0 , B 0 A1 , B 1 B6 B7 A6 , B 6 A7 , B 7 B5 B4 A5 , B 5 A4 , B 4

A4 A4 A5 B4 B5 A6 A7 B6 B7 A0 A1 B0 B1 A2 A3 B2 B3

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A5 A5 A4 B5 B4 A7 A6 B7 B6 A1 A0 B1 B0 A3 A2 B3 B2

B4 B4 B5 A4 , B 4 A5 , B 5 B6 B7 A6 , B 6 A7 , B 7 B0 B1 A0 , B 0 A1 , B 1 B2 B3 A2 , B 2 A3 , B 3

B5 B5 B4 A5 , B 5 A4 , B 4 B7 B6 A7 , B 7 A6 , B 6 B1 B0 A1 , B 1 A0 , B 0 B3 B2 A3 , B 3 A2 , B 2

A6 A6 A7 B6 B7 A5 A4 B5 B4 A3 A2 B3 B2 A0 A1 B0 B1

A7 A7 A6 B7 B6 A4 A5 B4 B5 A2 A3 B2 B3 A1 A0 B1 B0

B6 B6 B7 A6 , B 6 A7 , B 7 B5 B4 A5 , B 5 A4 , B 4 B3 B2 A3 , B 3 A2 , B 2 B0 B1 A0 , B 0 A1 , B 1

B7 B7 B6 A7 , B 7 A6 , B 6 B4 B5 A4 , B 4 A5 , B 5 B2 B3 A2 , B 2 A3 , B 3 B1 B0 A1 , B 1 A0 , B 0

Thus, when we pass from ℘ to ℘/N , the box A0 A1 B0 B1

A1 B0 B1 A0 B1 B0 B1 A0 , B0 A1 , B1 B0 A1 , B1 A0 , B0

in Table 3.3.3 becomes the element N in Table 3.3.2, similarly, the box A3 A2 B3 B2

A2 B3 B2 A3 B2 B3 B2 A3 , B3 A2 , B2 B3 A2 , B2 A3 , B3

becomes the element A2 N , and so on. In this way, one can see that the formation of a factor polygroup ℘/N causes a systematic collapsing of the elements X collapse to the single polygroup element XN in ℘/N . 3.4

Isomorphism theorems of polygroups

We recall that a strong homomorphism ϕ : P1 −→ P2 is an isomorphism if ϕ is one to one and onto. We write P1 ∼ = P2 if P1 is isomorphic to P2 . Because P1 is a polygroup, e ∈ aa−1 for all a ∈ P1 , then we have ϕ(e1 ) ∈ ϕ(a) ∗ ϕ(a−1 ) or e2 ∈ ϕ(a) ∗ ϕ(a−1 ) which implies ϕ(a−1 ) ∈ ϕ(a)−1 ∗ e2 . Therefore, ϕ(a−1 ) = ϕ(a)−1 for all a ∈ P1 . Moreover, if ϕ is a strong homomorhism from P1 into P2 , then the kernel of ϕ is the set kerϕ = {x ∈ P1 | ϕ(x) = e2 }. It is trivial that kerϕ is a subpolygroup of P1 but in general is not normal in P1 . Lemma 3.4.1. Let ϕ be a strong homomorphism from P1 into P2 . Then, ϕ is injective if and only if kerϕ = {e1 }.

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Proof. Let y, z ∈ P1 be such that ϕ(y) = ϕ(z) Then, ϕ(y) ∗ ϕ(y −1 ) = ϕ(z) ∗ ϕ(y −1 ). It follows that ϕ(e1 ) ∈ ϕ(yy −1 ) = ϕ(zy −1 ), and so there exists x ∈ yz −1 such that e2 = ϕ(e1 ) = ϕ(x). Thus, if kerϕ = {e1 }, x = e1 , whence y = z. Now, let x ∈ kerϕ. Then, ϕ(x) = e2 = ϕ(e1 ). Thus, if ϕ is injective, we conclude that x = e1 .  We are now in a position to state and review the fundamental theorems in polygroup theory. Theorem 3.4.2 (First Isomorphism Theorem). Let ϕ be a strong homomorphism from P1 into P2 with kernel K such that K is a normal subpolygroup of P1 . Then, P1 /K ∼ = Imϕ. Proof. We define ψ : P1 /K −→ Imϕ by setting ψ(Kx) = ϕ(x) for all x ∈ P1 . It is easy to see that ψ is an isomorphism.  Theorem 3.4.3. (Second Isomorphism Theorem). If K and N are subpolygroups of a polygroup P , with N normal in P , then K/(N ∩ K) ∼ = N K/N. Proof. Since N is a normal subpolygroup of P , N K = KN . Consequently N K is a subpolygroup of P . Further N = N e ⊆ N K given that N is a normal subpolygroup of N K; consequently N K/N is defined. Define ϕ : K −→ N K/N by ϕ(k) = N k. ϕ is a strong homomorphism. Consider any N a ∈ N K/N , a ∈ N K. Now a ∈ N K given a ∈ nk for some n ∈ N, k ∈ K. Thus, by Corollary 3.3.3, N a = N nk = N k = ϕ(k). This shows that ϕ is also onto. If we can establish that kerϕ = N ∩ K, since N ∩ K is a normal subpolygroup of K, we shall get that K/N ∩ K ∼ = N K/N . For any k ∈ K, k ∈ kerϕ ⇐⇒ ϕ(k) = N ⇐⇒ N k = N ⇐⇒ k ∈ N ⇐⇒ k ∈ N ∩ K (since k ∈ K), i.e., k ∈ kerϕ ⇐⇒ k ∈ N ∩ K. This yields kerϕ = N ∩ K. Hence, that results follows.  Theorem 3.4.4. (Third Isomorphism Theorem). If K and N are normal subpolygroups of a polygroup P such that N ⊆ K, then K/N is a normal subpolygroup of P/N and (P/N )/(K/N ) ∼ = P/K. Proof. We leave it to reader to verify that K/N is a normal subpolygroup of P/N . Further ϕ : P/N −→ P/K defined by ϕ(N x) = Kx is a strong homomorphism of P/N onto P/K such that kerϕ = K/N .  Corollary 3.4.5. If N1 , N2 are normal subpolygroups of P1 , P2 respectively, then N1 ×N2 is a normal subpolygroup of P1 ×P2 and (P1 ×P2 )/(N1 ×N2 ) ∼ = P1 /N1 × P2 /N2 .

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Let P be a polygroup. We define the relation β ∗ as the smallest equivalence relation on P such that the quotient P/β ∗ , the set of all equivalence classes, is a group. In this case β ∗ is called the fundamental equivalence relation on P and P/β ∗ is called the fundamental group. The product ⊗ in P/β ∗ is defined as follows: β ∗ (x) ⊗ β ∗ (y) = β ∗ (z) for all z ∈ β ∗ (x)β ∗ (y). Let UP be the set of all finite products of elements of P . We define the relation β as follows: x β y if and only if {x, y} ⊆ u for some u ∈ UP . We have β ∗ = β for hypergroups. Since polygroups are certain subclasses of hypergroups, we have β ∗ = β. The kernel of the canonical map ϕ : P −→ P/β ∗ is called the core of P and is denoted by ωP . Here we also denote by ωP the unit of P/β ∗ . It is easy to prove that the following statements: ωP = β ∗ (e) and β ∗ (x)−1 = β ∗ (x−1 ) for all x ∈ P . Theorem 3.4.6. Let β1∗ , β2∗ and β ∗ be fundamental equivalence relations on polygroups P1 , P2 and P1 × P2 respectively, then (P1 × P2 )/β ∗ ∼ = P1 /β1∗ × P2 /β2∗ . Corollary 3.4.7. If N1 , N2 are normal subpolygroups of P1 , P2 respectively, and β1∗ , β2∗ and β ∗ fundamental equivalence relations on P1 /N1 , P2 /N2 and (P1 × P2 )/(N1 × N2 ) respectively, then ((P1 × P2 )/(N1 × N2 ))/β ∗ ∼ = (P1 /N1 )/β ∗ × (P2 /N2 )/β ∗ . 1

2

Definition 3.4.8. Let f be a strong homomorphism from P1 into P2 and let β1∗ , β2∗ be fundamental relations on P1 , P2 respectively. We define kerf = {β1∗ (x) | x ∈ P1 , β2∗ (f (x)) = ωP2 }. Lemma 3.4.9. kerf is a normal subgroup of the fundamental group P1 /β1∗ . Proof. If β1∗ (x), β1∗ (y) ∈ kerf , then for every z ∈ xy −1 we have β1∗ (z) = β1∗ (x) ⊗ β1∗ (y −1 ). On the other hand, we have β2∗ (f (z)) = β2∗ (f (x)f (y −1 )) = β2∗ (f (x)) ⊗ β2∗ (f (y −1 )) = ωP2 ⊗ ωP2 = ωP2 . Therefore, β1∗ (z) ∈ kerf . Now, let β1∗ (a) ∈ P1 /β1∗ and β1∗ (x) ∈ kerf . Then, for every z ∈ axa−1 we have β1∗ (z) = β1∗ (a) ⊗ β1∗ (x) ⊗ β1∗ (a−1 ). On the other hand, we have β2∗ (f (z)) = β2∗ (f (a)f (x)f (a−1 )) = β2∗ (f (a)) ⊗ β2∗ (f (x)) ⊗ β2∗ (f (a−1 )) = β2∗ (f (a)) ⊗ ωP2 ⊗ β2∗ (f (a−1 )) = β2∗ (f (aa−1 )) = β2∗ (f (e1 )) = β2∗ (e2 ) = ωP2 . Hence, we obtain β1∗ (z) ∈ kerf . This completes the proof. 

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Theorem 3.4.10. Let P be a polygroup, M, N be two normal subpolygroups of P with N ⊆ M and φ : P/N −→ P/M be the canonical map. Suppose ∗ ∗ that βM , βN are the fundamental equivalence relations on P/M, P/N , re∗ ∗ spectively. Then, ((P/N )/βN )/kerφ ∼ . = (P/M )/βM ∗ ∗ map ψ : (P/N )/βN −→ (P/M )/βM by ψ : (for all x ∈ P ). We must check that ψ is well∗ ∗ ∗ ∈ P and βN (N x) = βN (N y) then βM (M x) = ∗ = βN (N y) if and only if {N x, N y} ⊆ u for some n Q xi }. Thereu ∈ UP/N . We have u = N x1 N x2 . . . N xn = {N z | z ∈

Proof. We define the ∗ ∗ βN (N x) 7−→ βM (M x) defined, that is, if x, y ∗ ∗ βM (M y). Now βN (N x)

fore, for some z1 ∈

n Q

xi , z2 ∈

i=1 xz1−1

n Q

i=1

xi we have N x = N z1 and N y = N z2 .

i=1

So there exist a ∈ ∩ N and b ∈ yz2−1 ∩ N , then x ∈ az1 and y ∈ bz2 . Hence, M x ∈ M a M z1 and M y ∈ M b M z2 . Since a, b ∈ N ⊆ M , then M a = M , M b = M . Since M M z1 = M z1 and M M z2 = M z2 , we have n Q M x = M z1 and M y = M z2 . From {M z1 , M z2 } ⊆ {M z | z ∈ xi }, we get {M x, M y} ⊆ {M z | z ∈

n Q

i=1

xi } = M x1 M x2 . . . M xn . Therefore,

i=1

∗ ∗ (M y). This follows that ψ is well-defined. Moreover, ψ is a (M x) = βM βM strong homomorphism, for if x, y ∈ P . Then, ∗ ∗ ∗ ∗ ψ(βN (N x) ⊗ βN (N y)) = ψ(βN (N xy)) = βM (M xy) ∗ ∗ = βM (M x) ⊗ βM (M y) ∗ ∗ = ψ(βN (N x)) ⊗ ψ(βM (M y)), ∗ ∗ (M ) = ωP/M . Clearly, ψ is onto. Now, we and ψ(ωP/N ) = ψ(βN (N )) = βM show that kerψ = kerφ. We have ∗ ∗ (N x)) = ωP/N } kerψ = {βN (N x) | ψ(βN ∗ ∗ = {βN (N x) | βM (M x) = ωP/N } ∗ ∗ (N x) | βM (φ(N x)) = ωP/N } = {βN = kerφ. 

3.5

γ ∗ relation on polygroups

We recall the following definition. If H is a semihypergroup, then we set: γ1 = {(x, x) | x ∈ H} and, for every integer n > 1, the relation is defined as follows: n n Q Q xγn y ⇐⇒ ∃(z1 , z2 , . . . , zn ) ∈ H n , ∃σ ∈ Sn : x ∈ zi , y ∈ zσ(i) . i=1

i=1

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Obviously, for every n ≥ 1, the relations γn are symmetric, and the relation S γ= γn is reflexive and symmetric. Let γ ∗ be the transitive closure of n≥1

γ. Then, γ ∗ is the smallest strongly regular equivalence such that H/γ ∗ is a commutative semigroup. Also, in every hypergroup, the relation γ is transitive, that is γ ∗ = γ, and in this case, the quotient H/γ ∗ is a commutative group. The γ ∗ -relation is a generalization of the β ∗ -relation. Let P be a polygroup. We consider the relation γ ∗ on P . The product ⊗ in P/γ ∗ is defined as follows: γ ∗ (x) ⊗ γ ∗ (y) = γ ∗ (z) for all z ∈ γ ∗ (x)γ ∗ (y). Clearly, we have γ ∗ (e) = 1P/γ ∗ and (γ ∗ (x))−1 = γ ∗ (x−1 ) for all x ∈ P . Let φ : P −→ P/γ ∗ be the canonical projection. DP is called the derived hypergroup and we have DP = φ−1 (1P/γ ∗ ). For every non-empty subset M of polygroup P we have φ−1 (φ(M )) = DP M = M DP . Theorem 3.5.1. Let Γ∗ and γB∗ be the γ ∗ -relations defined on A[B] and B. Then, we have A[B]/Γ∗ ∼ = B/γB∗ . Proof. Note that all the elements of A belongs to the class Γ∗ (e).  Example 3.5.2. Let S3 be the symmetric group of degree 3 and let A denote the polygroup S3 //(1 2) (double coset algebra). The multiplication table for A is A B A A B B B {A, B} Now, let the polygroup B has the following multiplication table: e a b c d

e e a b c d

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

Then, A[B] is a polygroup. By Theorem 3.5.1, the Γ∗ -classes are x = {e, a}, y = {b}, z = {c} and t = {d}. Therefore, A[B]/Γ∗ is the Klein’s group. Theorem 3.5.3. Let P be a polygroup and a1 , . . . , ak , b1 , . . . , bk ∈ P such that aj γ ∗ bj for all j = 1, . . . k. Then, for all x ∈ aδ11 . . . aδkk and for all y ∈ b1δ1 . . . bkδk where δi ∈ {1, −1} (i = {1, . . . , k}), we have xγ ∗ y.

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Proof. Suppose that aj γ ∗ bj for all j = 1, . . . , k. Then, there exist nj ∈ N and (zj1 , . . . , zjnj ) ∈ P nj , and there exists σj ∈ Snj such that aj ∈

nj Q i=1

zji and bj ∈

nj Q

zjσj (i) .

i=1

Therefore, a1 . . . ak ⊆

n1 Q i=1

z1i . . .

nk Q i=1

zki and b1 . . . bk ⊆

n1 Q

z1σ1 (i) . . .

i=1

nk Q

zkσk (i) .

i=1

If we rename zij ’s, then we conclude that there exists Σ ∈ Sn1 +...+nk such that n1 +...+n n1 +...+n Q k Q k a1 . . . ak ⊆ zi and b1 . . . bk ⊆ zΣ(i) , i=1

i=1

and so for all x ∈ a1 . . . ak and for all y ∈ b1 . . . bk we obtain x γ ∗ y. Now, ∗ −1 note that if aj γ ∗ bj , then a−1 j γ bj .  The relational notation A ≈ B is used to assert that the sets A and B have at least one element in common. Theorem 3.5.4. Let P be a polygroup. Then, x γ ∗ y if and only if there exist A, A0 ⊆ γ ∗ (a) and B, B 0 ⊆ γ ∗ (b) for some a, b ∈ P such that xA ≈ B and yA0 ≈ B 0 . Proof. Suppose that there exist A, A0 ⊆ γ ∗ (a) and B, B 0 ⊆ γ ∗ (b) for some a, b ∈ P such that xA ≈ B and yA0 ≈ B 0 . Then, we have (γ ∗ (x) ⊗ {γ ∗ (y 0 ) | y 0 ∈ A}) ≈ {γ ∗ (z) | z ∈ B}, (γ ∗ (y) ⊗ {γ ∗ (y 00 ) | y 00 ∈ A0 }) ≈ {γ ∗ (z 00 ) | z 00 ∈ B 0 }. Therefore, we obtain γ ∗ (x) ⊗ γ ∗ (a) = γ ∗ (b) and γ ∗ (y) ⊗ γ ∗ (a) = γ ∗ (b), which implies that γ ∗ (x) = γ ∗ (y) = γ ∗ (b) ⊗ γ ∗ (a−1 ). Therefore, x γ ∗ y. For the converse, if x γ ∗ y, then we can take A = A0 = DP and B = 0 B = γ ∗ (x). This completes the proof.  Corollary 3.5.5. Let P be a polygroup. Then, x, y ∈ γ ∗ (e) if and only if there exist A, A0 ⊆ γ ∗ (z) and B, B 0 ⊆ γ ∗ (z) for some z ∈ P such that xA ≈ B and yA0 ≈ B 0 . Theorem 3.5.6. Let P be a finite polygroup. For every a ∈ P , there exists a power ar , we take the minimal one, which contains an element of a lower power, that is, there exists as such that ar ≈ as , 0 < s < r. Then, ar−s ⊆ γ ∗ (e).

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Proof. From ar ≈ as we have φ(ar ) = φ(as ), and so (γ ∗ (a))r = (γ ∗ (a))s . Since (γ ∗ (a))r and (γ ∗ (a))s are the elements of abelian group P/γ ∗ , then (γ ∗ (a))r−s = 1P/γ ∗ = γ ∗ (e) which implies that φ(ar−s ) = γ ∗ (e), and so ar−s ⊆ γ ∗ (e).  Let M be a non-empty subset of a polygroup P , we say that M is a γ-part of P , if for every n ∈ N, for every (z1 , . . . , zn ) ∈ P n and for every σ ∈ Sn , we have: n n Q Q zi ≈ M =⇒ zσ(i) ⊆ M. i=1

i=1

Let A be a non-empty subset of P . The intersection of γ-parts P which contain A is called the γ-closure of A in P . It will be denoted Cγ (A). We have (1) A ∈ P ∗ (P ), one has DP A = ADP = Cγ (A). (2) A ∈ P ∗ (P ), then A is a γ-part of P if and only if ADP = A. (3) If A is a γ-part of P then AB, BA are γ-parts of P for every B ∈ P ∗ (P ). (4) DP is a γ-part of P . Q Let P be a polygroup and (P ) be the set of hyperproducts of elements Q of P . Let X =< (P ), > be the set of non-empty subsets of P endowed with the hyperoperation defined as follows: Q A B = {C ∈ (P ) | C ⊆ AB} , Q for all A, B ∈ (P ) with A, B 6= {e}, and A {e} = {e} A = A. Then, we have Q Theorem 3.5.7. If P is a polygroup, then < (P ), > is a regular hypergroup. Q Proof. First, we show that on (P ) is associative. Let A1 , A2 , A3 ∈ Q (P ). Then, we have Q A1 (A2 A3 ) = A1 {C ∈ (P ) | C ⊆ A2 A3 } Q = {D ∈ (P ) | D ⊆ A1 (A2 A3 )} Q = {D ∈ (P ) | D ⊆ (A1 A2 )A3 } Q = {C ∈ (P ) | C ⊆ A1 A2 } A3 = (A1 A2 ) A3 . p q Q Q Let’s prove now the reproducibility. Let A = ai and B = bi be i=1 i=1 Q elements of (P ). By reproducibility of ·, there exists y1 ∈ P such that

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ap ∈ y1 · bq . Similarly, there is y2 such that y1 ∈ y2 · bq−1 , whence ap ∈ y2 ·bq−1 ·bq . Going up in the same way, one obtains yq such that yq−1 ∈ yq ·b1 . p−1 Q Hence, ap ∈ yq · b1 · b2 · . . . · bq . Therefore, if we let X = ai · yq , we have i=1

A ∈ X B. Similarly, we can find z1 , z2 , . . . , zq such that a1 ∈ b1 · z1 , z1 ∈ b2 · z2 , . . . , zq−1 ∈ bq ·zq , whence A = a1 ·a2 ·. . .·ap ⊆ b1 ·b2 ·. . .·bq ·zq ·a1 ·a2 ·. . .·ap . Q Now, let E = {e}, then for all A ∈ (P ) we have A E = E A = A. We define the unary operation −I as follows: Q Q −I : (P ) −→ (P ) −1 (x1 . . . xn )−I = x−1 n . . . x1 .  Q Corollary 3.5.8. If A is a subpolygroup of P and A belongs to (P ), then A is contained in DP . The following example show that not all subpolygroups of a polygroup Q P are in (P ). Example 3.5.9. Let P be a polygroup with the following table: a b c d

a a b c d

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

It is clear that A = {a, b} is a subpolygroup of P , but A 6∈ Q DP = cdd = P ∈ (P ).

Q (P ). Moreover

Note that the following example shows that, in general, by Theorem 3.5.7, we do not obtain a polygroup structure. Example 3.5.10. Let P = {a, b} with a−1 = a, b−1 = b and the following hyperoperation · a b a a b b b {a, b} If X = a · a, Y = b · b and Z = a · a, then X ∈ Y Z but Y 6∈ X z −I . Indeed, Theorem 6 in [37] holds, if we add the following condition in the definition of : (∗) X ∈ A B, if for every a ∈ A and b ∈ B, there exists x ∈ X such that x ∈ a · b.

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Theorem 3.5.11. Let P be a polygroup. Then < group, if satisfies in the condition (∗).

Q (P ), > is a poly-

p m n Q Q Q Proof. Now, let X = xi , Y = yi and Z = zi be elements of i=1 i=1 i=1 Q (P ) such that X ∈ Y Z. Hence, X ⊆ Y · Z and for every y ∈ Y and z ∈ Z there exists x ∈ X such that x ∈ y · z and this implies that y ∈ xz −1 , and so y ∈ (x1 . . . xn )(zp−1 . . . z1−1 ) for every y ∈ Y . Therefore, Y ⊆ (x1 . . . xn )(zp−1 . . . z1−1 ) or Y ∈ X Z −I . Similarly, we get Z ∈ Y −I X.  Q If P is a polygroup, we denote (P ) the set of hyperproducts of eleCγ Q ments of P , such that Cγ (A) = A, for all A ∈ (P ). Cγ

Theorem 3.5.12. Let P be a polygroup and (x1 , . . . , xn ) ∈ P n be such n Q Q that xi ∈ (P ). Then, there exists (y1 , . . . , yn ) ∈ P n such that i=1

Cγ

x1 . . . xn yn . . . y1 = DP . Proof. For 1 ≤ j ≤ n, let aj be an element of DP . Then, there exists yj ∈ P such that aj ∈ xj yj . Since DP is a γ-part, we have xj yj ⊆ DP . Therefore, x1 . . . xn yn = DP x1 . . . xn yn = x1 . . . xn−1 DP xn yn = x1 . . . xn−1 DP n−1 Q = DP xi , i=1

and so x1 . . . xn yn yn−1 = DP = DP

n−2 Q

 xi xn−1 yn−1

i=1 n−2 Q

xi .

i=1

Going on in the same way, one arrives to x1 . . . xn yn . . . y2 = DP x1 whence finally x1 . . . xn yn . . . y1 = DP x1 y1 = DP .  Theorem 3.5.13. Let P be a polygroup. If P \DP is a hyperproduct, then DP is a hyperproduct, too. Proof. Since DP is a γ-part, so P \DP is also a γ-part. Now, by using

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Theorem 3.5.12, the proof is completed.  Using the γ ∗ -relation we define the semidirect hyperproducts of polygroups. Definition 3.5.14. Let A =< A, ·, e1 ,−1 > and B =< B, ∗, e2 ,−1 > be ∗ two polygroups. We consider the group AutA and the group B/γB , let ∗ b: B/γB −→ AutA ∗ ∗ b γB (b) −→ γ\ B (b) = b

be a homomorphism of groups. Then, in A × B we define a hyperproduct as follows: (a1 , b1 ) ◦ (a2 , b2 ) = {(x, y)|x ∈ a1 · bb1 (a2 ), y ∈ b1 ∗ b2 } and we call this the semidirect hyperproduct of polygroups A and B. The above definition, first was introduced by Vougiouklis concerning hypergroups and the fundamental relation β ∗ . Theorem 3.5.15. A × B equipped with the semidirect hyperproduct is a polygroup. Proof. It is easy to see that the associativity is valid. Since a = a · bb(e1 ), a = e1 · eb2 (a) and b = b ∗ e2 = e2 ∗ b, we have (a, b) ◦ (e1 , e2 ) = (a, b) = (e1 , e2 ) ◦ (a, b), that is, (e1 , e2 ) is the identity element in A × B, and we can −1 (a−1 ), b−1 ) is the inverse element of (a, b) in A × B. Now, we check that (bd show that (z1 , z2 ) ∈ (x1 , x2 ) ◦ (y1 , y2 ) imply (x1 , x2 ) ∈ (z1 , z2 ) ◦ (y1 , y2 )−I and (y1 , y2 ) ∈ (x1 , x2 )−I ◦ (z1 , z2 ). We have (z1 , z2 ) ∈ (x1 , x2 ) ◦ (y1 , y2 ) = {(a, b) | a ∈ x1 · x c2 (y1 ), b ∈ x2 ∗ y2 }, which implies that z1 ∈ x1 · x c2 (y1 ) and z2 ∈ x2 ∗ y2 . Since z1 ∈ x1 · x c2 (y1 ), we get x1 ∈ z1 · x c2 (y1 )−1 or x1 ∈ z1 · x c2 (y1−1 ). Since z2 ∈ x2 ∗ y2 , ∗ ∗ ∗ ∗ (x2 ) = γB (z2 ) ⊗ γB (y2−1 ) and so γ\ x2 ∈ z2 ∗ y2−1 . Therefore, γB B (x2 ) = −1 −1 −1 \ ∗ (z )\ ∗ ∗ ∗ \ γB c2 = zb2 yd 2 ⊗ γB (y2 ) = γB (z2 ) · γB (y2 ) or x 2 . Therefore, we get −1 −1 x ∈ z · zb yd (y ). Now, we have 1

1

2 2

1

−1 −1 −1 (x1 , x2 ) ∈ {(a, b) | a ∈ z1 · zb2 yd 2 (y1 ), b ∈ z2 ∗ y2 }

or (x1 , x2 ) ∈ (z1 , z2 ) ◦ (y1 , y2 )−I . On the other hand, we have −1 −1 (x , x )−I ◦ (z , z ) = (xd (x ), x−1 ) ◦ (z , z )) 1

2

1

2

2

1

1

2

2

d −1 −1 −1 −1 = {(a, , b) | a ∈ xd 2 (x1 ) · x2 (z1 ), b ∈ x2 ∗ z2 )} d = {(a, b) | a ∈ x−1 (x−1 · z ), b ∈ x−1 ∗ z }. 2

1

1

2

2

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d −1 −1 Since z1 ∈ x1 ·c x2 (y1 ), x c2 (y1 ) ∈ x−1 1 ·z1 . Hence, y1 ∈ x2 (x1 ·z1 ). Therefore, (y1 , y2 ) ∈ (x1 , x2 )−I ◦ (z1 , z2 ).  Therefore, A × B equipped with the semidirect hyperproduct becomes b a polygroup which we denote A×B. ∗ ∗ Theorem 3.5.16. Let A and B be two polygroups, γA and γB be γ ∗ ∗ relations on A and B respectively. If we consider b : B/γB −→ AutA. Then, bb(γ ∗ (a)) = γ ∗ (bb(a)) for all a ∈ A, b ∈ B. A

A

∗ ∗ (a)). Then, there exists y ∈ γA (a) such Proof. Suppose that x ∈ bb(γA n that x = bb(y). So, there exist (z1 , z2 , . . . , zn ) ∈ A and σ ∈ Sn such that n n Q Q a∈ zi and y ∈ zσ(i) , which implies that i=1

i=1 n n bb(a) ∈ Q bb(zi ) and bb(y) ∈ Q bb(zσ(i) ), i=1

i=1

∗ b ∗ b that is, bb(a) γA b(y) or bb(y) = x ∈ γA (b(a)). ∗b ∗ b b(a) and so there exist Conversely, suppose that x ∈ γA (b(a)). Then, xγA n n Q Q n (z1 , z2 , . . . , zn ) ∈ A and σ ∈ Sn such that x ∈ zi and bb(a) ∈ zσ(i) , i=1

i=1

which yields n n bb−1 (x) ∈ Q bb−1 (zi ) and a ∈ Q bb−1 (zσ(i) ). i=1

i=1

∗ ∗ (a)) and this (a). Therefore, x ∈ bb(γA Hence, bb−1 (x)γ ∗ a or bb−1 (x) ∈ γA completes the proof. 

3.6

Generalized permutations

According to [69] a generalized permutation is defined as follows (see also [147; 148]). Definition 3.6.1. Let Ω be a non-empty set. A map f : Ω → P ∗ (Ω) is called a generalized permutation on Ω if S f (ω) = f (Ω) = Ω, ω∈Ω ∗

where P (Ω) is the set of all non-empty subsets of Ω. We write   x , f= f (x)

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for the generalized permutation f . Denote MΩ the set of all the generalized permutations on Ω. Proposition 3.6.2. Let Θ be a one to one function from a set Ω1 onto a set Ω2 . For a generalized permutation f on Ω1 , we define a function Θ(f ) on Ω2 by the formula Θ(f )(y) = Θ(f (Θ−1 (y))), for all y ∈ Ω2 . Then, Θ(f ) is a generalized permutation on Ω2 . Proof. For every y ∈ Ω2 , we have Θ−1 (y) ∈ Ω1 which implies that f (Θ−1 (y)) ⊆ Ω1 , and so Θ(f (Θ−1 (y))) ⊆ Ω2 . Furthermore,  S S  Θ(f )(y) = Θ f (Θ−1 (y)) y∈Ω2 y∈Ω2 ! S =Θ f (Θ−1 (y)) y∈Ω2

= Θ(Ω1 ) = Ω2 .  Definition 3.6.3. Let f1 , f2 ∈ MΩ . We say that f1 is a subpermutation of f2 , or f2 contains f1 , and write f1 ⊆ f2 , if f1 (x) ⊆ f2 (x) for every x in Ω. The mapping g for which g(x) = Ω for all x ∈ Ω, is called the universal generalized permutation and contains all the elements of MΩ . Every map f : Ω −→ P ∗ (Ω) which contains a generalized permutation is itself a generalized permutation. The map i : Ω −→ P ∗ (Ω) where i(x) = {x} =: x for all x ∈ Ω, is a generalized permutation. We can define operation ◦, the usual composition on MΩ , i.e., if f, g ∈ MΩ , then S f ◦ g(x) = f (y), for all x ∈ Ω. y∈g(x)

Now, we define a hyperoperation ∗ on MΩ as follows:     x x Definition 3.6.4. Let and be two elements of MΩ . We f (x) g(x) ∗ define ∗ : MΩ × MΩ → P (MΩ ) by setting        S x x x ∗ = | h ⊆ f ◦ g, h(x) = Ω . g(x) f (x) h(x) x∈Ω   x The generalized permutation serves as a scalar element, since i(x)

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       x x x ∗ = ∗ . i(x) i(x) f (x)

For f ∈ MΩ the inverse of f which is denoted by f , is the generalized permutation defined as follows: f (y) = {x ∈ Ω | y ∈ f (x)}. It is clear that           x x x x x ∈ ∗ ∩ ∗ . i(x) f (x) f (x) f (x) f (x) Proposition 3.6.5. The hyperoperation ∗ is associative. Proof. Since ◦ is associative, it follows that ∗ is associative.  Definition 3.6.6. < P, ·, e,−1 > is called a poly-monoid if the following conditions hold: (1) (x · y) · z = x · (y · z), for all x, y, z ∈ P , (2) x ∈ e · x = x · e, for all x ∈ P , (3) e ∈ x · x−1 ∩ x−1 · x, for all x ∈ P . Corollary 3.6.7. < MΩ , ∗, i,− > is a poly-monoid. Definition 3.6.8. Let M =< P, ·, e,−1 > be a polygroup and MΩ be the set of all generalized permutations on the non-empty set Ω. A mapping ψ : P → MΩ with the properties ψ(x · y) = ψ(x) ∗ ψ(y) and ψ(x−1 ) = ψ(x), for all x, y ∈ P , is called a representation of P by generalized permutations. 3.7

Permutation polygroups

Permutation polygroups are studied by Davvaz [49]. In this section, by using the concept of generalized permutation, we define permutation polygroups and some concepts related to it. In particular, we introduce a generalization of Cayley’s theorem. Definition 3.7.1 Let M =< P, ·, e,−1 > be a polygroup and Ω be a nonempty set. A map f : Ω × P → P ∗ (Ω) is called an action of P on Ω if the following axioms hold: (1) f (ω, e) = {ω} = ω, for all ω ∈ Ω, S (2) f (f (ω, g), h) = f (ω, α), for all g, h ∈ P and ω ∈ Ω, α∈g.h S (3) f (ω, g) = Ω, for all g ∈ P , ω∈Ω

(4) ∀g ∈ P, α ∈ f (β, g) ⇒ β ∈ f (α, g −1 ).

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From the second condition, we obtain S S f (ω0 , h) = f (ω, α). ω0 ∈f (ω,g)

α∈g.h

For ω ∈ Ω, we write ω g := f (ω, g). Then, we have (1) ω e = ω, S g S g (2) (ω g )h = ω g·h , where Ag = α and ω B = ω , for all A ⊆ Ω, α∈A

g∈B

B ⊆ P, S g (3) ω = Ω, ω∈Ω

(4) ∀g ∈ P , α ∈ β g ⇒ β ∈ αg

−1

.

In this case, we say that P is a permutation polygroup on a set Ω and it is said that P acts on Ω. It is easy to see that if P is a permutation polygroup on two sets Ω1 and Ω2 , then P is a permutation polygroup on the set Ω1 × Ω2 with the action defined by (ω1 , ω2 )g = {(a, b) | a ∈ ω1g , b ∈ ω2g }, for all (ω1 , ω2 ) ∈ Ω1 × Ω2 and g ∈ P . The polygroup P acts on itself as a permutation polygroup if we define xg = x · g or xg = g −1 · x, for all x, g ∈ P . Proposition 3.7.2. Let N be a normal subpolygroup of a polygroup P . Let Ω denote the set of all right cosets N x, where x ∈ P , and we define (N x)g = {N z | z ∈ N xg}, for all g ∈ P . Then, P is a permutation polygroup on Ω. Proof. It is easy to see that (N x)e = N x. Now, if g, h ∈ P , then ((N x)g )h = ({N z | z ∈ N xg})h S = {N t | t ∈ N zh} z∈N xg

= {N t | t ∈ N xgh} S = {N t | t ∈ N xα} α∈g·h

=

S

(N x)α

α∈g·h

= (N x)g·h .

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Therefore, the second condition of Definition 3.7.1 is satisfied. Now, we S prove that (N x)g = Ω. Suppose that N y ∈ Ω, where y ∈ P . Since P N x∈Ω

is a hypergroup, there exists a ∈ P such that y ∈ ag which implies that S y ∈ N ag, and so N y ∈ (N a)g . Therefore, N y ∈ (N x)g . Now, we show N x∈Ω

that N x ∈ (N y)g ⇒ N y ∈ (N x)g

−1

.

In order to prove this, we observe that since N x ∈ {N z | z ∈ N yg}, there exists z0 ∈ N yg such that N x = N z0 . From z0 ∈ N yg, we obtain g ∈ y −1 N z0 . Hence, y −1 ∈ gz0−1 N and so y ∈ N z0 g −1 . Therefore, y ∈ N xg −1 −1 which implies that N y ∈ (N x)g .  Definition 3.7.3. Let P be a polygroup and P acts on Ω. The kernel of action is defined as follows: H = {g ∈ P | ω g = {ω}, for all ω ∈ Ω}. Theorem 3.7.4. (Generalization of Cayley’s Theorem). Let P be a polygroup acting on a non-empty finite set Ω such that the action is faithful. Then, there is a subset of MΩ which is a polygroup under the induced action of P and is isomorph to P . Proof. We define the subset SΩ of MΩ as follows: ( ! ) α1 α2 · · · α|Ω| SΩ = |g∈P . g α1g α2g · · · α|Ω| The hyperoperation ◦ on SΩ is defined as follows: ! ! α1 α2 · · · α|Ω| α1 α2 · · · α|Ω| ◦ g h α1g α2g · · · α|Ω| α1h α2h · · · α|Ω| ! ) ( α1 α2 · · · α|Ω| = | f ∈g·h . f α1f α2f · · · α|Ω| Then, < SΩ , ◦, i,−I > is a polygroup, where ! !−I α1 α2 · · · α|Ω| α1 α2 · · · α|Ω| −1 −1 = . g g −1 α1g α2g · · · α|Ω| α1g α2g · · · α|Ω| Now, we define φ : P −→ SΩ by φ(g) =

α1 α2 · · · α|Ω| g α1g α2g · · · α|Ω|

! .

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It is easy to see that φ is well defined, one to one and onto. Moreover, φ is a strong homomorphism because, for every g, h ∈ P , we have φ(g · h) = {φ(f ) | f ∈ g · h} ( ! ) α1 α2 · · · α|Ω| = | f ∈g·h f α1f α2f · · · α|Ω| = φ(g) ◦ φ(h). Therefore, P ∼ = SΩ .  Note that the above theorem is true when Ω is infinite. Definition 3.7.5. Let P be a polygroup acting on non-empty sets Ω1 and Ω2 . A map Θ : Ω1 → Ω2 is called a P -map if Θ(xg ) = Θ(x)g , for all g ∈ P and x ∈ Ω1 . If Θ is also a one to one correspondence, then Θ is called a P isomorphism and Ω1 , Ω2 are called isomorphic. Proposition 3.7.6. If P is a polygroup and λ, ρ are the left and the right regular representations of P , i.e., λg : x −→ g −1 · x, ρg : x −→ x · g, then (P, λ) and (P, ρ) are isomorphic. Proof. We define Θ : P −→ P by Θ(x) = x−1 . Then, Θ(ρg x) = Θ(x · g) = {Θ(y) | y ∈ x · g} = {y −1 |y ∈ x · g} = (x · g)−1 = g −1 · x−1 = g −1 · Θ(x) = λg Θ(x). Since Θ is one to one and onto, so it is a P -isomorphism.  Definition 3.7.7. Let P be a permutation polygroup on a non-empty set Ω. For α, β ∈ Ω we define ∼ by α ∼ β if and only if α ∈ β g , for some g ∈ P . Lemma 3.7.8. The relation defined above is an equivalence relation on Ω. Now, let Ω=

S

4(α)

α∈I

be the partition of Ω with respect to this relation. Then, the sets 4(α), α ∈ I, are called orbits of P on Ω. Definition 3.7.9. Let P be a permutation polygroup on a non-empty set Ω. If P has only one orbit, i.e., if α ∼ β for every α, β ∈ Ω, we say that the

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polygroup is transitive on Ω. If |I| > 1, we say that P is intransitive. Theorem 3.7.10. Let P be a permutation polygroup on a non-empty set Ω. Then, the polygroup P is transitive on every orbit. Proof. Let 4(α) be the orbit containing α ∈ Ω. Clearly, for the set 4(α), conditions (1), (2) and (4) of Definition 3.7.1 hold. Therefore, we prove the third condition of Definition 3.7.1, i.e., S ω g = 4(α), for all g ∈ P . ω∈4(α)

In order to prove this, suppose that β ∈

ω g . Then, β ∈ ω0g for some

S ω∈4(α)

ω0 ∈ 4(α). So, β ∼ ω0 and ω0 ∼ α which imply that β ∼ α or β ∈ 4(α). S Therefore, ω g ⊆ 4(α). ω∈4(α)

If 4(αi ), i ∈ I are all the disjoint orbits, then S ω g ⊆ 4(αi ), for all i ∈ I. ω∈4(αi )

But,

S

4(αi ) = Ω and 4(αi ) ∩ 4(αj ) = ∅, for all i 6= j. So,

ω∈Ω

S

ωg =

ω∈4(αi )

4(αi ), for all i ∈ I. Therefore, P acts on 4(α). Now, suppose that β1 , β2 ∈ 4(α). Then, β1 ∈ αg , β2 ∈ αh for some −1 g, h ∈ P . By the forth condition of Definition 3.7.1, we have α ∈ β2h and so −1

−1

β1 ∈ (β2h )g = β2h

g

.

Therefore, there exists x ∈ h−1 g such that β1 ∈ β2x .  Corollary 3.7.11. Let P be a permutation polygroup on a finite set Ω. If 4 is an orbit of P such that |Ω| = |4|, then P is transitive on Ω. Definition 3.7.12. Let P be a permutation polygroup on a non-empty set Ω and let ω ∈ Ω. The set n o −1 Pω = g ∈ P | ω g = ω g = {ω} ⊆ P is called the stabilizer of ω. Corollary 3.7.13. The stabilizer Pω is a subpolgroup of P for each ω ∈ Ω. Definition 3.7.14. Let P be a permutation polygroup on a non-empty set Ω and ω1 , · · · , ωk ∈ Ω. Then, the stabilizer Pω1 ,··· ,ωk is the subpolygroup

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o n −1 Pω1 ,··· ,ωk = g ∈ P | ωig = ωig = {ωi }, for all i = 1, . . . , k . This may be phrased also by Pω1 ,...,ωk =

k T

Pωi .

i=1

It follows that if P acts on Ω and ω1 , ω2 ∈ Ω, then Pω1 and Pω2 act on Ω and (Pω1 )ω2 = Pω1 ,ω2 = (Pω2 )ω1 . Proposition 3.7.15. Let N be a normal subpolygroup of a polygroup P . If Ω is the set of all the right cosets of N in P , then P acts on Ω and the kernel of this action is T −1 H= x N x. x∈P

Proof. Suppose that g ∈ H. Then, (N x)g = {N z | z ∈ N xg} = {N x}. T −1 From z ∈ N xg, we obtain g ∈ x−1 N z = x−1 N x. Thus, g ∈ x N x. x∈P T −1 Now, let g ∈ x N x. Then, g ∈ x−1 N x, for every x ∈ P . We have x∈P

(N x)g = {N z | z ⊆ {N z | z ⊆ {N z | z = {N z | z = {N z | z = {N x}.

∈ N xg} ∈ N xx−1 N x} ∈ N xN } ∈ N N x} ∈ N x}

Therefore, g ∈ H.  3.8

Representation of polygroups

Davvaz and Poursalavati studied the concept of representation of polygroups [59]. They introduced matrix representations of polygroups over hyperrings and shown such representations induce representations of the fundamental group over the corresponding fundamental ring. A hyperring in the general sense is the largest class of multivalued systems that satisfies the ring-like axioms: (R, +, ·) is a hyperring if (R, +) is

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a hypergroup, “ · ” is associative hyperoperation and the distributive laws x · (y + z) = x · y + x · z and (x + y) · z = x · z + y · z are satisfied for every x, y, z ∈ R. (R, +, ·) is called a semihyperring if “ + ”, “ · ” are associative hyperoperations, where “ · ” is distributive with respect to “ + ”. There are different notions of hyperrings. If only the addition “ + ” is a hyperoperatiopn and the multiplication “ · ” is a usual operation, then R is called an additive hyperring. A special case of this type is the hyperring introduced by Krasner [95]. The second type of a hyperring was introduced by Rota [126]. The multiplication is a hyperoperation, while the addition is an operation, that is called multiplication hyperring. The monograph [56] is devoted especially to the study of hyperring theory. In this section, we introduced the representation of polygroups by hypermatrices, and we introduced a polystructure on matrices. Also, we obtain some results in this connection. A hypermatrix is a matrix with entries from a semihyperring. The hyperproduct of two hypermatrices (aij ), (bij ) which are of type m × n and n × r respectively, is defined in the usual manner   n P (aij )(bij ) = (cij ) | cij ∈ aik bkj . k=1

One of the important problems concerning representation of polygroups is as follows: For a given polygroup P =< P, ·, e,−1 >, find a semihyperring R with identity such that one gets a representation of P by hypermatrices with entries from R. Recall that if MR = {(aij )|aij ∈ R}, then a map T : P −→ MR is called a representation if (1) T (x1 · x2 ) = {T (x) | x ∈ x1 · x2 } = T (x1 )T (x2 ), for all x1 , x2 ∈ P , (2) T (e) = I, where I is the identity matrix. If instead of the first condition, we have the condition T (x1 · x2 ) ⊆ T (x1 )T (x2 ), for all x1 , x2 ∈ P , then T is called an inclusion representation. Example 3.8.1. Suppose that the multiplication table for the polygroup P = (P, ·, e,−1 ), where P = {e, a, b} is a b · e e e a b a a {e, b} {a, b} b b {a, b} {e, a}

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In Z3 , we define a hyperoperation ⊕ as follows: 1 ⊕ 1 = {0, 2}, 2 ⊕ 2 = {0, 1}, 1 ⊕ 2 = 2 ⊕ 1 = {1, 2} and ⊕ = + be the usual sum for the other cases, and let be the usual product in Z3 . One can see that (Z3 , ⊕, ) is a semihyperring. Then, the map T : P −→ MR with       100 101 102 T (e) =  0 1 0  , T (a) =  0 1 0  , T (b) =  0 1 0  001 001 001 is a representation of the polygroup P . Generally, if we choose i0 , j0 , i0 6= j0 , 0 ≤ i0 , j0 ≤ n and then put T (e) = In , T (a) = An and T (b) = Bn where   aii = 1 for i = 1, · · · , n An = (aij ) with a =1  i0 j 0 aij = 0 otherwise.  Bn = (bij ) with

bij = aij if i 6= i0 , j 6= j0 bi0 j0 = 2.

Then, T is a representation of P . Example 3.8.2. Suppose that < P1 , ·, e1 ,−1 > and < P2 , ∗, e2 ,−1 > are two polygroups. We know < P1 × P2 , ◦, E,−I > is a polygroup, where (x1 , y1 ) ◦ (x2 , y2 ) = {(x, y) | x ∈ x1 · x2 , y ∈ y1 ∗ y2 }, E = (e1 , e2 ), (x, y)−I = (x−1 , y −1 ). Now, if T1 : P1 −→ MR and T2 : P2 −→ MR are two representations of P1 and P2 respectively, then we have the following representation for P1 × P2 : T1 × T2 : P1 × P2 −→ MR ,   T1 (x) 0 T1 × T2 (x, y) = . 0 T2 (y) Proposition 3.8.3. Let A =< A, ·, e,−1 > and B =< B, ·, e,−1 > be two polygroups. Let T be a representation of B. Then, ϕ : A[B] −→ MR where  T (x) if x ∈ B ϕ(x) = In if x ∈ A is a representation of A[B]. Proof. If x, y ∈ B and y = x−1 , then ϕ(x)ϕ(y) = T (x)T (y) and ϕ(x ∗ y) = ϕ(x · y ∪ A) = ϕ(x · y) ∪ ϕ(A) = ϕ(x · y) ∪ {In }. Since e ∈ x · y, ϕ(e) ∈ ϕ(x · y) and so In ∈ ϕ(x · y). Therefore, ϕ(x ∗ y) = ϕ(x)ϕ(y). Other cases are obvious. 

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We recall the following definition from [56; 147]. Definition 3.8.4. Let (R, +, ·) be a hyperring. We define the relation Γ as follows: aΓb if and only if {a, b} ⊆ u, where u is a finite sum of finite products of elements of R. We denote the transitive closure of Γ by Γ∗ . The equivalence relation ∗ Γ is called the fundamental equivalence relation in R. We denote the equivalence class of the element a (also called the fundamental class of a) by Γ∗ (a). According to the distributive law, every set which is the value of a polynomial in elements of R is a subset of a sum of products in R. Let U be the set of all finite sums of products of elements of R. We can rewrite the definition of Γ∗ on R as follows: a Γ∗ b if and only if there exist z1 , . . . , zn+1 ∈ R with z1 =a, zn+1 =b and u1 , . . . , un ∈ U such that {zi , zi+1 } ⊆ ui for i ∈ {1, . . . , n}. Theorem 3.8.5. Let (R, +, ·) be a hyperring. Then, the relation Γ∗ is the smallest equivalence relation on R such that the quotient R/Γ∗ is a ring. R/Γ∗ is called the fundamental ring. Proof. First, we prove that R/Γ∗ is a ring. The product and the sum ⊕ in R/Γ∗ are defined as follows: Γ∗ (a) ⊕ Γ∗ (b) = {Γ∗ (c) | c ∈ Γ∗ (a) + Γ∗ (b)}, Γ∗ (a) Γ∗ (b) = {Γ∗ (d) | d ∈ Γ∗ (a) · Γ∗ (b)}. Let a0 ∈ Γ∗ (a), b0 ∈ Γ∗ (b). Hence, a0 Γ∗ a implies that there exist x1 , . . . , xm+1 with x1 =a0 , xm+1 =a and u1 , . . . , um ∈ U such that {xi , xi+1 } ⊆ ui for i ∈ {1, . . . , m}; b0 Γ∗ b implies that there exist y1 , . . . , yn+1 with y1 = b0 , yn+1 = b and v1 , . . . , vn ∈ U such that {yj , yj+1 } ⊆ vj for j ∈ {1, . . . , n}. We obtain {xi , xi+1 } + y1 ⊆ ui + v1 , i ∈ {1, . . . , m − 1} xm+1 + {yj , yj+1 } ⊆ um + vj , j ∈ {1, . . . , n}. The sets ui + v1 = ti , i ∈ {1, . . . , m − 1} and um + vj = tm+j−1 , j ∈ {1, . . . , n} are elements of U. We choose the elements z1 , . . . , zm+n such that zi ∈ xi + y1 , i ∈ {1, . . . , m} and zm+j ∈ xm+1 + yj+1 , j ∈ {1, . . . , n}.

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We obtain {zk , zk+1 } ⊆ tk , k ∈ {1, . . . , m + n − 1}. Hence, any element z1 ∈ x1 +y1 = a0 +b0 is Γ∗ equivalent to any element zm+n ∈ xm+1 +yn+1 = a+b. Thus, Γ∗ (a) ⊕ Γ∗ (b) is singleton and we have Γ∗ (a) ⊕ Γ∗ (b) = Γ∗ (c), for all c ∈ Γ∗ (a) + Γ∗ (b). According to the distributive law, we have u·v ∈ U for all u, v ∈ U. Similarly, we obtain Γ∗ (a) Γ∗ (b) = Γ∗ (d) for all d ∈ Γ∗ (a) · Γ∗ (b). Therefore, it is immediate that R/Γ∗ is a ring. Now, let ρ be an equivalence relation in R, such that R/ρ is a ring and let ρ(a) be the equivalence class of the element a. Then, ρ(a) ⊕ ρ(b) and ρ(a) ρ(b) are singletons for all a, b ∈ R, which means that for all a, b ∈ R, for all c ∈ ρ(a) + ρ(b), for all d ∈ ρ(a) · ρ(b) we have ρ(a) ⊕ ρ(b) = ρ(c), ρ(a) ρ(b) = ρ(d). The above equalities are called the fundamental properties in (R/ρ, ⊕, ). Hence, for all a, b ∈ R and A ⊆ ρ(a), B ⊆ ρ(b) we have ρ(a) ⊕ ρ(b) = ρ(a + b) = ρ(A + B) and ρ(a) ρ(b) = ρ(a · b) = ρ(A · B). By induction, we extend the above equalities to finite sums and products. Now, set u ∈ U, which means that there exist the finite sets of indices J and Ij and the elements xi ∈ R such that: ! P Q u= xi . j∈J

i∈Ij

Q

For all Ij , the set xi is a subset of one class, say ρ(aj ). Thus, for all i∈Ij P a∈ aj we have j∈J

! u⊆

P

ρ(aj ) = ρ

j∈J

P

aj

= ρ(a).

j∈J

Therefore, for all x, y ∈ R, x Γ y implies x ρ y, whence x Γ∗ y implies that x ρ y. Hence, for all a ∈ R, Γ∗ (a) ⊆ ρ(a), which means that Γ∗ as the smallest equivalence relation in R such that the quotient R/Γ∗ is a ring.  ! P Q Remark 3.8.6. If u = xi ∈ U, then for all z ∈ u, j∈J

i∈Ij

! Γ∗ (u) = ⊕

P j∈J



Q i∈Ij

Γ∗ (xi )

= Γ∗ (z),

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P

and

Q

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denote the sum and the product of classes.

In order to speak about canonical maps, we need the following notion: Definition 3.8.7. Let R1 and R2 be two hyperrings. The map f : R1 → R2 is called an inclusion homomorphism if for all x, y ∈ R, the following conditions hold: f (x + y) ⊆ f (x) + f (y) and f (x · y) ⊆ f (x) · f (y). f is called a strong homomorphism if for all x, y ∈ R, we have f (x + y) = f (x) + f (y) and f (x · y) = f (x) · f (y). Let R be a hyperring. We denote by β. and β+ the following binary relations: xβ. y if and only if there exist z1 , . . . , zn ∈ R such that {x, y} ⊆ z1 ·. . .·zn and xβ+ y if and only if there exist z1 , . . . , zn ∈ R such that {x, y} ⊆ z1 + . . . + zn . We denote the transitive closures of the relations β. and β+ by β.∗ and ∗ ∗ the fundamental equivalence relations with reβ+ , and we call β.∗ and β+ spect to multiplication and addition, respectively. For all a ∈ R we denote ∗ (a) and we have the corresponding equivalence classes of a by β.∗ (a) and β+ ∗ β.∗ (a) ⊆ Γ∗ (a), β+ (a) ⊆ Γ∗ (a).

Let us consider the following canonical maps ϕ· : R −→ R/β.∗ , ϕ· (x) = β.∗ (x), ∗ ∗ ϕ+ : R −→ R/β+ , ϕ+ (x) = β+ (x), ∗ ∗ ∗ ∗ ϕ : R −→ R/Γ , ϕ (x) = Γ (x). ∗ We notice that the maps ϕ+ : (R, +) −→ (R/β+ , ⊕), ϕ· : (R, ·) −→ ∗ ∗ ∗ (R/β. , ), ϕ : (R, +, ·) −→ (R/Γ , ⊕, ) are strong homomorphisms. We denote by ω+ , ω ∗ the kernels of ϕ+ , ϕ∗ , respectively. If ¯0 is the ∗ zero element of R/β+ or R/Γ∗ , then

ω+ = kerϕ+ = {x ∈ R : ϕ+ (x) = ¯0}, ω ∗ = kerϕ∗ = {x ∈ R : ϕ∗ (x) = ¯0}. We have ω+ ⊆ ω ∗ . Theorem 3.8.8. A necessary condition in order to have an inclusion representation T of polygroup < P, ·, e,−1 > by n × n hypermatrices over the hyperring (R, +, ·) is the following:

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• For every fundamental class β ∗ (x), there must exist elements aij ∈ R (i, j ∈ {1, . . . , n}) such that n o T (β ∗ (x)) ⊆ (a0ij ) | a0ij ∈ Γ∗ (a0ij ), i, j ∈ {1, . . . , n} . Proof. Let x β ∗ y, for x, y ∈ P and T (x) = (xij ), T (y) = (yij ), where xij , yij ∈ R for all i, j ∈ {1, . . . , n}. Then, there exist elements z1 , . . . , zr−1 , hq1 , . . . , hqr ∈ P and Q1 , . . . , Qr finite sets of indices, such that setting z0 = x, zr = y, then Q {zv−1 , zv } ⊆ hqv , v ∈ {1, . . . , r}. pv ,qv ∈Qv

Therefore, for all v ∈ {1, . . . , r}, we obtain ! Q

{T (zv−1 ), T (zv )} ⊆ T

hqv

p ,qv ∈Qv

Qv

⊆

T (hqv ).

pv ,qv ∈Qv

In the ij entry of the hypermatrix Q

hqv

pv ,qv ∈Qv

there are sets which are the union of sets Uµ formed by sums of n|Qv |−1 products of Qv elements each. Consequently, each of these sets Uµ belong to one Γ∗ fundamental class. Furthermore, for every Uµ , there exists another Uλ such that Uµ ∩ Uλ 6= ∅. Since Γ∗ is transitive, there exists an element av,ij ∈ R such that for all ij entries zv−1,ij and zv,ij of the hypermatrices T (zv−1 ) and T (zv ), respectively, we have {zv−1,ij , zv,ij } ⊆ Γ∗ (av,ij ), for all v ∈ {1, . . . , r}. Again, from the transitivity of Γ∗ , we have Γ∗ (a1,ij ) = . . . = Γ∗ (av,ij ) := Γ∗ (aij ). In particular, we obtain {xij , yij } ⊆ Γ∗ (aij ), for all i, j ∈ {1, . . . , n}.  Theorem 3.8.9. Every representation T (a) = (aij ) of a polygroup < P, ·, e,−1 > by n×n hypermatrices over the hyperring (R, +, ·) induces a n×n representation T ∗ of the fundamental group P/β ∗ over the fundamental ring R/Γ∗ by setting T ∗ (β ∗ (a)) = (Γ∗ (aij )), for all β ∗ (a) ∈ P/β ∗ .

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Proof. By using Theorem 3.8.8, it is straightforward.  Let P =< P, ·, e,−1 > be a polygroup. Two elements x, y ∈ P are said to be conjugate if there exists an elements z ∈ P such that y ∈ z −1 xz. Theorem 3.8.10. Let T be a representation of P over R of degree n and let In be the unit matrix over R/Γ∗ . Then, (1) T ∗ (β ∗ (e)) = In ; (2) T ∗ (β ∗ (x−1 )) = (T ∗ (β ∗ (x)))−1 , for all x ∈ P ; (3) If x, y ∈ P are conjugate, then T ∗ (β ∗ (x)), T ∗ (β ∗ (y)) are conjugate. Proof. (1) For every x ∈ P , we have β ∗ (x) = β ∗ (x) β ∗ (e) = β ∗ (e) β ∗ (x) and so we obtain T ∗ (β ∗ (x)) = T ∗ (β ∗ (x))T ∗ (β ∗ (e)) = T ∗ (β ∗ (e))T ∗ (β ∗ (x)). Therefore, T ∗ (β ∗ (e)) = In . (2) We have e ∈ x · x−1 and so β ∗ (e) = β ∗ (x · x−1 ) = β ∗ (x) β ∗ (x−1 ) which implies that T ∗ (β ∗ (e)) = T ∗ (β ∗ (x) β ∗ (x−1 )) = T ∗ (β ∗ (x))T ∗ (β ∗ (x−1 )). So, In = T ∗ (β ∗ (x))T ∗ (β ∗ (x−1 )). Therefore, T ∗ (β ∗ (x−1 )) = (T ∗ (β ∗ (x))−1 . (3) The proof is similar to the proofs of (1) and (2).  3.9

Polygroup hyperrings

Davvaz and Poursalavati introduced the notion of polygroup hyperrings as a generalization of group rings [59]. They established homomorphisms among various polygroup hyperrings. Let < P, ·, e,−1 > be a finite polygroup, and < R, +, 0, − > be a commutative polygroup and (R, +, ∗) be a hyperring with scalar unit and zero element. Suppose that R[P ] is the set of all the functions on P with values in R, i.e., R[P ] = {f | f : P −→ R is a function}. On R[P ] we consider the hyperoperations defined as follow: f ⊕ g = {h | h(x) ∈ f (x) + g(x)}, ( ) P f g = h | h(z) ∈ f (x) ∗ g(y) . z∈x·y

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We define the mapping −I : R[P ] −→ R[P ], where f −I : P −→ R is defined by f −I (p) = −f (p) for every p ∈ P and let f0 be the zero map. Our aim in the following lemma and theorem is to show that R[P ] is a hyperring with hyperoperations ⊕ and . Lemma 3.9.1. < R[P ], ⊕, f0 ,−I > is a polygroup. Proof. For every f, g, h ∈ R[P ], obviously we have (f ⊕ g) ⊕ h = f ⊕ (g ⊕ h) and f0 ⊕ f = f ⊕ f0 = f . Now, let f ∈ g ⊕ h. Then, for every x ∈ P we have f (x) ∈ g(x) + h(x) and so g(x) ∈ f (x) − h(x) and h(x) ∈ −g(x) + f (x). Therefore, g ∈ f ⊕ (−h) and h ∈ (−g) ⊕ f  Theorem 3.9.2. (R[P ], ⊕, ) is a hyperring. Proof. By Lemma 3.9.1, < R[P ], ⊕, f0 ,−I > is a polygroup. Let f1 , f2 , f3 ∈ R[P ]. Then, f1 (f2 f3 ) ( = f1

) P

f | f (z) ∈

f2 (x) ∗ f3 (y)

z∈x·y

=

S

f1 f where the union is over f with

f

f (z) ∈

P

f2 (x) ∗ f3 (y)

z∈x·y

=

S

  P g | g(a) ∈ f1 (b) ∗ f (c)

f

a∈b·c

( =

g | g(a) ∈

!) P

f1 (b) ∗

P

f2 (x) ∗ f3 (y)

c∈x·y

a∈b·c

( =

g | g(a) ∈

) P

P

f1 (b) ∗ (f2 (x) ∗ f3 (y))

a∈b·c c∈x·y

  P P = g | g(a) ∈ c∈x·y (f1 (b) ∗ f2 (x)) ∗ f3 (y) a∈b·c ( ) P = g | g(a) ∈ (f1 (b) ∗ f2 (x)) ∗ f3 (y) a∈b·(x·y)

( =

g | g(a) ∈

) P

(f1 (b) ∗ f2 (x)) ∗ f3 (y)

a∈(b·x)·y

( =

g | g(a) ∈

) P

P

a∈d·y d∈b·x

(f1 (b) ∗ f2 (x)) ∗ f3 (y)

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=

S

121

f f3 where the union is over f with

f

f (z) ∈

P

f1 (x) ∗ f2 (y)

z∈x·y

= (f1 f2 ) f3 . Similarly, we have o S S n P f (f1 ⊕ f2 ) = f g = h | h(z) ∈ z∈x·y f (x) ∗ g(y) g∈f1 ⊕f2 g∈f1 ⊕f2 ( ) P = h | h(z) ∈ f (x) ∗ (f1 (y) + f2 (y)) z∈x·y

( =

)

h | h(z) ∈

P

(f (x) ∗ f1 (y)) + (f (x) ∗ f2 (y))

z∈x·y

( =

!

h | h(z) ∈

P

f (x) ∗ f1 (y)

!) +

z∈x·y

=

S

P

f (x) ∗ f2 (y)

z∈x·y

{h | h(z) ∈ h1 (z) + h2 (z)}

h1 ,h2

where the union is over h1 ∈ f f1 and h2 ∈ f f2 ! S SS S S h1 ⊕ h2 = (h1 ⊕ h2 ) = h1 ⊕ h2 ) = h1 ,h2

=

S

h1 h2

h1

h2

(f f1 ) ⊕ h2 = (f f1 ) ⊕ (f f2 ).

h2

Hence, f (f1 ⊕ f2 ) = (f f1 ) ⊕ (f f2 ). Similarly it can be proved that (f1 ⊕ f2 ) f = (f1 f ) ⊕ (f2 f ). Consequently, (R[P ], ⊕, ) is a hyperring.  Now that we constructed the hyperring R[P ] from R and P we will study relation between the polygroup P and the hyperring R[P ]. We define E : R −→ R[P ] by r 7→ Er , where Er : P −→ R is defined by  r if g = e Er (g) = 0 if g 6= e. It is clear that E is a one to one function and we have E(r1 + r2 ) = E(r1 ) ⊕ E(r2 ), E(r1 ∗ r2 ) = E(r1 ) E(r2 ), E(0) = E0 := zero function. Therefore, R is imbedded in R[P ]. If H is a subpolygroup of P , then we write RhHi = {f ∈ R[P ] | {x | f (x) 6= 0} ⊆ H}.

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Then, there is a one to one polygroup homomorphism from RhHi to R[P ]. Proposition 3.9.3. Let P1 and P2 be two polygroups and ψ : P1 −→ P2 be a mapping. Then, there exists an inclusion homomorphism of polygroups ϕ : R[P1 ] −→ R[P2 ]. Proof. We define ϕ(f ) = f ◦ ψ. Obviously, ϕ is well defined. If h ∈ f1 ⊕ f2 , then for every x ∈ P1 , we have ϕ(h)(x) = h(ψ(x)) ∈ f1 (ψ(x)) + f2 (ψ(x)) or ϕ(h)(x) ∈ ϕ(f1 )(x) + ϕ(f2 (x)) which implies that ϕ(h) ∈ ϕ(f1 ) ⊕ ϕ(f2 ) and so ϕ(f1 ⊕ f2 ) ⊆ ϕ(f1 ) ⊕ ϕ(f2 ).  Proposition 3.9.4. Let ψ : R −→ S be a surjective inclusion homomorphism of hyperrings and T = kerψ. Then, the mapping ψ : R[P ] −→ S[P ] denoted by ψ(f ) = ψ ◦ f is a surjective inclusion homomorphism whose kernel is T [P ]. Proof. We have ψ(f1 ⊕ f2 ) = ψ ◦ (f1 ⊕ f2 ) = ψ ◦ {f | f ∈ f1 ⊕ f2 } = {ψ ◦ f | f ∈ f1 ⊕ f2 } = {h | h(p) = ψ(f (p)), f (p) ∈ f1 (p) + f2 (p), ∀p ∈ P } = {h | h(p) ∈ ψ(f1 (p) + f2 (p)), ∀p ∈ P } ⊆ {h | h(p) ∈ ψ(f1 (p)) + ψ(f2 (p))} = ψ ◦ f1 ⊕ ψ ◦ f2 , and so ( ψ(f1 f2 ) =

)

ψ(f ) | f (x) ∈

P

f1 (y) ∗ f2 (z)

x∈y·z

( =

)

ψ ◦ f | f (x) ∈

P

f1 (y) ∗ f2 (z)

x∈y·z

( ⊆

!) P

ψ ◦ f | ψ ◦ f (x) ∈ ψ

f1 (y) ∗ f2 (z)

x∈y·z

( ⊆

)

ψ ◦ f | ψ ◦ f (x) ∈

P

ψ(f1 (y)) ∗ ψ(f2 (z))

x∈y·z

= ψ(f1 ) ψ(f2 ). Therefore, ψ is an inclusion homomorphism. Obviously, ψ is onto, and kerψ = {f = {f = {f = {f

∈ R[P ] ∈ R[P ] ∈ R[P ] ∈ R[P ]

| | | |

ψ ◦ f = f0 }, where f0 is the zero function ψ(f (x)) = 0, ∀x ∈ P } f (x) ∈ kerψ, ∀x ∈ P } f (x) ∈ T, ∀x ∈ P } = T [P ]. 

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Let Γ∗1 and Γ∗2 be the fundamental relations on R and R[P ], respectively. Let U1 and U2 denote the sets of all finite polynomials of elements of R and R[P ] over natural numbers, respectively. In the following theorem we will construct a homomorphism between R/Γ∗1 and R[P ]/Γ∗2 . Theorem 3.9.5. There is a homomorphism g : R/Γ∗1 −→ R[P ]/Γ∗2 . Proof. We define g(Γ∗1 (r)) = Γ∗2 (Er ). First, we prove that g is well defined. We know a Γ∗1 b if and only if there exist x1 , . . . , xm+1 ∈ R; u1 , . . . , um ∈ U1 with x1 = a, xm+1 = b such that {xi , xi+1 } ⊆ ui , i = 1, . . . , m. Then, this implies E({xi , xi+1 }) ⊆ E(ui ) or {E(xi ), E(xi+1 )} ⊆ E(ui ) ∈ U2 and so E(xi )Γ∗2 E(xi+1 ), i = 1, . . . , m. Therefore, E(a)Γ∗2 E(b) that is to say Γ∗2 (Ea ) = Γ∗2 (Eb ). Now, we will show that g is a homomorphism. This is because g(Γ∗1 (a) ⊕ Γ∗1 (b)) = g(Γ∗1 (a + b)) = Γ∗2 (E(a + b)) = Γ∗2 (E(a) ⊕ E(b)) = Γ∗2 (E(a)) ⊕ Γ∗2 (E(b)) = g(Γ∗1 (a)) ⊕ g(Γ∗1 (b)). Similarly, we obtain g(Γ∗1 (a) Γ∗1 (b)) = g(Γ∗1 (a)) g(Γ∗1 (b)).  Corollary 3.9.6. The following diagram is commutative, i.e., ϕ2 E = gϕ1 , where ϕ1 and ϕ2 are canonical maps. R

E

/ R[P ]

g

 / R[P ]/Γ∗2

ϕ1

 R/Γ∗1

3.10

ϕ2

Solvable polygroups

Aghabozorgi, Davvaz and Jafarpour in [2] introduced the concept of solvable polygroups. The purpose of this section is to provide a detailed structure description of derived subpolygroups of polygroups. We investigate the concept of perfect and solvable polygroups and we give some results in this respect. Finally, we discuss on τ -multi-semi-direct hyperproduct of polygroups. Definition 3.10.1. Let H be a hypergroup. We define (1) [x, y]r = {h ∈ H | x · y ∩ y · x · h 6= ∅} ; (2) [x, y]l = {h ∈ H | x · y ∩ h · y · x 6= ∅} ;

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(3) [x, y] = [x, y]r ∪ [x, y]l . From now on we call [x, y]r , [x, y]l and [x, y] right commutator x and y, left commutator x and y and commutator x and y, respectively. Also, we will denote [H, H]r , [H, H]l and [H, H] the set of all right commutators, left commutators and commutators, respectively. Proposition 3.10.2. If H is a group, then [y, x]−1 = [x, y]r = r [x−1 , y −1 ]l = [y −1 , x−1 ]−1 , for every x, y in H. l Proof. It is straightforward.  Example 3.10.3. Suppose that H = {e, a, b}. Consider the hypergroup (H, ·), where · is defined on H as follows: a b · e e {a, b} e e a e a b b e {a, b} {a, b} It is easy to see that {a} = [a, a]r 6= [a, a]l = {a, b} = [a−1 , a−1 ]l , where a−1 is the inverse of a in H. Proposition 3.10.4. If H is a commutative hypergroup, then [x, y]r = [x, y]l = [x, y], for all (x, y) ∈ H 2 . Proof. It is straightforward.  Let X be a non-empty subset of a polygroup hP, ·, e,−1 i. Let {Ai | i ∈ J} T be the family of all subpolygroups of P containing X. Then, Ai is i∈J

called the subpolygroup generated by X. This subpolygroup is denoted by < X > and we have < X >= ∪{xε11 · . . . · xεkk | xi ∈ X, k ∈ N, εi ∈ {−1, 1}}. If X = {x1 , x2 , . . . , xn }, then the subpolygroup < X > is denoted < x1 , x2 , . . . , xn >. In a special case < [P, P ]r >, < [P, P ]l > and < [P, P ] > are shown by Pr0 , Pl0 and P 0 , respectively. Proposition 3.10.5. Let hP, ·, e,−1 i be a polygroup and (x, y) ∈ P 2 . Then, (1) [x, y]r = [x−1 , y −1 ]l ; (2) P 0 = Pr0 = Pl0 ; (3) x ∈ P 0 ⇒ x−1 ∈ P 0 . Proof. (1) Suppose that u ∈ [x, y]r . Then, x · y ∩ y · x · u 6= ∅ so there exists t ∈ P such that t ∈ x · y ∩ y · x · u. Thus, t ∈ x · y ∩ v · u for some v ∈ y · x. Since P is a polygroup, we have v −1 ∈ x−1 · y −1 ∩ u · y −1 · x−1 . Therefore,

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u ∈ [x, y]l . Hence, [x, y]r ⊆ [x, y]l . Similarly, we have [x, y]l ⊆ [x, y]r . (2) It follows from (1). (3) Suppose that a ∈ [x, y]r . Then, x · y ∩ y · x · a 6= ∅ and so y −1 · x−1 ∩ −1 a · x−1 · y −1 6= ∅. Hence, a−1 ∈ [y −1 , x−1 ]l . Now, let x ∈ P 0 . Then, we have x ∈ x1 · x2 · . . . · xn , where xi ∈ [ai , bi ]r and 1 ≤ i ≤ n. Therefore, by −1 −1 0 (2) x−1 ∈ x−1 n · xn−1 · . . . · x1 ⊆ P follows. Corollary 3.10.6. If hP, ·, e,−1 i is a polygroup, then P 0 is a subpolygroup of P. From now on we call P 0 the derived subpolygroup of P. Proposition 3.10.7. Let hP, ·, e,−1 i be a polygroup. Then, P 0 = {e} if and only if P be an abelian group. Proof. Suppose that P 0 = {e}. Then, we have a · b · a−1 · b−1 = e, for all (a, b) ∈ P 2 and hence there exists x ∈ P such that e = a · x. Thus, x = a−1 and so a · a−1 = e. Now, let (x, y) ∈ P 2 and z ∈ x · y. We obtain y ∈ x−1 · z. Therefore, x · y = z and hence P is an abelian group.  Definition 3.10.8. Let hP, ·, e,−1 i be a polygroup. Then, we define N (P 0 ) = {u ∈ P 0 | u · P 0 = P 0 · u} and is called the normalizer P 0 in P . Example 3.10.9. Suppose that P = {e, a, b, c}. We consider the noncommutative polygroup hP, ·, e,−1 i, where · is defined on P as follow: · e a b c

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

b c b c P c b {b, c} c P

In this case, we can see that P 0 = P . Recall that a subhypergroup K of a hypergroup (H, ·) is invertible on the right if and only if K \ H = {K · x | x ∈ H} is a partition of H. If K is an invertible to the left subhypergroup of a hypergroup H, then the quotient K \ H by the hyperoperation K · x ⊗ K · y = {K · z | z ∈ x · K · y}, is a hypergroup. Theorem 3.10.10. If hP, ·, e,−1 i is a polygroup, then

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(1) (2) (3) (4)

P 0 is invertible; N = N (P 0 ) is subpolygroup of hP, ·, e,−1 i; P 0 E N (P ) and (P 0 \ N, ⊗) is commutative polygroup; if K is a complete subpolygroup of P such that K \ P is commutative, then P 0 ⊆ K.

Proof. (1) It is obvious. (2) Suppose that x, y ∈ N and a ∈ x · y. We have a · P 0 ⊆ x · y · P 0 = x · P 0 · y = P 0 · x · y. Hence, a · P 0 = P 0 · b, for some b ∈ x · y. Therefore, a ∈ P 0 · b and so P 0 · a = P 0 · b = a · P 0 which means a ∈ N . Moreover, if x ∈ N we can easily see that x−1 ∈ N . (3) It is easy to see that the hyperoperation ⊗ is well defined and also P 0 \ N is a polygroup. For commutativity, since P is a polygroup, we have [x, y]l = x · y · x−1 · y −1 , for all (x, y) ∈ P 2 . Hence, x · y ⊆ [x, y]l · y · x and so N · x ⊗ N · y = N · x · y = N · y · x = N · y ⊗ N · x. (4) Suppose that (x, y) ∈ P 2 and a ∈ x · y · x−1 · y −1 = [x, y]l . Since K ⊆ K · x · y · x−1 · y −1 , we have x · y · x−1 · y −1 ∩ K 6= ∅. Hence, x · y · x−1 · y −1 ⊆ K and so a ∈ K. Therefore, P 0 ⊆ K.  Definition 3.10.11. A polygroup P is called perfect if P 0 = P . Definition 3.10.12. A polygroup P is called solvable if P (n) = ωP , for some n ∈ N, where P (1) = P 0 and P (n+1) = (P (n) )0 . Proposition 3.10.13. Every non-trivial perfect group is not solvable. Proof. It is straightforward.  In the following, we show that the above proposition is not true for the class of polygroups. Example 3.10.14. Suppose that P = {e, a, b, c}. Consider the commutative polygroup hP, ·, e,−1 i, where · is defined on P as follows: · e a b c

e a b c e a b c a P {a, b, c} {a, b, c} b {a, b, c} P {a, b, c} c {a, b, c} {a, b, c} P

We can easily see that P is a perfect and solvable polygroup. Notice that P 0 = P = ωP .

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Example 3.10.15. Suppose that P = {e, a, b, c}. Consider the noncommutative polygroup hP, ·, e,−1 i, where · is defined on P as follows: · e a b c

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

b c b c P c b {b, c} c P

In this case, we can see that P 0 = P = ωP . In the following theorem, consider G//H as a double coset algebra (see Example 3.1.2). Theorem 3.10.16. Let (G, ·) be a group and H be a subgroup of G. We set HG0 H = {HgH | g ∈ G0 }. Then, (1) HG0 H ⊆ (G//H)0 ; (2) If G0 · H = G then (G//H) is a perfect polygroup; (3) If HG0 H = (G//H) then G0 · H = G. Proof. (1) We have [HaH, HbH] = (HaH) ∗ (HbH) ∗ (HaH)−I ∗ (HbH)−I = {Hah1 bh2 a−1 h3 b−1 H | h1 , h2 , h3 ∈ H}, since H is a subgroup of G, e ∈ H. Now, suppose that h1 = h2 = h3 = e. Then, H[a, b]H = Haba−1 b−1 H ∈ [HaH, HbH]. Therefore, HG0 H ⊆ (G//H)0 . (2) Suppose that HxH ∈ (G//H). Then, x ∈ G. Hence, there exists a, b ∈ G and h ∈ H such that x = [a, b]h. So, we have HxH = H[a, b]hH = H[a, b]H ∈ HG0 H ⊆ (G//H)0 . Thus, (G//H) is a perfect polygroup. (3) Since G0 is a normal subgroup of G, then G0 · H ⊆ G. Now, suppose that x ∈ G. Then, there exists (a, b) ∈ G2 such that HxH = H[a, b]H. Therefore, there exists (h1 , h2 ) ∈ H 2 such that x = h1 [a, b]h2 = −1 −1 0 0 h1 [a, b]h−1 1 h1 h2 = [h1 ah1 , h1 bh1 ]h1 h2 ∈ G · H. Hence, G = G · H.  S Definition 3.10.17. Let P be a hypergroup. Suppose that τ = τm , m≥1

where τ1 is the diagonal relation and for every integer m > 1, τm is the relation defined as follows: / P 0 such that x τm y ⇔ ∃(z1 , . . . , zm ) ∈ P m , ∃σ ∈ Sm : σ(i) = i if zi ∈ m m Q Q x∈ zi and y ∈ zσ(i) . i=1

i=1

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Obviously, the relation τ is reflexive and symmetric. Now, let τ ∗ be the transitive closure of τ . Theorem 3.10.18. The relation τ ∗ is a strongly regular relation. Proof. It is easy to see that τ ∗ is an equivalence relation. In order to prove that it is strongly regular, first we show that: =

=

xτ y ⇒ x · z τ ∗ y · z, z · x τ ∗ z · y, for every z ∈ P . Suppose that xτ y. Then, there exists m ∈ N such that xτm y. Hence, there exist (z1 , . . . , zm ) ∈ P m , σ ∈ Sm with σ(i) = i if m m Q Q zi and y ∈ zσ(i) . Suppose that z ∈ P . We zi ∈ / P 0 , such that x ∈ m Q

have x · z ⊆ (

i=1

zi ) · z, y · z ⊆ (

i=1

m Q

i=1

zσ(i) ) · z and σ(i) = i if zi ∈ / P 0 . Now,

i=1

suppose that zi+1 = z and we define the permutation σ 0 ∈ Sm+1 as follows: σ 0 (i) = σ(i), for all 1 ≤ i ≤ m and σ 0 (m + 1) = m + 1. Thus, x · z ⊆

m+1 Q

zi and y · z ⊆

i=1 =

m+1 Q

zσ0 (i) such that σ 0 (i) = i if zi ∈ / P 0.

i=1

=

Therefore, x · z τ ∗ y · z. Similarly, we have z · x τ ∗ z · y. Now, if xτ ∗ y then there exists k ∈ N and (x = u0 , u1 , . . . , uk = y) ∈ P k+1 such that x = u0 τ u1 τ . . . τ uk−1 τ uk = y. Hence, by the above results, we obtain =

=

=

=

=

x · z = u0 · z τ ∗ u1 · z τ ∗ u2 · z τ ∗ . . . τ ∗ uk−1 · z τ ∗ uk · z = y · z =

and so x · z τ ∗ y · z.

=

Similarly, we can prove that z · x τ ∗ z · y. Therefore, τ ∗ is a strongly regular relation on P .  Definition 3.10.19. Let (H, ·) and (K, ∗) be hypergroups. A map f : H → P ∗ (K) is called a good multihomomorphism if f (x · y) = f (x) ∗ f (y), S for all x, y ∈ H. If (H, ·) = (K, ∗) and f (h) = H, then f is called a h∈H

generalized automorphism. Moreover, we will denote by GAut(H) the set of all generalized automorphisms of (H, ·). Proposition 3.10.20. (GAut(H), ◦) is a monoid, where ◦ is defined as follows: S (f ◦ g)(h) = f (a), a∈g(h)

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for all f, g ∈ GAut(H) and h ∈ H. Moreover, Aut(H) (i.e., the group of automorphism of H) is a subgroup of GAut(H). Proof. It is straightforward.  Definition 3.10.21. Let (H, ·) and (K, ∗) be two hypergroups. We consider the monoid GAut(H) and the group τK∗ . Let: ϕ:

K → GAut(H), τ∗ τ ∗ (x) → ϕτ ∗ (x)

be a homomorphism. Then, we define a hyperoperation in H ×K as follows: (x1 , y1 ) ◦ (x2 , y2 ) = {(x, y) | x ∈ x1 · ϕτ ∗ (y1 ) (x2 ), y ∈ y1 ∗ y2 }. We call it a τ -multisemi-direct hyperproduct of hypergroups H and K through ϕ and we denote it by H ×ϕ K. Moreover, we call a τ -multisemidirect hyperproduct is special if Im(ϕ) ⊆ Aut(H). Lemma 3.10.22. Let H and K be two hypergroups. Then, H ×K equipped with the τ -multisemi-hyperproduct is a hypergroup. Proof. Suppose that (x1 , y1 ), (x2 , y2 ) and (x3 , y3 ) are elements in H × K. If (s, t) ∈ [(x1 , y1 ) ◦ (x2 , y2 )] ◦ (x3 , y3 ), then (s, t) ∈ (u, v) ◦ (x3 , y3 ) for some (u, v) ∈ (x1 , y1 ) ◦ (x2 , y2 ). Therefore, s ∈ u · ϕτ ∗ (v) (x3 ), t ∈ v ∗ y3 and u ∈ x1 · ϕτ ∗ (y1 ) (x2 ), v ∈ y1 ∗ y2 . Thus, s ∈ (x1 · ϕτ ∗ (y1 ) (x2 )) · ϕτ ∗ (v) (x3 ) and t ∈ (y1 ∗ y2 ) ∗ y3 . By the associativity of · and ∗, we have s ∈ x1 · (ϕτ ∗ (y1 ) (x2 ) · ϕτ ∗ (v) (x3 )) and t ∈ y1 ∗ (y2 ∗ y3 ). Since v ∈ y1 ∗ y2 , we conclude that s ∈ x1 · (ϕτ ∗ (y1 ) (x2 ) · ϕτ ∗ (y1 )τ ∗ (y2 ) (x3 )). On the other hand, if (s, t) ∈ (x1 , y1 ) ◦ [(x2 , y2 ) ◦ (x3 , y3 )], then (s0 , t0 ) ∈ (x1 , y1 ) ◦ (u0 , v 0 ) for some (u0 , v 0 ) ∈ (x2 , y2 ) ◦ (x3 , y3 ). Therefore, s0 ∈ x1 · ϕτ ∗ (y1 ) (u0 ), t0 ∈ y1 ∗ v 0 and u0 ∈ x2 · ϕτ ∗ (y2 ) (x3 ), v 0 ∈ y2 ∗ y3 . Thus, s0 ∈ x1 · ϕτ ∗ (y1 ) (x2 .ϕτ ∗ (y2 ) (x3 )) = x1 · (ϕτ ∗ (y1 ) (x2 ) · ϕτ ∗ (y1 )τ ∗ (y2 ) (x3 )) and t0 ∈ y1 ∗ (y2 ∗ y3 ). By the above results, we conclude the associativity of ◦. Now, suppose that (a, x) ∈ H 2 and (b, y) ∈ K 2 . Since H and K are hypergroups, there exists (u, w) ∈ H × K such that a ∈ x · u and b ∈ y ∗ w. Since u ∈ H = ϕτ ∗ (y) (K), we conclude that there exists t ∈ K such that u ∈ ϕτ ∗ (y) (t) and so we have (a, b) ∈ (x, y) ◦ (t, w). Similarly, there exists (t0 , w0 ) ∈ H × K such that (a, b) ∈ (t0 , w0 ) ◦ (x, y).  Theorem 3.10.23. Let P1 and P2 be two polygroups. Then, P1 × P2

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equipped with a special τ -multisemi-hyperproduct is a ploygroup. Proof. Suppose that ϕ is a special τ -multisemi-hyperproduct. According to the previous lemma, P1 ×ϕ P2 is a hypergroup. Since ϕ is special, a = a · ϕτ ∗ (b) (e1 ) and a = e1 · ϕτ ∗ (e1 ) (b). Hence, (a, b) ◦ (e1 , e2 ) = (a, b) = (e1 , e2 )◦(a, b), that is, (e1 , e2 ) is the identity element in P1 ×ϕ P2 . Moreover, we can check that (ϕτ ∗ (b−1 ) (a−1 ), b−1 ) is the inverse element of (a, b) ∈ P1 × P2 . Now, we check that if (z1 , z2 ) ∈ (x1 , x2 ) ◦ (y1 , y2 ), then (x1 , x2 ) ∈ (z1 , z2 ) ◦ (y1 , y2 )−1 . Since z2 ∈ x2 ∗ y2 and z1 ∈ x1 · ϕτ ∗ (x2 ) (y1 ), we have (z1 , z2 ) ◦ (y1 , y2 )−1 = {(a, b)|a ∈ z1 · ϕτ ∗ (z2 ) (ϕτ ∗ (y−1 ) (y1−1 )), b ∈ z2 ∗ y2−1 } 2

= {(a, b)|a ∈ z1 · ϕτ ∗ (z

∗ −1 2 )τ (y2 )

(y1−1 ), b ∈ z2 ∗ y2−1 }

= {(a, b)|a ∈ z1 · ϕτ ∗ (x2 ) (y1−1 ), b ∈ z2 ∗ y2−1 }. Hence, x1 ∈ z1 · ϕτ ∗ (x2 ) (y1 )−1 = z1 · ϕτ ∗ (x2 ) (y1−1 ) and x2 ∈ z2 ∗ y2−1 as we want. The τ -multisemi-direct hyperproduct of polygroups P1 and P2 through zero homomorphism ϕ0 , i.e., P2 → GAut(P1 ), ϕ0 : τ∗ ϕ0 (τ ∗ (x)) = iAut(P1 ) which we denote it by P1 × P2 and is called τ -direct hyperproduct of P1 and P2 . Proposition 3.10.24. Let P1 , P2 be two polygroups. Then, (P1 × P2 )0 = P10 × P20 . Proof. Suppose that ((x1 , y1 ), (x2 , y2 )) ∈ (P1 × P2 )2 . Then, we have −1 −1 −1 [(x1 , y1 ), (x2 , y2 )] = (x1 , y1 ) ⊗ (x2 , y2 ) ⊗ (x−1 1 , y1 ) ⊗ (x2 , y2 ) −1 −1 −1 = {(x, y)|x ∈ x1 · x2 · x−1 1 · x2 , y ∈ y1 ◦ y2 ◦ y1 ◦ y2 }

= {(x, y)|x ∈ [x1 , x2 ], y ∈ [y1 , y2 ]} = [x1 , x2 ] × [y1 , y2 ], and the proof completes.  Corollary 3.10.25. τ -direct hyperproduct of two polygroups P1 and P2 is perfect if and only if P1 and P2 are perfect. Let H be a regular hypergroup. For n ∈ N, let a1 , . . . , an be elements in H, and a01 , . . . , a0n are their inverses in H, respectively. The set a1 · a2 · . . . · an · a0n · a0n−1 · . . . · a01

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is called a product of type zero and denote with N (0) the union of all products of type 0. Theorem 3.10.26. Let H be a regular and reversible hypergroup and e n Q be a bilateral identity. If e ∈ zi , then there exist inverses of z1 , . . . , zn i=1

0 0 respectively z10 , . . . , zn0 such that zn ∈ zn−1 · zn−2 · . . . · z10 · e.

Proof. The theorem is true for n = 2. Indeed, for the hypothesis of reversibility, e ∈ z1 · z2 implies that there exists at least one inverse of z1 , like z10 such that z2 ∈ z10 · e. Suppose that for for every k < n and for every y1 , . . . , yk ∈ H, the implication of the theorem is satisfied that is 0 0 there exist y10 , . . . , yk−1 such that yk ∈ yk−1 · . . . · y10 · e. The hypothesis n n−1 Q Q e∈ zi implies that there exists u ∈ zn−1 ·zn such that e ∈ zi ·u. From i=1

i=1

0 0 u ∈ zn−1 · zn follow that there exists zn−1 such that zn ∈ zn−1 · u. Since n−2 Q 0 e∈ zi · u, the inductive hypothesis implies that there exist z10 , . . . , zn−2 i=1

0 0 0 such that u ∈ zn−2 · . . . · z10 · e. Therefore, zn ∈ zn−1 · zn−2 · . . . · z10 · e. 

Theorem 3.10.27. If H is a regular and reversible hypergroup, then the heart of H is the union of the products of type zero (i.e., ωH = N (0)). Proof. It is clear that N (0) ⊆ ωH . We prove the converse. Let x ∈ ωH . Then, if e is a bilateral identity, one has xβ ∗ e. Now, it follows that there n Q exists n ≥ 1 and z1 , . . . , zn such that {e, x} ⊆ zi . By Theorem 3.10.26, i=1

0 0 there exist z10 , . . . , zn−1 such that x ∈ z1 · . . . · zn−1 · zn−1 · . . . z10 · e ⊆ 0 0 e · z1 · . . . · zn−1 · zn−1 · . . . · z1 · e ⊆ N (0). 

Corollary 3.10.28. Let ωP1 , ωP2 and ωP1 ×P2 be the hearts of P1 , P2 and P1 × P2 , respectively. Then, ωP1 ×P2 = ωP1 × ωP2 . Proposition 3.10.29. Let P1 , P2 be two polygroups. If P1 × P2 is solvable, then P1 and P2 are solvable. Proof. The proof follows from Proposition 3.10.24 and Corollary 3.10.28.  3.11

Nilpotent polygroups

Jafarpour, Aghabozorgi and Davvaz in [84] introduced the concept of nilpotent polygroups. In this section, we study the notions of nilpotent poly-

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groups by using the notion of hearts of polygroups. In particular, we obtain a necessary and sufficient condition between nilpotent (solvable) polygroups and fundamental groups. Let (G, ·) be a group and PG = G ∪ {a}, where a ∈ / G. We define on PG the hyperoperations ◦ as follows: (1) (2) (3) (4) (5)

a ◦ a = e; e ◦ x = x ◦ e = x, for every x ∈ PG ; a ◦ x = x ◦ a = x, for every x ∈ PG \ {e, a}; x ◦ y = x · y, for every (x, y) ∈ G2 such that y 6= x−1 ; x ◦ x−1 = {e, a}, for every x ∈ PG \ {e, a}.

Proposition 3.11.1. If G is a group, then hPG , ◦, e,−1 i is a polygroup. Proof. First of all, we prove the associativity of ◦. Suppose that (x, y, z) ∈ PG3 . (i) If {x, y, z} ∩ {e, a} = ∅, then we have two following cases. Case1. x 6= y −1 6= z and x 6= z −1 . In this case (x ◦ y) ◦ z = (x · y) · z = x · (y · z) = x ◦ (y ◦ z). Case2. There exists {u, v} ⊆ {x, y, z} such that u = v −1 . Without losing generality suppose that x = u, y = v. Thus, (x ◦ y) ◦ z = {e, a} ◦ z. Hence, {e, a} ◦ z = z. On the other hand, if y = z −1 , then x ◦ (y ◦ z) = x ◦ {e, a} = x = y −1 = z and if y 6= z −1 we have x ◦ (y ◦ z) = x ◦ (y · z) = x · (y · z) = (x · x−1 ) · z = e · z = z. (ii) If {x, y, z} ∩ {e, a} 6= ∅. Let e ∈ {x, y, z}. It is easy to see that the associativity condition holds. Now suppose that {x, y, z} ∩ {e, a} = {a}. Without losing generality let x = a, in this case we have  a if y = a, z = a     if y = a, z 6= a z (x ◦ y) ◦ z = x ◦ (y ◦ z) = y if y 6= a, z = a    y · z if y 6= z −1 , y 6= a 6= z   {e, a} if y = z −1 , y 6= a 6= z. According to the structure of ◦ we conclude that e is the identity element of PG and the other conditions for being polygroup hold too.  Proposition 3.11.2. If G is a group, then PG /β ∗ ∼ = G. Proof. It is straightforward.  Definition 3.11.3.

A polygroup hP, ·, e,−1 i is said to be nilpotent if

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`n (P ) ⊆ ωP or equivalently `n (P ) · ωP = ωP , for some integer n, where `0 (P ) = P and `k+1 (P ) = h{h ∈ P | x · y ∩ h · y · x 6= ∅, such that x ∈ `k (P ) and y ∈ P }i. The smallest integer c such that `c (P ) · ωP = ωP is called the nilpotency class or for simplicity the class of P . Notice that P = `0 (P ) ⊇ `1 (P ) ⊇ `2 (P ) ⊇ . . . that is {`k (P )}k≥o is a decreasing sequence which we call it generalized descending central series. Proposition 3.11.4. Every commutative polygroup is nilpotent of class 1. Proof. Suppose that hP, ·, e,−1 i is a commutative polygroup and h ∈ `1 (P ). Then, there exists (x, y) ∈ P 2 such that x · y ∩ h · y · x 6= ∅. Since P is ¯ xy¯ and so h ¯ = e which commutative we have x · y ∩ h · x · y 6= ∅. So, x ¯y¯ = h¯ means that h ∈ ωP . Therefore, `1 (P ) · ωP = ωP  A polygroup is called proper if it not a group. Proposition 3.11.5. Every proper polygroup of order less than 7 is nilpotent of class 1. Proof. Suppose that hP, ·, e,−1 i is a polygroup of order less than 7. Then, P/β ∗ is an abelian group of order less that 6. Now, let h ∈ `1 (P ). Then, ¯ yx ¯ xy¯ there exists (x, y) ∈ P 2 such that x · y ∩ h · x · y 6= ∅. Thus, x ¯y¯ = h¯ ¯ = h¯ which implies that h ∈ ωP . Therefore, `1 (P ) ⊆ ωP and consequently `1 (P ) · ωP = ωP .  Corollary 3.11.6. The symmetric group S3 is the smallest non-nilpotent polygroup. Example 3.11.7. Let P = {e, a, b, c, d, f, g}. We consider the proper noncommutative polygroup hP, ·, e,−1 i, where · is defined on P as follows: · e a b c d f g

e e a b c d f g

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

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It is easy to see that ωP = {e, a} while `n (P ) = {e, a, f, g} and hence `n (P ) · ωP 6= ωP for all n ∈ N. Thus, P is not a nilpotent polygroup of order 7. Proposition 3.11.8. Let hP, ·, e,−1 i be a polygroup and G = for all k ≥ 1 `k (G) = ht¯ | t ∈ `k (P )i.

P β∗ .

Then,

Proof. Suppose that hP, ·, e,−1 i is a polygroup and G = βP∗ . Then, we do the proof by induction on k. For k = 0, we have ht¯ | t ∈ `0 (P ) = P i = `0 (G). Now, suppose that a ¯ ∈ ht¯ | t ∈ `k+1 (P )i. Then, a ∈ `k+1 (P ) and so there exist x ∈ `k (P ) and y ∈ P such that xy ∩ ayx 6= ∅. Thus, x ¯y¯ = a ¯y¯x ¯. By hypothesis of induction we conclude that a ¯ ∈ `k+1 (G). Conversely, let a ¯ ∈ `k+1 (G). Without losing generality suppose that a ¯=x ¯y¯x ¯−1 y¯−1 , where x ¯ ∈ `k (G), y¯ ∈ G, which implies that x ¯y¯ = a ¯y¯x ¯. Thus, there exist c ∈ xy ¯ Since P is a polygroup there exists u ∈ P such and d ∈ ayx such that c¯ = d. that c ∈ xy ∩ uyx. From x ¯ ∈ `k (G), y¯ ∈ G and the hypothesis of induction we have x ∈ `k (P ), y ∈ P . Thus, u ∈ `k+1 (P ) and a ¯y¯x ¯ = d¯ = c¯ = x ¯y¯ = u ¯y¯x ¯. ¯ Therefore, a ¯=u ¯ ∈ ht | t ∈ `k+1 (P )i.  Theorem 3.11.9. Let hP, ·, e,−1 i be a polygroup. Then, P is nilpotent if and only if G = βP∗ is nilpotent. Proof. Suppose that P is a nilpotent polygroup so there exists k ∈ N such that `k (P ) ⊆ ωp . According to the previous proposition, we have `k (G) = ht¯ | t ∈ `k (P ) ⊆ ωp i = {eG } = ωG , and so G = βP∗ is a nilpotent group. Similarly, we can see the converse.  Corollary 3.11.10. Let G be a group. Then, PG is nilpotent if and only if G is nilpotent. Theorem 3.11.11. Let hP, ·, e,−1 i be a polygroup and N be a normal P ) = `n (PN)·N , for all n ≥ 0. subpolygroup of P . Then, `n ( N P Proof. By induction on n we show that `n ( N ) ⊆ `n (PN)·N and `n (PN)·N ⊆ P `n ( N ). For n = 0, the inclusions are obvious. Now, suppose that yN ∈ P P P `n+1 ( N ). Then, yN ∈ [aN, bN ], where aN ∈ `n ( N ) and bN ∈ N . By the 0 0 hypothesis of induction we have aN = a N , where a ∈ `n (P ). Therefore, (P )·N (P )·N yN = y 0 N , where y 0 ∈ [a0 , b]. Thus, yN ∈ `n+1N . If yN ∈ `n+1N , 00 00 then yN = y N, where y ∈ `n+1 (P ). So, there exist a ∈ `n (P ) and b ∈ P P such that y 00 ∈ [a, b]. Hence, aN = yN ∈ [aN, bN ], where aN ∈ `n ( N ) P which means that yN ∈ `n+1 ( N ) and our proof is completed. 

Corollary 3.11.12. If hP, ·, e,−1 i is a nilpotent polygroup, then

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P N

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Definition 3.11.13. Let hP, ·, e,−1 i be a polygroup. We define Z0 (P ) = ωP and Zn (P ) = h{x|x · y · Zn−1 (P ) = y · x · Zn−1 (P ), ∀y ∈ P }i, for all n ∈ N. Notice that ωP = Z0 (P ) ⊆ Z1 (P ) ⊆ Z2 (P ) ⊆ . . . that is {Zm (P )}m≥0 is an increasing sequence which we call it generalized ascending central series. Moreover, Zn (P ) is a subpolygroup of P , for every n ≥ 0. Proposition 3.11.14. If hP, ·, e,−1 i is a polygroup and n ≥ 0, then (1) Zn (P ) is a complete subpolygroup of P ; (2) g · g −1 ⊆ Zn (P ), for every g ∈ P ; (3) Zn (P ) is a normal subpolygroup of P. Proof. (1) Since ωP ⊆ Zn (P ), we conclude that C(Zn (P )) = Zn (P ) · ωP = Zn (P ), which means that Zn (P ) is complete. (2) Let g ∈ P . Since e ∈ g · g −1 ∩ Zn (P ) and Zn (P ) is complete, g · g −1 ⊆ Zn (P ). (3) Let g ∈ P be an arbitrary element and x ∈ Zn (P ). Then, g · x · g −1 · Zn−1 (P ) = g · g −1 · x · Zn−1 (P ) ⊆ g · g −1 · Zn (P ) = Zn (P ). Hence, g · x · g −1 ⊆ Zn (P ).  Theorem 3.11.15. Let hP, ·, e,−1 i be a polygroup. Then, P is nilpotent if and only if there exists r ≥ 0 such that Zr (P ) = P . Proof. Suppose that there exists r ≥ 0 such that Zr (P ) = P . In order to prove `r (P ) ⊆ ωP , by induction we show that `i (P ) ⊆ Zr−i (P ). For i = 0 we have `0 (P ) = P ⊆ P = Zr (P ). Now, if a ∈ `i+1 (P ), then without loss generality suppose that x · y ∩ a · y · x 6= ∅, where x ∈ `i (P ) and y ∈ P . By the hypothesis of induction we conclude that x ∈ Zr−i (P ). Hence, x · y · Zr−i−1 (P ) = y · x · Zr−i−1 (P ) and so a ∈ Zr−i−1 (P ). Now if i = r, then `r (P ) ⊆ Z0 (P ) = ωP . For the converse, suppose that `r (P ) ⊆ ωP . It is enough to show that `r−i (P ) ⊆ Zi (P ), for all 0 ≤ i ≤ n. For i = 0, we have `r (P ) ⊆ ωP = Z0 (P ). Let a ∈ `r−i−1 (P ) and b ∈ P . Then, [a, b] ⊆ `r−i (P ). By using the hypothesis of induction, we have [a, b] ⊆ Zi (P ). Therefore, a · b · Zi (P ) = b · a · Zi (P ) and so a ∈ Zi+1 (P ) as we need. If we take i = r, then we get our claim.  Corollary 3.11.16. Let hP, ·, e,−1 i be a polygroup. Then, `c (P ) ⊆ ωP if and only if Zc (P ) = P, that is P is nilpotent of class c if and only if

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Zc (P ) = P. Proposition 3.11.17. Let P1 and P2 be two polygroups. Then, for all k>0 `k (P1 × P2 ) = `k (P1 ) × `k (P2 ). Proof. We prove our claim by induction on k. For k = 0, it is obvious. Now, suppose that (a, b) ∈ `k+1 (P1 × P2 ). Then, there exist (u, v) ∈ `k (P1 × P2 ) and (s, t) ∈ P1 × P2 such that (u, v) ∗ (s, t) ∩ (a, b) ∗ (s, t) ∗ (u, v) 6= ∅ that is u · s ∩ a · s · u 6= ∅ and v ◦ t ∩ b ◦ t ◦ v 6= ∅. By using the hypothesis of induction, we conclude that a ∈ `k+1 (P1 ) and b ∈ `k+1 (P2 ). Thus, (a, b) ∈ `k+1 (P1 ) × `k+1 (P2 ). Similarly, we obtain the converse.  Proposition 3.11.18. Let P1 , P2 be two polygroups. Then, P1 × P2 is nilpotent if and only if P1 and P2 are nilpotent. Proof. If P1 , P2 are nilpotent, then there exist k1 and k2 such that `k1 (P1 ) ⊆ ωP1 and `k2 (P2 ) ⊆ ωP2 . Suppose that k = lcm(k1 , k2 ). Hence, `k (P1 ) ⊆ `k1 (P1 ) and `k (P2 ) ⊆ `k2 (P2 ) and so we obtain `k (P1 × P2 ) = `k (P1 ) × `k (P2 ) ⊆ `k1 (P1 ) × `k2 (P2 ) ⊆ ωP1 × ωP2 = ωP1 ×P2 . Conversely, suppose that P1 × P2 is nilpotent. Then, there exists k such that `k (P1 ) × `k (P2 ) = `k (P1 × P2 ) ⊆ ωP1 ×P2 = ωP1 × ωP2 . Hence, `k (P1 ) ⊆ ωP1 and `k (P2 ) ⊆ ωP2 . Therefore, P1 , P2 are nilpotent.  Example 3.11.19. Let P1 = {e, a, b, c} be the polygroup in Example 3.10.9 and P2 = {0, 1} be the cyclic group of order two. Consider the noncommutative polygroup P ∼ = P1 × P2 , where · is defined on P as follows: · e a b c d f g h

e e a b c d f g h

a b a b e c c b b c f {e, b, d} d {a, c, f } h {b, g} g {c, h}

c d f g h c d f g h b f d h g c {e, b, d, g} {a, c, f, h} g h b {a, c, f, h} {e, b, d, g} h g a, c, f d f {d, g} {f, h} {e, b, d} f d {f, h} {d, g} {c, h} g h {e, b, d, g} {a, c, f, h} {b, g} h g {a, c, f, h} {e, b, d, g}

It is easy to see that `1 (P1 ) = P10 = ωP1 and `1 (P2 ) = ωP2 = {0}. Hence, P is a nilpotent polygroup of class 1. Proposition 3.11.20. Let hP1 , ·, e1 ,−1 i and hP2 , ◦, e2 ,−I i be two poly/ P2 be a good homomorphism. If φ is one to one groups, and φ : P1

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and K1 is a nilpotent subpolygroup of P1 , then φ(K1 ) is a nilpotent subpolygroup of P2 . Proof. (1) By induction on n we show that `n (φ(K1 )) = φ(`n (K1 )). For n = 0 it is obvious. Let z ∈ `n+1 (φ(K1 )). Then, there exist x ∈ `n (φ(K1 )) and y ∈ φ(K1 ) such that x ◦ y ∩ z ◦ y ◦ x 6= ∅. Since {x, y, z} ⊆ φ(K1 ), there exist a ∈ `n (φ(K1 )), b ∈ K1 , c ∈ K1 such that φ(a) = x, φ(b) = y, φ(c) = z. Therefore, we have x ◦ y ∩ z ◦ y ◦ x = φ(a) ◦ φ(b) ∩ φ(c) ◦ φ(b) ◦ φ(a) = φ(a · b) ∩ φ(c · b · a) 6= ∅. Since φ is one to one, we conclude that φ(a·b∩c·b·a) 6= ∅, so a·b∩c·b·a 6= ∅. By the hypothesis of induction we have c ∈ `n+1 (K1 ). Thus, z = φ(c) ∈ φ(`n+1 (K1 ). Conversely, let z ∈ φ(`n+1 (K1 )). Then, there exists c ∈ `n+1 (K1 ) such that φ(c) = z. So, from c ∈ `n+1 (K1 ) we conclude that there exist a ∈ `n (K1 ) and b ∈ K1 such that a · b ∩ c · b · a 6= ∅. Thus, φ(a) ◦ φ(b) ∩ φ(c) ◦ φ(b) ◦ φ(a) 6= ∅. By the hypothesis of induction we have φ(a) ∈ φ(`n (K1 ) = `n (φ(K1 )) and φ(b) ∈ φ(K1 ). So, z = φ(c) ∈ `n+1 (φ(K1 )). Now, let K1 be a nilpotent subpolygroup of P1 . Then, there exists m such that `m (K1 ) ⊆ ωK1 . By the previous theorem, we have `m (φ(K1 )) = φ(`m (K1 ) ⊆ φ(ωK1 ) ⊆ ωφ(K1 ) , and the proof is completed.  Proposition 3.11.21. Every commutative polygroup is solvable of length 1. Proof. It is straightforward.  Proposition 3.11.22. Let P be a polygroup and G = k≥1

P β∗ .

Then, for all

ık (G) = ht¯ | t ∈ ık (P )i. Proof. Suppose that P is a polygroup and G = βP∗ . Then, we do the proof by induction on k. For k = 0, we have ht¯ | t ∈ ı0 (P ) = P i = ı0 (G). Now, suppose that a ¯ ∈ ht¯ | t ∈ ık+1 (P )i. Then, a ∈ ık+1 (P ) and so there exist x, y ∈ ık (P ) such that xy ∩ ayx 6= ∅. Thus, x ¯y¯ = a ¯y¯x ¯. By the hypothesis of induction we conclude that a ¯ ∈ ık+1 (G). Conversely, let a ¯ ∈ ık+1 (G). With−1 −1 out losing generality suppose that a ¯=x ¯y¯x ¯ y¯ , where x ¯, y¯ ∈ ık (G), which implies that x ¯y¯ = a ¯y¯x ¯. Thus, there exist c ∈ xy and d ∈ ayx such that ¯ Since P is a polygroup, there exists u ∈ P such that c ∈ xy ∩ uyx. c¯ = d. By the hypothesis of induction we have x, y ∈ ık (P ) which implies that u ∈ ık+1 (P ) and a ¯y¯x ¯ = d¯ = c¯ = x ¯y¯ = u ¯y¯x ¯, so a ¯=u ¯ ∈ ht¯ | t ∈ ık+1 (P )i. 

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Theorem 3.11.23. Let P be a polygroup. Then, P is solvable if and only if G = βP∗ is solvable. Proof. Suppose that P is a solvable polygroup. Then, there exists k ∈ N such that ık (P ) ⊆ ωp . According to the previous proposition, we have ık (G) = ht¯ | t ∈ ık (P )i = {eG } = ωG , and so G = βP∗ is a nilpotent group. Similarly, we can see the converse.  Corollary 3.11.24. Every nilpotent polygroup is solvable. Proposition 3.11.25. Every proper polygroup of order less than 61 is solvable. Proof. Suppose that hP, ·, e,−1 i is a proper polygroup of order less than 61. Then, P/β ∗ is a group of order less that 60. Thus, P/β ∗ is not isomorphic to the smallest non-solvable group A5 (alternating group of degree 5). Hence, P is solvable.  In the following example, we introduce one of the smallest proper polygroups of order 61. Example Let A5 be the alternating group of degree 5 and P = A5 ∪ {a}, where a ∈ / A5 . We define on P the hyperoperations ◦, as follows: (1) (2) (3) (4) (5)

a ◦ a = {e, a}; e ◦ x = x ◦ e = x, for every x ∈ P ; a ◦ x = x ◦ a = x, for every x ∈ P \ {e, a}; x ◦ y = x · y, for every x, y ∈ A5 such that y 6= x−1 ; x ◦ x−1 = {e, a}, for every x ∈ P \ {e, a}.

It is easy to see that (P, ◦) is a polygroup. Moreover, P/β ∗ ∼ = A5 and hence P is not solvable.

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4.1

Weak hyperstructures

Weak hyperstructures or Hv -structures were introduced by Vougiouklis at the Fourth AHA congress (1990)[149]. 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 then the study of Hv -structure theory has been pursued in many ˇ directions by T. Vougiouklis, B. Davvaz, S. Spartalis, A. Dramalidis, S. Hoˇskov´ a, and others. In this section, firstly, we present some definitions and theorems on weak hyperstructures [143; 146]. Definition 4.1.1. 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. In an obvious way, the Hv -subgroup of an Hv -group is defined. All the weak properties for hyperstructures can be applied for subsets. 139

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For example, if (H, ·) is a weak commutative Hv -group, then for all nonempty subsets A, B, 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. Definition 4.1.2. Let (H1 , ·), (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 . Finally, f is called a strong homomorphism or a good 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 4.1.3. Let (H, ·), (H, ∗) be two Hv -groups defined on the same set H. We say that “ · ” less than or equal to “ ∗ ” 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 [144], the set of all Hv -groups with a scalar unit defined on a set with three elements is determined using this property. Theorem 4.1.4. Greater hyperoperation from the one of a given Hv -group defines an Hv -group. The weak commutativity is also valid for every greater hyperoperation. Proof. Obvious from the definition.  We remark that the above theorem is not true for hypergroups. 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. If U denotes the set of all finite

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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. 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)| c ∈ β ∗ (a) · β ∗ (b)} for all a, b ∈ H. It is not difficult to see that β ∗ (a) β ∗ (b) is the singleton {β ∗ (c)} for all c ∈ β ∗ (a) · β ∗ (b). In this way H/β ∗ becomes a group. A motivation to study the above structures is given by the following examples: Example 4.1.5. (1) Let (G, ·) be a group and R be an equivalence relation on G. In G/R consider the hyperoperation defined by x y = {z| z ∈ x · y}, where x denotes the equivalence class of the element x. Then, (G, ) is an Hv -group which is not always a hypergroup. (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. 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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(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, m2 , . . . , 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 and we have Zn /β ∗ ∼ = Zm1 × Zm2 × . . . × Zmn . 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 4.1.6. 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 4.1.7. 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 4.1.8. 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

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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. Corollary 4.1.9. 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.  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 [136]. 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, S il (x) = il (x, e) is the set of all left inverses of the element x. If A is e∈E S a non-empty subset of H, then il (A) = il (a). Similar notations hold a∈A

for the right inverses, too. The following definition is similar to Definition 2.6.2. Definition 4.1.10. 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).

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Similarly, H is called right completely reversible in itself if (2) c ∈ a · b implies a ∈ c · v, for all v ∈ ir (b). Lemma 4.1.11. 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 4.1.12. 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 4.1.11 (3), we have il (a) ⊆ il (il (il (a, e))) ⊆ il (a, e) ⊆ il (a). 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

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is satisfied: il (a, e) = il (a) = il (il (il (a, e))) = il (il (a)). Proposition 4.1.13. Let H be left completely reversible and a ∈ H such T that 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 4.1.12 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 4.1.14. 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 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 4.1.13 (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 [66] and studied by Spartalis [136].

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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 4.1.13 (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 4.1.13 (1), this hyperoperation is equivalent to the following one x ˆ yˆ = {w ˆ | w ∈ x · y}. ˆ ) is an Hv -group. Lemma 4.1.15. (H, Proof. It is straightforward.  Proposition 4.1.16. Let H be left completely reversible and e ∈ E. Then, ˆ is a left reversible in itself Hv -group and eˆ = E is the unique identity of H ˆ Moreover, eˆ is a right scalar, H ˆ is regular and each x ˆ has a unique H. ˆ∈H inverse. Proof. Suppose that a ∈ E. By Lemma 4.1.11 (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 ˆ identity or t is a right identity of H. Then, eˆ ∈ sˆ eˆ∩ eˆ tˆ and so there exist 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}. 0

0

If w ∈ x · e and e ∈ 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 It is easy to prove that each x ˆ ∈ H has a unique inverse, i.e., i(ˆ 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, there exists y 0 ∈ yˆ such that z ∈ x · y 0 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:

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(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 4.1.17. Let H be left completely reversible and K be an Hv subgroup of K. 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 4.1.18. 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 condition hold: (1) E is a total Hv -subgroup of H; (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 4.1.13 (2) and Lemma 4.1.11 (2), for all a ∈ H, e ∈ E and u ∈ i(a), we obtain i(a) = u · a = i(a, e).

(?)

(1) Suppose that e ∈ E. Then, from the previous relation, we obtain i(e) = u · e0 , for all u ∈ i(e) and e0 ∈ E.

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Moreover, according to Lemma 4.1.11 (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 (?) we obtain x · E = i(u) = x · e and hence H/E = {x · e | x ∈ H}. Furtheremore, 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.  4.2

Weak polygroups as a generalization of polygroups

In this section, we study the concept of weak polygroups which is a generalization of polygroups [46; 50]. Definition 4.2.1. A multivalued system M =< P, ·, e,−1 >, where e ∈ P, −1 : P −→ P, · : P × P −→ P ∗ (P ) is called a weak polygroup if the following axioms hold for all x, y, z ∈ P : (1) (x · y) · z ∩ x · (y · z) 6= ∅ (weak associative), (2) x · e = x = e · x, (3) x ∈ y · z implies y ∈ x · z −1 and z ∈ y −1 · x. The following elementary facts about weak polygroups follow easily from the axioms: e ∈ x · x−1 ∩ x−1 · x, e−1 = e, (x−1 )−1 = x. Example 4.2.2. Consider P = {e, a, b, c} and define ∗ on P with the help of the following table: ∗ e a b c

e e a b c

a a {e, a} c b

b b c {e, b} a

c c b a {e, c}

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Then, < P, ∗, e, −1 >, where x−1 = x for every x ∈ P , is a weak polygroup which is not a polygroup. Indeed, we have (a ∗ b) ∗ c = c ∗ c = {e, c} and a ∗ (b ∗ c) = a ∗ a = {e, a}. Therefore, ∗ is not associative. Proposition 4.2.3. Let (G, ·) be a group and θ be an equivalence relation on G such that (1) xθe implies x = e, (2) xθy implies x−1 θy −1 . Let θ(x) be the equivalence class of the element x ∈ G. If G/θ = {θ(x) | x ∈ G}, then < G/θ, , θ(e),−I > is a weak polygroup, where the hyperoperation is defined as follows: : G/θ × G/θ → P ∗ (G/θ) θ(x) θ(y) = {θ(z)|z ∈ θ(x).θ(y)}, and θ(x)−I = θ(x−1 ). Proof. For all x, y, z ∈ G, we have x · (y · z) ∈ θ(x) (θ(y) θ(z)), (x · y) · z ∈ (θ(x) θ(y)) θ(z). Therefore, is weak associative. It is easy to see that θ(e) is the identity element in G/θ and θ(x−1 ) is the inverse of θ(x) in G/θ. Now, we show that: θ(z) ∈ θ(x) θ(y) implies θ(x) ∈ θ(z) θ(y −1 ) and θ(y) ∈ θ(x−1 ) θ(z). We have θ(z) ∈ θ(x) θ(y) = {θ(a) | a ∈ θ(x)·θ(y)}. Hence, θ(z) = θ(a) for some a ∈ θ(x).θ(y). Therefore, there exist b ∈ θ(x) and c ∈ θ(y) such that a = b · c, so b = a · c−1 which implies that θ(b) = θ(a.c−1 ) ∈ θ(a) θ(c−1 ). Therefore, θ(x) ∈ θ(z) θ(y −1 ). By a similar way, we obtain θ(y) ∈ θ(x−1 ) θ(z). Therefore, < G/θ, , θ(e),−I > is a weak polygroup.  An extension of polygroups by polygroups introduced in Section 3.2. We

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can consider an extension of a weak polygroup by another weak polygroup in a similar way. Theorem 4.2.4. Let A =< A, ·, e,−1 > and B =< B, ·, e,−1 > be weak polygroups whose elements have been renamed so that A ∩ B = {e}, where e is the identity of both A and B. A new system A[B] =< M, ∗, e,−I >, which is called the extension of A by B is a weak polygroup. Proof. The verification of the third condition of Definition 4.2.1 is similar to the proof of Theorem 3.2.4. Therefore, we show that weak associativity is valid. For all x, y, z in M , we consider the following cases: (1) (2) (3) (4) (5) (6) (7) (8)

If If If If If If If If

x, y, z ∈ A, then (x · y) · z = (x ∗ y) ∗ z and x · (y · z) = x ∗ (y ∗ z), x, y, z ∈ B, then (x · y) · z ⊆ (x ∗ y) ∗ z and x · (y · z) ⊆ x ∗ (y ∗ z), x ∈ A, y, z ∈ B, then (y · z) ⊆ (x ∗ y) ∗ z and y · z ⊆ x ∗ (y ∗ z), x ∈ A, y ∈ B, z ∈ A, then y ∈ (x ∗ y) ∗ z and y ∈ x ∗ (y ∗ z), x ∈ A, y ∈ A, z ∈ B, then z ∈ (x ∗ y) ∗ z and z ∈ x ∗ (y ∗ z), x ∈ B, y, z ∈ A, then x ∈ (x ∗ y) ∗ z and x ∈ x ∗ (y ∗ z), x ∈ B, y ∈ B, z ∈ A, then x · y ⊆ (x ∗ y) ∗ z and x · y ⊆ x ∗ (y ∗ z), x ∈ B, y ∈ A, z ∈ B, then x · z ⊆ (x ∗ y) ∗ z and x · z ⊆ x ∗ (y ∗ z).

Thus, ∗ is weak associative.  The following definition, first defined in Section 3.3 for polygroups. The equivalence relation θ on a weak polygroup M is called a (full) conjugation on M if (1) xθy implies x−1 θy −1 , (2) z ∈ x · y and z1 θz imply z1 ∈ x1 · y1 for some x1 and y1 where θ(x1 ) = θ(x) and θ(y1 ) = θ(y). The collection of all θ-classes, with the induced operation from M, forms a weak polygroup. Corollary 4.2.5. Let M be a weak polygroup, then θ is a conjugation on M if and only if (1) (θ(x))−1 = θ(x−1 ); (2) θ(θ(x)y) = θ(x)θ(y). Proof. The proof is similar to the proof of Lemma 3.3.11.  Definition 4.2.6. Let A =< A, ., e1 ,−1 > and B =< B, ∗, e2 ,−1 > be two weak polygroups, and let f be a mapping from A into B, such that f (e1 ) = e2 . Then, f is called

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(1) a weak homomorphism, if f (x · y) ∩ f (x) ∗ f (y) 6= ∅, for all x, y ∈ A, (2) an inclusion homomorphism, if f (x·y) ⊆ f (x)∗f (y), for all x, y ∈ A, (3) a strong homomorphism, if f (x · y) = f (x) ∗ f (y), for all x, y ∈ A. If f is one to one, onto and strong homomorphism, then it is called an isomorphism. Moreover, if f is defined on the same weak polygroup, then it is called an automorphism. The set of all automorphism of A, written AutA, is a group. Definition 4.2.7. If A is a subpolygroup of a weak polygroup P , then we define the relation a ≡ b (modA) if and only if there exists a set {c0 , c1 , . . . , ck+1 } ⊆ P , where c0 = a, ck+1 = b such that −1 −1 a · c−1 ∩ A 6= ∅. 1 ∩ A 6= ∅, c1 · c2 ∩ A 6= ∅, . . . , ck · b

This relation is denoted by aA∗P b. Lemma 4.2.8. The relation A∗P is an equivalence relation. Proof. (1) Since e ∈ a·a−1 ∩A for all a ∈ P ; then aA∗P a, i.e., A∗P is reflexive. (2) Suppose that aA∗P b. Then, there exists {c0 , c1 , . . . , ck+1 } ⊆ P where c0 = a, ck+1 = b such that −1 −1 a · c−1 ∩ A 6= ∅. 1 ∩ A 6= ∅, c1 · c2 ∩ A 6= ∅, . . . , ck · b

Therefore, there exists xi ∈ ci · c−1 (i = 0, . . . , k) which implies i+1 ∩ A −1 −1 that x−1 ∈ c · c and x ∈ A, this means that b A∗P a, and so A∗P is i+1 i i i symmetric. (3) Let a A∗P b and b A∗P c, where a, b, c ∈ P . Then, there exist {c0 , c1 , . . . , ck+1 } ⊆ P and {d0 , d1 , . . . , dr+1 } ⊆ P, where c0 = a, ck+1 = b = d0 , dr+1 = c such that −1 −1 a · c−1 ∩ A 6= ∅, 1 ∩ A 6= ∅, c1 · c2 ∩ A 6= ∅, . . . , ck · b −1 −1 b · d−1 ∩ A 6= ∅. 1 ∩ A 6= ∅, d1 · d2 ∩ A 6= ∅, . . . , dr · c

We take {c0 , c1 , . . . , ck+1 , d1 , d2 , . . . , dr+1 } ⊆ P which satisfies the condition for a A∗P c, and so A∗P is transitive. Therefore, A∗P is an equivalence relation.  We denote A∗P [x] the equivalence class with representative x. Theorem 4.2.9. Let P be a weak polygroup. If A is a subpolygroup of P , then on the set [P : A] = {A∗P [a] | a ∈ P } we define the hyperoperation as follows: A∗P [a] A∗P [b] = {A∗P [c] | c ∈ A∗P [a] · A∗P [b]},

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what gives the weak polygroup < [P : A], , A∗P [e],−I >, where A∗P [a]−I = A∗P [a−1 ]. Proof. For a, b, c ∈ P , we have (a · b) · c ⊆ (A∗P [a] · A∗P [b]) · A∗P [c], a · (b · c) ⊆ A∗P [a] · (A∗P [b] · A∗P [c]). Thus, is weak associative. Now, we show that A is the unit element in [P : A]. Obviously, we have A ⊆ A∗P [e]. On the other hand, if a ∈ A∗P [e], then there exists a set {c0 , c1 , . . . , cn+1 } ⊆ P where c0 = a, cn+1 = e such that −1 −1 a · c−1 ∩ A 6= ∅. 1 ∩ A 6= ∅, c1 · c2 ∩ A 6= ∅, . . . , cn · e −1 So cn ∈ A. Since cn−1 · c−1 n ∩ A 6= ∅, there exists x ∈ cn−1 · cn ∩ A which implies cn−1 ∈ x · cn , and so cn−1 ∈ A. By induction, we obtain a ∈ A. Therefore, A∗P [e] = A. Now, we show that A∗P [a] A∗P [e] = A∗P [a]. Suppose that A∗P [z] ∈ A∗P [a] A∗P [e]. We claim that A∗P [z] = A∗P [a]. We have z ∈ A∗P [a] · A∗P [e]. Hence, there exist x ∈ A∗P [a] and y ∈ A such that z ∈ x · y which implies that y ∈ x−1 · z. Then, x−1 · z ∩ A 6= ∅, and so A∗P [x] = A∗P [z]. Therefore, A∗P [z] = A∗P [a]. It is easy to see that A∗P [a−1 ] is the inverse of A∗P [a] in [P : A]. Now, we show that A∗P [c] ∈ A∗P [a] A∗P [b] implies A∗P [a] ∈ A∗P [c] A∗P [b−1 ] and A∗P [b] ∈ A∗P [a−1 ] A∗P [c]. Since A∗P [c] ∈ A∗P [a] A∗P [b], we have A∗P [c] = A∗P [x] for some x ∈ A∗P [a] · A∗P [b]. Therefore, there exist y ∈ A∗P [a] and z ∈ A∗P [b] such that x ∈ y · z, so y ∈ x · z −1 . This implies that A∗P [y] ∈ A∗P [x] A∗P [z −1 ], and so A∗P [a] ∈ A∗P [c] A∗P [b−1 ]. Similarly, we get A∗P [b] ∈ A∗P [a−1 ] A∗P [c]. Therefore, < [P : A], , A,−I > is a weak polygroup. 

If A is a subpolygroup of a weak polygroup P , then the weak polygroup [P : A], is called the quotient weak polygroup of P by A. Corollary 4.2.10. Let ρ be a strong homomorphism from a weak polygroup P1 into a weak polygroup P2 . Then, the following propositions hold: (1) For all a ∈ P1 , ρ(a−1 ) = ρ(a)−1 ; (2) The kernel of ρ is a subpolygroup of P1 ; (3) Let A be a subpolygroup of P1 . The image ρ(A) = {ρ(x) | x ∈ A} is a subpolygroup of P2 . For a subpolygroup B of P2 , the inverse image ρ−1 (B) = {x | x ∈ P1 , ρ(x) ∈ B} is a subpolygroup of P1 . Let P1 , P2 be two weak polygroups and ρ a strong homomorphism of P1 onto P2 . If K is the kernel of ρ then we can form [P1 : K]. It is fairly

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natural to expect that there should be a very close relationship between P2 and [P1 : K]. The fundamental homomorphism theorem, which we are about to prove, spells out this relationship in exact detail. Theorem 4.2.11. (Fundamental Homomorphism Theorem). Let ρ be a strong homomorphism from P1 onto P2 with kernel K. Then, [P1 : K] ∼ = P2 . Proof. We define ϕ : [P1 : K] −→ P2 as follows: ϕ(KP∗1 [x]) = ρ(x), for all x ∈ P1 . This mapping is well defined, because if KP∗1 [x] = KP∗1 [y], then there exists {z0 , z1 , . . . , zk+1 } ⊆ P1 where z0 = x, zk+1 = y such that x · z1−1 ∩ K 6= ∅, z1 · z2−1 ∩ K 6= ∅, . . . , zk · y −1 ∩ K 6= ∅. Thus, e2 ∈ ρ(x · z1−1 ), e2 ∈ ρ(z1 · z2−1 ), . . . , e2 ∈ ρ(zk · y −1 ) or e2 ∈ ρ(x) ∗ ρ(z1 )−1 , e2 ∈ ρ(z1 ) ∗ ρ(z2 )−1 , . . . , e2 ∈ ρ(zk ) ∗ ρ(y)−1 and so ρ(x) = ρ(y). Now, for every KP∗1 [x], KP∗1 [y] ∈ [P1 : K], we have ϕ(KP∗1 [x] KP∗1 [y]) = ϕ({KP∗1 [z] | z ∈ KP∗1 [x] · KP∗1 [y]} = {ρ(z) | z ∈ KP∗1 [x] · KP∗1 [y]} = ρ(KP∗1 [x] · KP∗1 [y]) = ρ(KP∗1 [x]) ∗ ρ(KP∗1 [y]) = ρ(x) ∗ ρ(y) = ϕ(KP∗1 [x]) ∗ ϕ(KP∗1 [y]), ∗ and ϕ(K) = ϕ(KP1 [e1 ]) = ρ(e1 ) = e2 . Therefore, ϕ is a strong homomorphism. Furthermore, if ϕ(KP∗1 [x]) = ϕ(KP∗1 [y]), then ρ(x) = ρ(y) which implies that x · y −1 ∩ K 6= ∅, and so KP∗1 [x] = KP∗1 [y]. Thus, ϕ is a one to one mapping. 

4.3

Fundamental relations on weak polygroups

In this section, we consider the fundamental relation β ∗ defined on weak polygroups and present some results in this respect. Moreover, we define a semi-direct hyperproduct of two weak polygroups in order to obtain an extension of weak polygroups by weak polygroups. Let M =< P, ·, e,−1 > be a weak polygroup. We can define the relation ∗ β as the smallest equivalence relation on P such that quotient P/β ∗ is a group. In this case β ∗ is called the fundamental equivalence relation on P and P/β ∗ is called the fundamental group. Let UP be the set of all finite products of elements of P and define the relation β on P as follows:

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xβy

if and only if {x, y} ⊆ u, for some u ∈ UP .

One can prove that the fundamental relation β ∗ is the transitive closure of the relation β. The kernel of the canonical map ϕ : P → P/β ∗ is denoted by ωP . It is easy to prove that the following statements: (1) ωP = β ∗ (e); (2) β ∗ (x)−1 = β ∗ (x−1 ), for all x ∈ P ; (3) β ∗ (β ∗ (x)y) = β ∗ (x) β ∗ (y), for all x, y ∈ P . An element x ∈ P will be called single if its equivalence class with respect to β ∗ is singleton, i.e., β ∗ (x) = {x}. We denote the set of all the single elements of P by SP . It is straightforward to prove that for a ∈ P and x ∈ SP , if x ∈ a · y for some y ∈ P , then β ∗ (a) = {z ∈ P | zy = x}. Let M1 =< P1 , ., e1 ,−1 > and M2 =< P2 , ∗, e2 ,−I > be two weak polygroups. Then, on P1 × P2 we can define a hyperproduct similar to the hyperproduct of polygroups as follows: (x1 , y1 ) ◦ (x2 , y2 ) = {(a, b) | a ∈ x1 · x2 , b ∈ y1 ∗ y2 }. We call this the direct product of P1 and P2 . It is easy to see that P1 × P2 equipped with the usual direct product operation becomes a weak polygroup. Lemma 4.3.1. We have UP1 ×P2 = UP1 × UP2 . Proof. It is straightforward.  Corollary 4.3.2. Let β1∗ , β2∗ and β ∗ be the fundamental equivalence relations on P1 , P2 and P1 × P2 respectively. Then, (a, b) βP∗ 1 ×P2 (c, d) ⇐⇒ aβP∗ 1 c and b βP∗ 2 d. Theorem 4.3.3. Let β1∗ , β2∗ and β ∗ be the fundamental equivalence relations on P1 , P2 and P1 × P2 respectively. Then, (P1 × P2 )/β ∗ ∼ = P1 /β1∗ × P2 /β2∗ . Proof. We consider the map f : P1 /β1∗ × P2 /β2∗ −→ (P1 × P2 )/β ∗ with f (β1∗ (x), β2∗ (y)) = β ∗ (x, y). It is easy to see that f is an isomorphism. 

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Similar to polygroups and using the fundamental equivalence relation, we can define semidirect hyperproduct of weak polygroups. Definition 4.3.4. Let A =< A, ·, e1 ,−1 > and B =< B, ∗, e2 ,−1 > be weak polygroups. We consider the group AutA and the fundamental group ∗ B/βB , let ∗ : B/βB → AutA ∗ [ β (b) → β (b) = bb c

∗

be a homomorphism of groups. Then, on A × B we define a hyperproduct as follows: (a1 , b1 ) ◦ (a2 , b2 ) = {(x, y) | x ∈ a1 · bb1 (a2 ), y ∈ b1 ∗ b2 } and we call this the semidirect hyperproduct of weak polygroups A and B. Theorem 4.3.5. A × B equipped with the semidirect hyperproduct is a weak polygroup. Proof. The proof is similar to the proof of Theorem 3.5.12.  Lemma 4.3.6. Let f : P1 −→ P2 be a strong homomorphism of weak polygroups and β1∗ , β2∗ fundamental equivalence relations on P1 , P2 respectively. Then, the map F : P1 /β1∗ −→ P2 /β2∗ defined by F (β1∗ (x)) = β2∗ (f (x)) is a homomorphism of fundamental groups. Proof. First, we show that F is well-defined. Suppose that β1∗ (x) = β1∗ (y). Then, there exist x1 , x2 , . . . , xn+1 ∈ P1 with x1 = x, xn+1 = y and u1 , . . . , un ∈ UP1 such that {xi , xi+1 } ⊆ ui (i = 1, . . . , n). Since f is a strong homomorphism and ui ∈ UP1 , we get f (ui ) ∈ UP2 . Therefore, f (x)β2∗ f (y) which implies that β2∗ (f (x)) = β2∗ (f (y)), and so F (β1∗ (x)) = F (β1∗ (y)). Thus, F is well-defined. Now, we have F (β1∗ (x) ⊗ β1∗ (y)) = F (β1∗ (x · y)) = β2∗ (f (x · y)) = β2∗ (f (x) · f (y)) = β2∗ (f (x)) ⊗ β2∗ (f (y)) = F (β1∗ (x)) ⊗ F (β1∗ (y)).  Definition 4.3.7. Let f be a strong homomorphism from P1 into P2 and let β1∗ , β2∗ be the fundamental relations on P1 , P2 respectively. Then, we define kerf = {β1∗ (x) | x ∈ P1 , β2∗ (f (x)) = ωP2 }. The next corollary summarize the results of Lemma 4.3.6. Corollary 4.3.8. (1) kerf is a normal subgroup of the fundamental group

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P1 /β1∗ . (2) If f is onto, then (P1 /β1∗ )/kerf ∼ = P2 /β2∗ . Let P be the set of all weak polygroups and all strong homomorphisms. One can show that P is a category. We set Pβ ∗ the category of fundamental groups and homomorphisms of groups. Then, we have the following theorem: Theorem 4.3.9. Let F be the function from P into Pβ ∗ defined by F(P ) = P/β ∗ and when f : P1 −→ P2 is a strong homomorphism F(f ) : P1 /β1∗ −→ P2 /β2∗ , β1∗ (x) −→ β2∗ (f (x)), where β1∗ , β2∗ are the fundamental relations on P1 , P2 respectively. Then, F is a functor. Proof. Clearly, F is well-defined. Now, we have the following: (1) If P1 ∈ objP, then P1 /β1∗ ∈ objPβ ∗ . (2) If f : P1 −→ P2 is a strong homomorphism, by Lemma 4.3.6, F(f ) is a homomorphism of groups. f g (3) Suppose β3∗ is the fundamental relation on P3 . If P1 −→ P2 −→ P3 is a sequence of strong homomorphisms in P, then F(gf )(β1∗ (x)) = β3∗ (gf (x)) = β3∗ (g(f (x))) = F(g)β2∗ (f (x)) = F(g)(F(f )(β1∗ (x))), so F(gf ) = F(g)F(f ). (4) For every P1 ∈ objP, we have F(1P1 )(β1∗ (x)) = β1∗ (1P1 (x)) = β1∗ (x). Thus, F(1P1 ) = 1F (P1 ) . Therefore, F is a functor.  4.4

Small weak polygroups

A weak polygroup is called commutative weak polygroup if the usual commutative axiom is valid, i.e., x · y = y · x, for all x, y ∈ P . Proposition 4.4.1. All weak polygroups of order 3 are commutative. Proof. Suppose that < P = {e, a, b}, ·, e,−1 > is a weak polygroup. It is enough to show that for every x, y ∈ P \{e} with x 6= y, we have x·y = y ·x. We consider the following cases:

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(1)

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−1

: P −→ P be the identity map. Suppose that t ∈ x · y. Then, t 6= e and so t = x or t = y. If t = x, then x ∈ x · y, which implies that y ∈ x−1 ·x = x·x. Hence, x ∈ y ·x−1 = y ·x, and so x·y ⊆ y ·x. Similarly, we have y · x ⊆ x · y. Therefore, x · y = y · x. (2) −1 : P −→ P be the following map: e −→ e,

a −→ b,

b −→ a.

Suppose that t ∈ x·y. If t = e, then y = x−1 and e ∈ x·x−1 ∩x−1 ·x which implies that t = e ∈ y · x. Now, if t 6= e, then t = x or t = y. If t = x, then x ∈ x · y and so x ∈ x · y −1 = x · x. Hence, x ∈ x−1 · x = y · x. Therefore, x · y ⊆ y · x. Similarly, y · x ⊆ x · y, and so x · y = y · x.  Lemma 4.4.2. Let e be a unit of a hyperstructure (H, ·). If x, y, z ∈ H, such that e ∈ {x, y, z}, then (x · y) · z ∩ x · (y · z) 6= ∅. Proof. It is straightforward.  Corollary 4.4.3. Let ({e, a, b}, ·) be a commutative hyperstructure with a scalar unit e. If a · (a · b) ∩ (a · a) · b 6= ∅

and

b · (b · a) ∩ (b · b) · a 6= ∅,

then ({e, a, b}, ·) is weak associative. In [144], one can find all Hv -groups with three elements which contain a scalar unit element e. Now, we determine the set of all weak polygroups defined on a set with three elements and then we conclude that all weak polygroups with three elements are polygroups. For computing these, we program the algorithm by Mathematica 4.0. Problem 4.4.4. Find all weak polygroups with three elements. For P = {e, a, b}, consider the following Cayley table: · e a b

e e a b

a a 1 3

b b 2 4

and then find all the fours, of non-empty subsets of P , in the place of the four (1, 2, 3, 4) such that all conditions of Definition 4.2.1 are valid. Since by Proposition 4.4.1, every weak polygroup on the set P = {e, a, b} is commutative, so there exist 73 = 343 Cayley tables for candidate of a weak

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polygroup. The following algorithm illustrate one method of doing so. We use the function p for finding the product of two singletons and the function prod for finding the product of two arbitrary sets. set = {{e}, {a}, {b}}; prodtable = {}; pt = {{{e}, {a}, {b}}, {{a}, , }, {{b}, , }}; powerset = {{e}, {a}, {b}, {e, a}, {e, b}, {a, b}, {e, a, b}}; For [s0=1, s0 ≤ 7, s0++, pt[[2,2]]=powerset[[s0]]; For [s1=1,s1 ≤ 7, s1++, pt[[2,3]]=powerset[[s1]]; pt[[3,2]]=pt[[2,3]]; For [s3=1,s3 ≤ 7, s3++, pt[[3,3]]=powerset[[s3]]; AppendTo[prodtable,pt]; ]; ]; ]; index[ele ]:=Module[{x=ele,y=set,t}, For [t=1,t¡=3,t++, If [x==y[[t]], Return [t]]; ]; ]; p[ele1 , ele2 , ind , collect ]:=collect[[ind]][[index[ele1], index[ele2]]]; set2 ,q , ptable ]:=Module[{f=set1, g=set2, prod[set1 , prd=ptable,u={},m,n}, For [m=1, m ≤ Length[f],m++, For [n=1, n ≤ Length[g],n++, u=Union [u,p[{f [[m]]}, {g[[n]]}, h, prd]]; ]; ]; Return[u]; ];

h=q,

The following subroutine checks the weak associativity. We put the Cayley tables which are weak associative in newpt variable. perm = {{a, a, b}, {b, b, a}};

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associative = {}; For [l=1,l≤ Length[prodtable],l++, asctv=True; For [k=1, k≤ 2,k++, asctv = asctv && ( Intersection[prod[prod[{perm[[k,1]]}, {perm[[k,2]]}, l, prodtable], {perm[[k,3]]}, l, prodtable], prod[{perm[[k,1]]}, prod[{perm[[k,2]]}, {perm[[k,3]]}, l, prodtable], l, prodtable]]=!={}); If [!asctv, Break[]]; ]; If [asctv, AppendTo[associative, l]]; ]; newpt=prodtable[[associative]];

The following subroutine checks the third condition of Definition 4.1.1.

funs = {{{e}, {a}, {b}}, {{e}, {b}, {a}}}; newpt2 ={,}; For [o1=1,o1≤ 2,o1++, inv={}; For [o2=1, o2≤ Length[newpt], o2++, mem3=True; For [o3=1, o3≤ 3, o3++, mem4=True; For [o4=1, o4≤ 3, o4++, mem5=True; For [o5=1, o5≤ Length[newpt[[o2]][[o3]][[o4]]], o5++, mem5=mem5 && (MemberQ[prod[{newpt[[o2]][[o3]][[o4]][[o5]]}, funs[[o1]][[o4]], o2, newpt], set[[o3]][[1]]])&& (MemberQ[prod[funs[[o1]][[o3]],{newpt[[o2]][[o3]][[o4]][[o5]]}, o2, newpt], set[[o4]][[1]]]); If [!mem5, Break[]]; ]; mem4=mem4 && mem5; If [!mem4, Break[]];

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]; mem3=mem3 && mem4; If [!mem3, Break[]]; ]; If [mem3, AppendTo[inv,o2]]; ]; newpt2[[o1]]=newpt[[inv]]; For [u=1,u≤ Length[newpt2[[o1]]], u++, print[MatrixForm[newpt2[[o1]][[u]]]];]; ]; After running this program, we obtain 15 weak polygroups as follows: · e a b · e a b

e e a b e e a b

· e a b · e a b · e a b

e e a b e e a b e e a b

a a e b a a e, a b a a e, b a, b a a e, a, b a

a a a e, a, b

b b b e, a

· e a b

b b b e, a, b b b a, b e, a b b a e, b b b e, a, b b

e e a b

· e a b

a a e b

e e a b

b b b e, a, b

a a e, b a

· e a b

e e a b

a a e, b a, b

· e a b

e e a b

a a e, a, b a, b

· e a b

e e a b

a a b e

· e a b

e e a b

a a e, a b

b b b e, a

b b a e

· e a b

e e a b

a a e, b a

b b a e, b

b b a, b e, a, b

· e a b

e e a b

a a e, a, b a

b b a, b e, a b b e a

b b a e

· e a b

e e a b

a a e, a, b a, b

b b a, b e, a, b

· e a b

e e a b

a a a, b e, a, b

b b e, a, b a, b

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On the set P only one non identity automorphism can be defined:  f:

e ←→ e a ←→ b

So, two weak polygroups (P, ·) and (P, ∗) 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

· e a b

e e a b

a a 1 3

b · b e take the table 2 a 4 b

e e a b

a a 4 2

b b 3 1

and then replace a by b and b by a. The result is the isomorphic weak polygroup to the first one. Therefore, we have: Theorem 4.4.5. There are only 10 weak polygroups on the set P = {e, a, b} up to isomorphism as follows:

· e a b

e e a b

a a e b

b b b e, a

· e a b

e e a b

a a e b

b b b e, a, b

· e a b

e e a b

a a e, a b

b b b e, a

· e a b

e e a b

a a e, a b

b b b e, a, b

· e a b

e e a b

a a e, b a, b

b b a, b e, a

· e a b

e e a b

a a e, b a, b

b b a, b e, a, b

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· e a b

e e a b

a a e, a, b a, b

b b a, b e, a, b

· e a b

e e a b

a a a e, a, b

b b e, a, b b

· e a b

e e a b

b b e a

· e a b

e e a b

a a a, b e, a, b

b b e, a, b a, b

a a b e

Corollary 4.4.6. All the weak polygroups with three elements are polygroups.

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Chapter 5

Combinatorial Aspects of Polygroups

5.1

Chromatic polygroups

An important class of polygroups is derived from color schemes, a notion that extends D.G. Higman’s homogeneous coherent configuration [77]. Chromatic polygroups are studied by Comer in [14; 15; 16; 18; 20] and they are obtained from certain edge colored complete graph by making multi-valued algebra out of the set of colors. The results of this section are obtained by Comer. In the definition of a color scheme presented below the relative product (or composition) of two relations is denoted by k and the inverse (or converse) of a relation is denoted by `. Suppose that C is a set (of colors) and  is an involution of C. Definition 5.1.1. A color scheme is a system V =< V, Cx >x∈C , where (1) (2) (3) (4)

{Cx | x ∈ C} partitions V 2 − I = {(a, b) ∈ V 2 | a 6= b}; Cx` = C(x) , for all x ∈ C; for all x ∈ C, a ∈ V there exist b ∈ V such that (a, b) ∈ Cx ; Cx ∩ (Cy k Cz ) 6= ∅ implies Cx ⊆ Cy k Cz , i.e., the existence of a path colored (y, z) between two vertices joined by an edge colored x is independent of the two vertices.

Given a color scheme V, choose a new symbol I 6∈ C. (Think of I as the identity relation on V ). The algebra (color algebra) of V is the system MV =< C ∪ {I}, ·, I,−1 >, where x−1 = (x) for x ∈ C, I −1 = I, x · I = x = I · x, for all x ∈ C ∪ {I}, and for x, y, z ∈ C, x · y = {z ∈ C | Cz ⊆ Cx k Cy } ∪ {I | y = x−1 }. It is straightforward to verify MV is a polygroup. 163

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Definition 5.1.2. A polygroup is called chromatic if it is isomorphic to an algebra MV , for some color scheme V. Example 5.1.3. A natural example of a chromatic polygroup is the system G//H of all double cosets of a group G modulo a subgroup H, see Example 3.1.2 (1). In order to see that G//H is chromatic, consider the color scheme V =< V, Cx >x∈C , where C = (G//H) \ {H}, V = {Ha | a ∈ G} and for x ∈ C, Cx = {(Ha, Hb) | ab−1 ∈ x}. It is not difficult to verify that G//H is isomorphic to MV . Proposition 5.1.4. Suppose that A =< A, ·, e,−1 > and B =< B, ·, e,−1 > are two polygroups whose elements have been renamed so that A ∩ B = {e}. If A ∼ = MV and B ∼ = MW , then the extension of A by B, i.e., A[B] is also chromatic. Proof. Suppose that V =< V, Ca >a∈A\{e} . First, we introduce a family of pairwise disjoint color schemes {Vw | w ∈ W }, where each Vw is isomorphic to V. Assume that the vertex set of Vw is Vw and the isomorphism of V onto Vw sends x to xw . We construct a scheme V[W] in the following way. Replace each vertex w of the scheme W by the copy of V with vertex set Vw . Thus, the set of all vertices of V[W] is just the union of all Vw ’s. An edge coloring using the elements of (A ∪ B) \ {e} as colors is introduced in the following way. For a ∈ A \ {e} and b ∈ B \ {e}, let (xu , yv ) ∈ Ca if and only if u = v and (x, y) ∈ Ca (in V) (xu , yv ) ∈ Cb if and only if (u, v) ∈ Cb (in W). It is easily seen that V[W] is a scheme that represents A[B].  The double coset construction generalizes to the idea of a double quotient. This idea will not be needed in this section for a general polygroup but only for ordinary groups. The general notation (see Definition 3.3.10) is equivalent to the following when restricted to groups. Definition 5.1.5. An equivalence relation θ on a group G is called a (full) conjugation on G if (1) θ(x)−1 = θ(x−1 ), for all x ∈ G, (2) θ(xy) ⊆ θ(x)θ(y), for all x, y ∈ G. The natural quotient system G//θ is a chromatic polygroup and we define Q2 (Group)= {G//θ | θ is a conjugation on some group G}. A conjugation θ is called special if it satisfies

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(3) xθe implies x = e. The class of all polygroups isomorphic to double quotients of groups via special conjugation is denoted by Q2s (Group). Example 5.1.6. Some of the ways to obtain conjugations on a group G are indicated below: (1) A congruence relation θ on G is a conjugation. G//θ is just the usual quotient group in this case. (2) If H is a subgroup of G and we define xθH y if and only if x and y generate the same double coset (i.e., HxH = HyH), then θH is a conjugation on G and G//θH = G//H. (3) Define xθy if and only if x and y are conjugate in the usual sense with respect to a subgroup H (i.e., there exists h ∈ H such that y = h−1 xh). Then, θ is a special conjugation. This example generalizes as follows. (4) Suppose that K is a group of automorphisms of G. Define xθy if and only if y = σ(x), for some σ ∈ K. This is also a special conjugation on G. Proposition 5.1.7. If θ is a conjugation on a group G, then θ(e) = H is a subgroup of G and G//θ ∼ = (G//H)//ψ for some special conjugation ψ on G//H. Proof. Suppose that x, y ∈ θ(e) and z = xy −1 . Then, x = zy. Hence, we have e ∈ θ(e) = θ(x) = θ(zy) ⊆ θ(z)θ(y). Therefore, there exist z 0 ∈ θ(z) and y 0 ∈ θ(y) such that e = z 0 y 0 which implies that z 0 = y 0−1 . Now, we obtain θ(xy −1 ) = θ(z) = θ(z 0 ) = θ(y 0−1 ) = θ(y −1 ) = θ(e). Thus, xy −1 ∈ θ(e). For the second part, it is enough to define HxH ψ HyH ⇐⇒ x θ y.  Double quotients of groups are related to chromatic polygroups. Theorem 5.1.8. Every polygroup in Q2 (Group) is chromatic. Proof. Suppose that θ is a conjugation on a group G. By Proposition 5.1.7, θ(e) = H is a subgroup of G. Let C = {θ(g) | θ(g) 6= H}, (a) = a−1 for a ∈ C, V = {Hx | x ∈ G}, and for each a ∈ C, set Ca = {(Hx, Hy) ∈ V 2 | xy −1 ∈ a}.

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It is easy to see that V =< V, Ca >a∈C is a color scheme. If c ∈ a · b (in G//θ) and (Hu, Hv) ∈ Cc , there exist r ∈ a and s ∈ b with uv −1 = rs. Letting z = r−1 u it follows that (Hu, Hz) ∈ Ca and (Hz, Hv) ∈ Cb , so (Hu, Hv) ∈ Ca k Cb . Conversely, if Cc ⊆ Ca k Cb and x ∈ c, then (Hx, H) ∈ Cc , so there exists z ∈ b with xz −1 ∈ a. Hence, x ∈ a · b and so c ⊆ a · b (in G//θ). It now easily follows that the natural map from G//θ onto MV that sends θ(g) to θ(g) and θ(e) to I (=identity of MV ) is an isomorphism.  The color scheme used in Theorem 5.1.8, is called the regular color scheme representation of G//θ. Definition 5.1.9. An automorphism of a color scheme V =< V, Ca >a∈C is a permutation σ of V such that for all a ∈ C, x, y ∈ V , (x, y) ∈ Ca if and only if (σ(x), σ(y)) ∈ Ca . Theorem 5.1.10. Let M =< M, ·, e,−1 > be a polygroup. Then, M ∈ Q2 (Group) if and only if M ∼ = MV , for some color scheme V with Aut(V) transitive on vertices. Proof. First note that if V is the regular color scheme representing G//θ, then G acts transitively on the coset space V by multiplication. Now, suppose that M is represented by a color scheme V =< V, Ca >a∈M with Aut(V) = G transitive on V . Fix x ∈ V and partition V = ∪{Va | a ∈ M } where Ve = {x} and Va = {y ∈ V | (x, y) ∈ Ca } for a 6= e. Define an equivalence relation θV on G by: σθV τ if and only if for every a ∈ M, σ(x) ∈ Va if and only if τ (x) ∈ Va . The θV -classes correspond one to one with elements of M. Namely, a ∈ M corresponds to the θV -class Gxa = {σ ∈ G | σ −1 (x) ∈ Va } where, of course, Gxe = Gx the stabilizer of G at x. The elements of V correspond in a natural way to cosets of Gx . Namely, for y ∈ V , Gxy = {σ ∈ G | y = σ(x)} = Gx τ , where τ is any element of Gxy . We obtain that θV is a conjugation on G. It is easily to see that θV preserves inverses. We suppose that σ0 = σ2 σ1 and σ00 θV σ0 and show σ00 = σ20 σ10 for some σ10 θV σ1 and σ20 θV σ2 . Suppose that σ0 ∈ Gxc , σ1 ∈ Gxa and σ2 ∈ Gxb . We have

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c

bc

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bc σ0 (x)

a

b

bc σ2 (x)

in V. For, if (σ0 (x), σ2 (x)) ∈ Cd for some d, applying σ2−1 yields (σ1 (x), x) ∈ Cd (since σ0 = σ2 σ1 ); but (σ1 (x), x) ∈ Ca so d = a. Consequently, the colored triangle (a, b, c) is locally realizable in V. Now, σ00 θV σ0 . Let y = σ00 (x). Since (a, b, c) is locally realizable and (y, x) ∈ Cc , there exists z such that (y, z) ∈ Ca and (z, x) ∈ Cb . G vertextransitive implies that z = τ (x), for some τ ∈ Gxb . Choose µ ∈ G so that µ(x) = τ −1 (y) (G vertex-transitive). Now, (y, τ (x)) ∈ Ca implies that (µ(x), x) ∈ Ca , so µ ∈ Gxa . Since τ µ(x) = y, both τ µ and σ00 belong to the coset Gxy . Thus, σ00 = τ µv for some v ∈ Gx from which it follows that σ00 = σ20 σ10 , where σ10 = µvθV σ1 and σ20 = τ θV σ2 as required. In order to finish the theorem, we must show the natural bijection that sends a ∈ M to Gxa ∈ G//θV is an isomorphism. First, we check that e ∈ a · b if and only if Gx ⊆ Gxb Gxa (so that inverses correspond). If e ∈ a · b (so a = b−1 ) and σ ∈ Gxa , then (x, σ(x)) ∈ Ca−1 = Cb . So, 1V = σ −1 σ ∈ Gxb Gxa and therefore Gx ⊆ Gxb Gxa . Conversely, if Gx ⊆ Gxb Gxa , then 1V = τ σ for some σ ∈ Gxa and τ ∈ Gxb . Since (σ(x), x) ∈ Ca , applying τ yields (x, τ (x)) ∈ Ca . Therefore, (τ (x), x) ∈ Ca−1 ∩ Cb so b = a−1 . It remains to show c ∈ a · b if and only if Gxc ⊆ Gxb Gxa , where we may assume a, b, c 6= e. The argument from left to right is essentially the same as used in the above. We suppose that Gxc ⊆ Gxb Gxa . By the product definition in MV it suffices to realize an (a, b, c) triangle on some (x, y) ∈ Cc . Choose σ such that x = σ(y) (by transitivity of G). Then, σ ∈ Gxc ⊆ Gxb Gxa so σ = µτ for some τ ∈ Gxa and µ ∈ Gxb . Let z = τ −1 (x). Then, we have

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c

x bc

bc

y

b

a

bc z

in V. Clearly, (x, z) ∈ Ca . Suppose that (z, y) = (τ −1 (x), τ −1 µ−1 (x)) ∈ Cd for some d. Applying τ we obtain (x, µ−1 (x)) ∈ Cd from which d = b follows. Thus, c ∈ a · b as desired.  A group G of automorphisms of a graph is called strongly transitive on edges if, for every pair of edges (x, y) and (u, v) there exists σ ∈ G such that σ(x) = u and σ(y) = v. Theorem 5.1.11. ∼ MV (1) M is isomorphic to a double coset algebra if and only if M = for some color scheme V where Aut(V) is strongly transitive on the edges of each monochrome subgraph. (2) M ∈ Q2s (Group) if and only if M ∼ = MV for some color scheme V for which there exists G ⊆ Aut(V) such that (i) G is vertex-transitive, (ii) Gx = {1V }, for all x ∈ V . Proof. (1) We consider the regular color scheme representing G//H. If (Hx, H), (Hy, H) ∈ Ca , both x and y belong to the double coset a. So Hy can be obtained from Hx by right multiplication by an element of H. Thus, the regular color scheme has the desired properties. Conversely, if Aut(V) is vertex-transitive, MV ∼ = G//θ (Theorem 5.1.10) where G = Aut(V) and each θ-block Gxa is a union of cosets Gx τ (where xτ −1 ∈ Va ). Gxa is a single double coset since Gx is transitive on Va . (2) If M = G//θ where θ(e) = {e}, the vertex set in the regular color scheme representing M is indentifiable with G and the action of G on G is regular (i.e., Gx = {1} for all x). Conversely, suppose that V representing M has (i) and (ii). Notice that the definition of θV in Theorem 5.1.10 depends only on G being a transitive subgroup of Aut(V). Thus, (i) implies

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that M ∼ = G//θ and (ii) implies that θ is special.  The next three results show that various important special classes of chromatic polygroups are closed under the extension operation A[B]. Theorem 5.1.12. If A, B ∈ Q2 (Group), then A[B] ∈ Q2 (Group). Proof. Suppose that A ∼ = MV , B ∼ = MW , and V[W] is the color scheme constructed from V and W in the proof of Proposition 5.1.4. Automorphism τ on V and σ on W induce automorphisms on V[W] in the following way: (1) For σ ∈ Aut(W) define σ ˆ on V[W] by σ ˆ (xw ) = xσ(w) , for all w ∈ W . (2) For τ ∈ Aut(V) and w ∈ W define τ w on V[W] so that by τ w acts like τ on Vw and is the identity otherwise. It is easy to see that the maps σ ˆ and τ w described in (1) and (2) are automorphisms of V[W]. From Theorem 5.1.10, we may assume that Aut(V) and Aut(W) are transitive. Using maps of type (1) and (2) it easily follows that Aut(V[W]) is transitive on vertices so Theorem 5.1.10 yields the desired conclusion.  Theorem 5.1.13. If A, B ∈ Q2s (Group), then A[B] ∈ Q2s (Group). Proof. Suppose that A = G1 //θ1 and B = G2 //θ2 where θ1 and θ2 are special conjugations on G1 and G2 respectively. Let G = G1 × G2 and define θ on G by (g1 , g2 )θ(g10 , g20 ) ⇐⇒ (g2 = g20 = e and g1 θ1 g10 ) or (g2 , g20 6= e and g2 θ2 g20 ). Notice that the θ-classes are θ(g, e) = {(h, e) | hθ1 g} and for h 6= e, θ(e, h) = {(g, h0 ) | h0 θ2 h}. In order to show θ is a special conjugation, conditions (1), (2) and (3) need to be checked. First, (3) holds because θ1 special implies that θ(e, e) = {(e, e)}. Also (1), θ(g, h)−1 = (θ(g, h))−1 , holds since θ1 and θ2 have similar properties. It remains to check (2) θ((g1 , h1 )(g2 , h2 )) ⊆ θ(g1 , h1 )θ(g2 , h2 ). Suppose that (g, h) ∈ θ(g1 g2 , h1 h2 ) = θ((g1 , h1 )(g2 , h2 )). The definition of θ gives two cases: Case (1) h = h1 h2 = e and gθ1 (g1 g2 ). Since θ1 is a conjugation, g = g10 g20 for some g10 θ1 g1 and g20 θ1 g2 . Then, (g, e) = (g10 , h1 )(g20 , h2 ), so it is suffices to show that (gi0 , hi )θ(gi , hi ) for i = 1, 2. The conclusion follows from gi0 θ1 gi when h1 = h2 = e while it follows from hi θ2 hi if h1 , h2 6= e.

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Case (2) h, h1 h2 6= e and hθ2 h1 h2 . Since θ2 is a conjugation, h = h01 h02 for some h01 θ2 h1 and h02 θ2 h2 . This yields (g, h01 )θ(g1 , h1 ) and (e, h02 )θ(g2 , h2 ) whenever h01 , h02 6= e; so (g, h) = (g, h01 )(e, h02 ) ∈ θ(g1 , h1 )θ(g2 , h2 ). On the other hand, suppose that one of the h01 , h02 is e, say h01 = e and h02 = h 6= e. Then, (g1 , e)θ(g1 , h1 ) and (g1−1 g, h)θ(g2 , h) since h 6= e from which it follows that (g, h) belongs to θ(g1 , h1 )θ(g2 , h2 ). Thus, θ is a special conjugation on the group G1 × G2 . A bijection F between the elements of A[B] and G1 × G2 //θ is defined in the following way. Let F (e) = θ(e, e) and for a = θ1 (g1 ) 6= e in A \ {e}, let F (a) = θ(g1 , e), and for b = θ2 (g2 ) 6= e in B \ {e}, let F (b) = θ(e, g2 ). From the description of the θ-classes above it is clear that the map F from A[B] to G1 × G2 //θ is a bijection. By properties (1) and (2) of θ the inverse and identity elements correspond. Computation, as in the proof of (2), show that F preserves products in case at least one factor belong to A. When both factors belong to B there are two cases. First, if θ2 (g2 ), θ2 (g20 ) ∈ B\{e} and θ2 (g2 ) 6= (θ2 (g20 ))−1 , then F (θ2 (g2 )θ2 (g20 )) = F ({θ2 (g) | e 6= g ∈ θ2 (g2 )θ2 (g20 )}) = {θ(e, g) | g ∈ θ2 (g2 )θ2 (g20 )} = {(h, g) | g ∈ θ2 (g2 )θ2 (g20 )} = θ(e, g2 )θ(e, g20 ) = F (θ2 (g2 ))F (θ2 (g20 )). Finally, suppose that θ2 (g2 ), θ2 (g20 ) ∈ B − {e} and θ2 (g20 ) = (θ2 (g2 ))−1 . Then, e ∈ θ( g2 )θ2 (g20 ) so F (θ2 (g2 )θ2 (g20 )) = F ({θ2 (g) |g ∈ θ2 (g2 )θ2 (g20 )} ∪ A) = {θ(e, g) | g ∈ θ2 (g2 )θ2 (g20 )} ∪ (G1 × {e}) = {(h, g) | g ∈ θ2 (g2 )θ2 (g20 )} since e ∈ θ( g2 )θ2 (g20 ) = θ(e, g2 )θ(e, g20 ) = F (θ2 (g2 ))F (θ2 (g20 )). Now, the theorem follows from the fact F is an isomorphism.  The next result shows the class of double coset algebras is closed under

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the extension construction. Theorem 5.1.14. If A ∼ = G1 //H1 and B ∼ = G2 //H2 , then there exist b b b b such that A[B] ∼ b H. b groups G and H with H ⊆ G = G// b = GX ×ϕ G2 , the Proof. Let X = G2 /H2 = {H2 g | g ∈ G2 } and let G 1 X semidirect product of GX 1 by G2 , where ϕ, mapping G2 into (Aut(G1 ), is ∗ X the homomorphism given by ϕg (f ) = g f , for all f ∈ G1 . XA AA AA A ϕg (f ) AA

g∗

G1

/X } } }} }} f } }~

∗

i.e., g ∈ G2 induces g : X −→ X by right multiplication, so ϕg (f )(x) = X−{H2 } b f (xg) for g ∈ G2 , f ∈ GX )×ϕ H2 , 1 and x ∈ X. Also, let H = (H1 ×G1 where X−{H2 }

ϕ : H2 −→ Aut(H1 × G1

)

∗

is defined, as above, by ϕg (f ) = g f . b = H ×ϕ H2 , where H = {f ∈ GX | f (H2 ) ∈ H1 }. Clearly, Notice that H 1 b b so it remains to show that A[B] ∼ b H. b H is a subgroup of G, = G// b H. b For g ∈ G1 let gb = (f, 1) where First, we identify A with part of G//

1 is the identity element of G2 and let f ∈ GX 1 is defined by  g if H2 x = H2 f (H2 x) = e if H2 x 6= H2 . Then, b gH b = (H×ϕ H2 )(f, 1)(H×ϕ H2 ) = (Hf ×ϕ H2 )(H×ϕ H2 ) = (Hf H)×ϕ H2 , Hb where the second equality holds because, for h ∈ H2 , ϕh fixes the “H2 coordinate” of f . Thus, b ∼ (1) (GX 1 ×ϕ H2 )//H = G1 //H1 . Now, we consider the elements in G2 //H2 (∼ = B). For g ∈ G2 let g = (E, g) where E is the identity element of GX 1 . Then, b = (H ×ϕ H2 )g = H ×ϕ H2 g Hg and, for g 6∈ H2 , b H b = (H ×ϕ H2 g)(H ×ϕ H2 ) = GX Hg 1 ×ϕ (H2 gH2 ) since, for f1 , f2 ∈ H, (f1 · f2hg )(x) = f1 (x)f2 (xhg) will produce any element ∗ of GX 1 . In order to see this observe that g 6∈ H2 means g is a permutation of X that moves H2 , so for all x ∈ X, either f1 (x) or f2 (xhg) can be any element of G1 . Thus,

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b = S{GX ×ϕ b | b ∈ G2 //H2 }. (2) G 1 A one to one correspondence between the nonidentity elements of b H b is introduced as follows: to every noniden(G1 //H1 )[G2 //H2 ] and G// tity element a ∈ G1 //H1 assign the element a ×ϕ H2 where a = {f ∈ GX 1 | f (H2 ) ∈ a} and to every nonidentity element b ∈ G2 //H2 correlate GX 1 ×ϕ b. In view of (1), to show this correspondence is an isomorphism it is enough to check: b gb1 H)( b = Hg b 2 H, b for g1 ∈ G1 , g2 ∈ G2 \ H2 , b Hg b 2 H) (3) (H b H b gb1 H) b = Hg b 2 H, b for g1 ∈ G1 , g2 ∈ G2 \ H2 , b 2 H)( (4) (Hg b 1 H)( b Hg b 2 H) b = GX ×ϕ (H2 g1 H2 g2 H2 ), for g1 , g2 ∈ G2 \ H2 . (5) (Hg 1 To establish (3), let gb1 = (g10 , 1). Then, b gb1 H)( b = ((Hg 0 H) ×ϕ H2 )(GX ×ϕ H2 g2 H2 ) b Hg b 2 H) (H 1 1 = GX 1 ×ϕ H2 g2 H2 b 2H b = Hg since, for h ∈ H2 , ϕh fixes the “H2 -coordinate” and permutes all others. The verification of (4) is easier: b H b gb1 H) b = (GX ×ϕ H2 g2 H2 )(Hg 0 H ×ϕ H2 ) b 2 H)( (Hg 1 1 = GX 1 ×ϕ H2 g2 H2 . Finally, we check (5): b 1 H)( b Hg b 2 H) b = (GX ×ϕ H2 g1 H2 )(GX ×ϕ H2 g2 H2 ) (Hg 1 1 = GX 1 ×ϕ (H2 g1 H2 )(H2 g2 H2 ). b H b ∼ Now, it follows that G// = A[B] as desired.  Now, we consider the converse of the properties established in Theorems 5.1.4, 5.1.12, 5.1.13 and 5.1.14. The basic idea for establishing the converses is illustrated by the proof of the following result. Theorem 5.1.15. If A[B] is chromatic, then both A and B are chromatic. Proof. Let A[B] ∼ = MW for some color scheme W =< W, Cx >x∈C . Recall that C = (A ∪ B) \ {e}. We define a relation ≈ on W by w ≈ w0 if and only if w = w0 or (w, w0 ) ∈ Ca for some a ∈ A \ {e}. It is easy to see that ≈ is an equivalence relation on W and each ≈-block, say [p] = {w | w ≈ p} for a fixed p ∈ W , inherites the structure of a color scheme from W. The color algebra of this scheme is exactly A, so A is

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chromatic. In order to treat B we form a new scheme W/ ≈ on the set {[w] | w ∈ W } using the elements of B − {e} as colors. For distinct vertices [v] and [w] set ([v], [w]) ∈ Cb if and only if (v, w) ∈ Cb (in W). Since a1 ba2 = b holds in A[B] for a1 , a2 ∈ A and b ∈ B, it follows that the assignment of a color to the edge ([v], [w]) is independent of the ≈representations. It is not difficult to check that MW / ≈∼ = B as desired.  Using the idea above an analysis of the proofs of Theorems 5.1.12, 5.1.13 and 5.1.14 give a hint of how to construct their converses. We leave the details to the reader. Theorem 5.1.16. If A[B] is a double coset algebra (in Q2 (Group), Q2s (Group)), then both A and B are double coset algebras (in Q2 (Group), Q2s (Group), respectively). Suppose that V0 =< V0 , Cα >α∈A−{e} and V1 =< V1 , Cβ >β∈B−{e} are color schemes. It is convenient to assume the identity relation occurs among the colors, i.e., Ce = IV0 (in V0 ) and Ce = IV1 (in V1 ). Consider the product scheme V0 × V1 =< V0 × V1 , C(α,β) >(α,β)∈A×B−{(e,e)} where C(α,β) = {((a, i), (b, j)) ∈ (V0 × V1 )2 | (a, b) ∈ Cα and (i, j) ∈ Cβ }. Theorem 5.1.17. V0 × V1 is a color scheme and MV0 ×V1 = MV0 × MV1 . Proof. Most of the color scheme properties are trivial. We show (4). Suppose that ((a, i), (b, j)) belongs to C(α0 ,β0 ) ∩ (C(α1 ,β1 ) k C(α2 ,β2 ) ). If αk = e or βk = e for some k ∈ {0, 1, 2}, the condition above degenerates and the conclusion is clear. Assume that all α’s and β’s are 6= e. Then, (a, b) ∈ Cα0 and (i, j) ∈ Cβ0 and there exists (c, k) such that (a, c) ∈ Cα1 , (i, k) ∈ Cβ1 , (c, b) ∈ Cα2 and (k, j) ∈ Cβ2 . Then, Cα0 ∩(Cα1 k Cα2 ) 6= ∅ and Cβ0 ∩(Cβ1 k Cβ2 ) 6= ∅, so Cα0 ⊆ Cα1 k Cα2 and Cβ0 ⊆ Cβ1 k Cβ2 . It follows that C(α0 ,β0 ) ⊆ C(α1 ,β1 ) k C(α2 ,β2 ) . In order to see that MV0 ×V1 = MV0 × MV1 , notice that allowing α = e and β = e simplifies the definition of ∗ in MV0 ×V1 . Namely, (α0 , β0 ) ∗ (α1 , β1 ) = {(α2 , β2 ) | C(α2 ,β2 ) ∩ (C(α0 ,β0 ) k C(α1 ,β1 ) ) 6= ∅}. As in the proof of (4) above, the product (α2 , β2 ) ∈ (α0 , β0 ) ∗ (α1 , β1 )

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is equivalent to products α2 ∈ α0 · α1 and β2 ∈ β0 · β1 on the factors. This, in turn, is equivalent to the product in MV0 × MV1 .  What other classes are closed under direct product? For color scheme V0 and V1 it is not hard to see that Lemma 5.1.18. Aut(V0 ) × Aut(V1 ) ∼ = Aut(V0 × V1 ). Proof. Consider the map that sends (σ0 , σ1 ) ∈ Aut(V0 ) × Aut(V1 ) to the automorphism [σ0 , σ1 ] defined as [σ0 , σ1 ](a, i) = (σ0 (a), σ1 (i)) for all (a, i) ∈ V0 × V1 .  Theorem 5.1.19. The following classes are closed under direct product: (1) Q2 (Group); (2) Q2s (Group); (3) double coset algebras. Proof. (1) By Theorem 5.1.10, M ∈ Q2 (Group) if and only if M ∼ = MV for some color scheme V with Aut(V) transitive on vertices. Given V0 , V1 both with such automorphism groups Lemma 5.1.18 shows that Aut(V0 × V1 ) is also transitive on vertices. By Theorem 5.1.17, MV0 × MV1 ∈ Q2 (Group). The proof of (2) and (3) is similar using Theorem 5.1.11 items (1) and (2).  We mention one application of direct product. We associate a polygroup M(L) with each modular lattice L =< L, ∨, ∧ > with a minimum element e. Namely, M(L) =< L, ·, −1 , e > where x−1 = x and x · y = {z ∈ L | x ∨ z = y ∨ z = x ∨ y}. It is not hard to check that M(L1 × L2 ) = M(L1 ) × M(L2 ). Theorem 5.1.17 implies that whenever the M(L) construction associates a chromatic polygroup to latices L1 and L2 , the product also gives rise to a chromatic polygroup. A similar conclusion holds for the classes listed in Theorem 5.1.19. Recall that the ordered sum L0 ⊕ L1 of two bounded lattices is the lattice obtained by identifying the minimum element of L1 with the maximal element L0 . Corollary 5.1.20. For bounded modular lattices L0 and L1 , M(L0 ⊕L1 ) = M(L0 )[M(L1 )]. Thus, L0 ⊕ L1 yields a chromatic polygroup whenever L0 and L1 do. Composition series play an important role in the study of groups. Polygroups also exhibit similar series. The core of a polygroup M, written

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Core(M), is the subpolygroup generated by ∪{a · a−1 | a ∈ M }. For a subpolygroup N of M, we introduce a conjugation ΘN on M by aΘN b if and only if b ∈ N aN . Then, M//ΘN is a polygroup. Definition 5.1.21. An ultragroup is a polygroup M for which exists a chain of subpolygroups {e} = Nk ⊆ Nk−1 ⊆ . . . ⊆ N0 = M, where Ni //ΘNi−1 is a group for all i < k. The groups Ni //ΘNi−1 are called the factors of the series. Note that Core(A[B]) = A ∪ Core(B) so if B is a group, Core(A[B]) = A. Proposition 5.1.22. Given groups G0 , G1 , . . . , Gk+1 and 0 < i < k, let Ni denote the extension (. . . (G0 [G1 ]) . . .)[Gi−1 ]. Then, (1) Nk is an ultragroup with factors Gk−1 , . . . , G0 ; (2) Nk is a double coset algebra. Proof. (1) Since Ni = Ni−1 [Gi−1 ] and Core(Ni ) = Ni−1 , Nk ⊃ Nk−1 ⊃ . . . ⊃ N1 ⊃ {e} gives the lower ultra-series for Nk . Moreover, note that for a, b ∈ Ni , a ΘNi−1 b if and only if a, b ∈ Ni−1 or a = b. Then, Ni //ΘNi−1 = Ni //Ni−1 = Gi−1 . (2) is obvious.  5.2

Polygroups derived from cogroups

Cogroups were introduced by Eaton [70] in an attempt to axiomatize Dhypergroups, i.e., systems obtainable from groups by right coset decompositions with respect to, not necessarily normal, subgroups. Eaton’s axioms apply only to finite systems. Utimi [141] formulated a general notion and gave an example of a cogroup not isomorphic to a D-hypergroup. The definition of a cogroup given by Comer [14] is equivalent to the one formulated by Utuni [141]. The cardinality axiom assumed by both Utumi and Eaton does not play a role in Comer’s development. The apparently weaker notion of weak cogroup is obtained by removing this assumption. The main reference for this section is [14]. Definition 5.2.1. A weak cogroup is a system < A, ·, −1 , e >, where e ∈ A; x · y is a non-empty subset of A for x, y ∈ A; x−1 is a non-empty subset of A for x ∈ A; and the following axioms hold for all x, y, z ∈ A;

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(1) (2) (3) (4) (5)

(x · y) · z = x · (y · z), e · x = x, y ∈ x−1 ⇐⇒ e ∈ x · y, x ∈ y · z =⇒ y ∈ x · z −1 and z ∈ y −1 · x, x · y ∩ z · y 6= ∅ =⇒ x ∈ z · e.

A weak cogroup is called a cogroup if, in addition, it satisfies the axiom (6) |x · y| = |x · z|, for all x, y, z ∈ A. If H is a subgroup of a group G, the system G/H =< {Hg | g ∈ G}, ·, −1 , H > of all right cosets becomes a cogroup using the operation (Hg) · (Hk) = {Hghk | h ∈ H} and (Hg)−1 = {Hg −1 h | h ∈ H}. The system G/H is known as a D-hypergroup. Elements x and y in a weak cogroup are called e-conjugates, in symbols x ≈ y if and only if x ∈ y · e. It is easy to see that ≈ is an equivalence relation on A, the ≈-class of e is {e}, and x ∈ y · e ⇔ x · e = y · e ⇔ x · z = y · z for all z. The product x · e is the ≈-class that contains x. The canonical example of a cogroup is G/H while that of a polygroup is G//H. Every element in G//H is a ≈-class of G/H. This suggests a way to construct polygroups from arbitrary weak cogroups. Lemma 5.2.2. Suppose that A is a weak cogroup and ≈ is the relation of e-conjugation. Then, (1) (a·e)·(b·e) = (a·b)·e = {c·e | c ∈ a·b} and (a·e)−1 = a−1 ·e = a−1 . (2) The system A/ ≈ of all ≈-classes, with operations inherited from A, is a polygroup. Proof. It is straightforward. As an example consider a−1 ·e ⊆ a−1 . Suppose that b ∈ c · e, for some c ∈ a−1 . Then, e ∈ a · c, which implies that e ∈ c · a = b · a. Hence, e ∈ a · b and so b ∈ a−1 .  The system A/ ≈ is called the polygroup derived from A. The following definition and lemma are essentially due to Utumi [141]. For an equivalence relation θ on a weak cogroup < A, ·, −1 , e >, let A∗θ =< A, ∗,

−1

, e∗ >

where e∗ = e, x ∗ y = θ(x) · y and x−1 = θ(x)−1 , i.e., x−1 = ∪{y −1 | y ∈ θ(x)}. A∗θ is called a scalar partition hypergroupoid with respect to θ. We refer to the structure as A∗ whenever θ is understood.

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Lemma 5.2.3. Suppose that A is a weak cogroup and θ is an equivalence relation on A with x · e ⊆ θ(x), for all x. Then, A∗ is a weak cogroup if and only if (1) θ(e) = {e}, (2) θ(x−1 ) = (θ(x))−1 , (3) θ(θ(x)y) = θ(x)θ(y). Moreover, if A is a cogroup, so is A∗ . Proof. It is straightforward.  The condition x · e ⊆ θ(x) in Lemma 5.2.3 is only needed to establish the implication ⇒. For either a weak cogroup A or a polygroup M, an equivalence relation θ on A that satisfies conditions (1),(2), (3) in Lemma 5.2.3 and has x · e ⊆ θ(x) for all x ∈ A is called an Utumi partition. The condition x · e ⊆ θ(x) for all x ∈ A, is redundant when A is a polygroup. These partitions are closely related to special conjugations. In Definition 3.3.10, we introduced the notion of conjugation and in Lemma 3.3.11, we gave a necessary and sufficient condition for an equivalence relation be conjugation. Notice that θ is a special conjugation if and only if θ is a Utumi partition. Corollary 5.2.4. If θ is a special conjugation on a group G, then G∗θ is a cogroup and G∗θ / ≈= G//θ. Proof. The equality of quotients follows because x ≈ y if and only if x ∈ y ∗ e = θ(y) if and only if θ(x) = θ(y).  The corollary will be generalized to conjugation. Suppose that H is a subgroup of a group G and π is a Utumi partition on G/H. Let G[H, π] = (G/H)∗π / ≈ . Also, we define π on G2 by g1 πg2 ⇐⇒ (π(Hg1 ))H = (π(Hg2 ))H. For readability, (π(A))B is written as π(A)B. Lemmas 5.2.2 and 5.2.3 show that G[H, π] is a polygroup. Also, in (G/H)∗ , Hg1 ≈ Hg2 if and only if π(Hg1 )H = π(Hg2 )H. Thus, π(Hg)H is the ≈-class of Hg. Also, π(Hg)H = π(Hg) in (G/H)∗ since a Utumi

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partition satisfies x · e ⊆ π(x). Thus, ≈ coincides with π. Lemma 5.2.5. π is a conjugation on G and G[H, π] ∼ = G//π. Proof. Clearly, π is an equivalence relation on G. In order to check condition (1) for a conjugation, suppose that xπy. Then, Hx ≈ Hy in (G/H)∗ . Since Hx−1 is an inverse of Hx, e-conjugate elements have the same inverses, and every two inverses of an element are e-conjugate, Hx−1 ≈ Hy −1 . Thus, x−1 πy −1 and (1) holds. In order to check the condition (2) for π to be a conjugation, assume that x0 πx and x = y · z. Then, Hx0 ∈ π(Hx) and Hx ∈ (Hy)(Hz); so, in G/H, Hx0 ∈ π(Hx) ⊆ π(Hy)(Hz)) ⊆ π(π(Hy)Hz) = π(Hy)π(Hz) by the Utumi properties. Thus, Hx0 ⊆ (Hy 00 )(Hz 00 ) for some Hy 00 ∈ π(Hy) and Hz 00 ∈ π(Hy). Then, x0 = y 0 · z 0 for some y 0 ∈ Hy 00 and z 0 ∈ Hz 00 . Since π(Hy 0 ) = π(Hy 00 ) = π(Hy), y 0 πy. Similarly, z 0 πz. Thus, (2) holds and π is a conjugation on G. The correspondence that sends π to Hg/ ≈= π(Hg)H = π(Hg) is clearly a bijection between polygroups G//π and G[H, π]. The identity elements correspond as well inverses since (π(x))−1 = π(x−1 ) and (π(Hx))−1 = π(Hx−1 ). In order to see that the correspondence is an isomorphism observed that π(Hx) ∈ π(Hy) · π(Hz) is equivalent to π(x) ∈ π(y) · π(z). The lemma above shows that polygroups derived from Utumi partitions on D-hypergroups are double quotients of groups. The next result establishes the converse. Namely, every double quotient of a group is derivable from a D-hypergroup with a Utumi partition. Theorem 5.2.6. For any G//θ ∈ Q2 (Group) there exist a subgroup H of G and a Utumi partitions π on G/H such that G//θ ∼ = G[H, π]. Proof. Given G and θ, H = θ(e) is a subgroup of G. Define π on G?H by (Hg1 )π(Hg2 ) if and only if g1 θg2 . Since θ is a conjugation on G, Hg1 = Hg2 implies that g1 θg2 , from which it follows that π is well defined (i.e., it factors through the quotient mod H). Clearly, π is an equivalence relation on G/H and, since a conjugation θ has the property gθgh for all h ∈ H, (Hg)H ⊆ π(Hg). We claim that (1) π is a Utumi partition on G/H.

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Since H = θ(e), (Hg)πH easily implies that Hg = H. It remains to check (2) and (3) of Lemma 5.2.3. First, we show that (π(Hg))−1 = π(Hg −1 ). Suppose that Hg0 ∈ (π(Hg))−1 . Then, H ∈ Hg0 Hg 0 for some g 0 θg which yields that g 0−1 θg0 . Since θ is a conjugation. g 0−1 θg −1 , from which we obtain Hg0 ∈ π(Hg −1 ). The converse is similar. Now, we consider π(π(Hg1 ))Hg2 ) = π(Hg1 )π(Hg2 ). In order to establish ⊆ assume that (Hg)π(Hg 0 ) ∈ (π(Hg1 ))(Hg2 ) for some g 0 θg. It follows that g 0 = hg10 h0 g2 = g 00 g2 for some h, h0 ∈ H and g10 θg1 , where g 00 = hg10 h0 . Since θ is a conjugation, it follows that g = g3 g20 for some g3 θg 00 θg10 θg1 and g20 θg2 . Thus, Hg = Hg3 · g20 ∈ π(Hg1 ) · π(Hg2 ) as desired. A similar argument yields ⊇. This completes the proof of (1). It follows from (1), Lemmas 5.2.2 and 5.2.3 that (2) (G/H)∗∗ is a cogroup and G[H, π] is a polygroup. In (G/H)∗∗ , Hg1 ≈ Hg2 ⇐⇒ Hg1 ∈ π(Hg2 )H ⇐⇒ Hg1 = Hg20 h for some g20 θg2 and h ∈ H. Since θ is a conjugation and H = θ(e), g20 hθg20 , which implies that g1 θg2 . On the other hand g1 θg2 implies that (Hg1 )π(Hg), which yields Hg1 ≈ Hg2 . Thus, we obtain (3) Hg1 ≈ Hg2 if and only if g1 θg2 . By (3), θ = π (introduced in Lemma 5.2.5), so Lemma 5.2.5 yields G//θ ∼ = G[H, π] which completes the proof of theorem.  As noted before the previous two results yield a factorization of double quotients. Corollary 5.2.7. M ∈ Q2 (Group) if and only if M ∼ = G[H, π] for some subgroup H of a group G and Utumi partition π on G/H. Denote the class of all polygroups derived from weak cogroups (respectively, cogroups) by the construction in Lemma 5.2.2 as D(w-cogroup) (respectively, D(cogroup)). Then, Theorem 5.2.6 says Corollary 5.2.8. Q2 (Group)⊆ D(cogroup). We conclude by observing that all polygroups derived from weak

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cogroups are chromatic. Theorem 5.2.9. For every weak cogroup A, M = A/ ≈ is chromatic. Proof. Let C = {X ∈ M | X 6= {e}} and for X ∈ C let CX = {(a, b) ∈ A2 | a ∈ Xb}. It is not difficult to show that V =< A, Cx >x∈C is a color scheme. As a sample of the argument consider CX ∩ (CY k XZ ) 6= ∅ implies CX ⊆ CY k CZ . Suppose that a ∈ Xb, a ∈ Y c and c ∈ Zb for some c ∈ A. Then, a ∈ Y Zb, from which it follows that a ∈ ub for some u ∈ y · z, where y ∈ Y and z ∈ Z. Also, a ∈ xb for some x ∈ X. Thus, condition (5) of Definition 5.2.1 yields u ≈ x and so X ⊆ Y Z. Now, for any (r, s) ∈ CX , r ∈ Xs ⊆ Y Zs, which gives (r, s) ∈ CY k CZ as desired. The other conditions are verified in a similar way. In particular, the argument used to show (4) also shows that for X, Y, Z 6= {e}, X ∈ Y · Z (in MV ) if and only if CX ⊆ CY k CZ if and only if X ⊆ Y Z (in M). This is the key step in verifying the natural map from MV to M is an isomorphism.  5.3

Conjugation lattice

Comer described a few elementary properties of the lattice of conjugation relations of a group. A decomposition of a group into double cosets as well as its decomposition into ordinary conjugation classes give examples of conjugation relations. Comer considered conjugations derivable from subsystems of polygroups and techniques for creating other conjugation from these. A lot of information about a group is coded into its conjugation lattice. In this section, we study conjugation relations on polygroups. The main reference for this section is [10]. For a polygroup P let Conj(P ) denote the collection of all conjugation relations on P and let ConjS (P ) denote the collection of all special conjugations (see Definition 3.3.10). The smallest conjugation relation is the identity relation, denoted by δP and the largest conjugation relation is P 2 which is denotes by 1P . Let 1SP denote the special conjugation relation which identifies all elements of P different from the identity e. When the polygroup P is understood δ, 1 and 1S will be written instead of δP , 1P and 1SP .

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Definition 5.3.1. A partially ordered set (L, ≤) is a complete lattice if every subset A of L has both a greatest lower bound (the infimum, also called the meet) and a least upper bound (the supremum, also called the join) in (L, ≤). The meet of a and b is denoted by a ∧ b and the join by a ∨ b. Definition 5.3.2. A non-empty subset I of a complete lattice (L, ≤) is an ideal, if the following conditions hold: (1) for every x, y in I, x ∨ y in I; (2) for every x in I, y ≤ x implies that y is in I. The smallest ideal that contains a given element a is a principal ideal and a is said to be a principal element of the ideal in this situation. Proposition 5.3.3. If P is a polygroup, then (1) Conj(P ) forms a complete lattice whose join is the same as the join in the lattice of all equivalence relations on P ; (2) ConjS (P ) is the principal ideal in Conj(P ) determined by 1S . Proof. (1) It suffices to show, for every non-empty set S of Conj(P ) W that the join S of S (in the lattice of all equivalence relations on P ) W is again a conjugation. Suppose that z 0 ( S)z and z ∈ x · y. Then, zθ0 z1 θ1 z2 . . . θn−1 zn = z 0 for some z1 , . . . , zn−1 ∈ P and θ0 , . . . , θn−1 ∈ S. Because θ0 ∈ Conj(P ), z1 ∈ x1 · y1 for some x1 θ0 x and y1 θ0 y by Definition 3.3.10. Repeat for θ1 , . . . , θn−1 to obtain x1 , . . . , xn and y1 , . . . , yn such that xθ0 x1 θ1 . . . θn−1 xn , yθ0 y1 θ1 . . . θn−1 yn and zi ∈ xi · yi for all i ≤ n. Hence, W W xn ( S)x, yn ( S)y and z 0 ∈ xn · yn ; so the second condition of Definition W 3.3.10 holds for S. The verification of the first condition is routine. (2) It is obvious, since θ ∈ Conj(P ) is special if and only if θ ≤ 1S .  There are situations when it is desirable to regard a conjugation on P as a partition of P instead of an equivalence relation. Partitions and equivalence relations will be interchanged freely. Partitions will be written in the form {A; B; . . .}, where A, B, . . . are the blocks of the partition. The largest special conjugation relation 1SP denotes the partition {{e}; P \ {e}}. Notice that neither Conj(P ) nor ConjS (P ) is a sublattice of the partition lattice, in general, because the intersection of two conjugation relations is not necessarily a conjugation relation. In the following we give an example involving conjugations on S3 . Example 5.3.4. Let θ = {{0}; {(1 3)}; {(2 3), (1 2), (1 2 3), (1 3 2)}}

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and ϕ = {{e}; {(1 2}; {(2 3), (1 3), (1 2 3), (1 3 2)}}. Then, θ and ϕ are special conjugation relations on S3 , but θ ∩ ϕ = {{e}; {(1 3)}; {(1 2)}; {(2 3), (1 2 3), (1 3 2)}} is not a conjugation relation because (2 3)(θ ∩ ϕ)(1 2 3) and (1 2 3) = (1 2)((1 3) and (2 3) 6= (1 2)((1 3). The following lemma and the fact that S3 is generated by {(1 3), ((1 2)} shows that for θ and ϕ above, θ ∧ ϕ is the identity conjugation relation. Lemma 5.3.5. If G is a group and θ ∈ ConjS (G), then H = {x ∈ G | |θ(x)| = 1} is a subgroup of G. Proof. Clearly, e ∈ H and H is closed under inverses. If x, y ∈ H, then θ(xy) ⊆ θ(x)θ(y) ⊆ {xy}, so H is closed under products also.  Definition 5.3.6. An equivalence relation θ on a polygroup P is called a congruence relation if (1) θ is a regular relation (see Definition 2.5.1); (2) xθy implies x−1 θy −1 , for all x, y ∈ P . The lattice of all congruences on P is denoted by Con(P ). In the following example, we give a conjugation relation associated with a subpolygroup H of a polygroup P . Example 5.3.7. (1) For a subpolygroup H of P define a relation θH for x, y ∈ P by x θH y if and only if HxH = HyH. (2) If H is a subgroup of Aut(P ), define a relation θH for x, y ∈ P by x θH y if and only if σ(x) = y for some σ ∈ H. The relation θH is a conjugation relation and the relation θH is a special conjugation on P . In Theorem 5.3.8, we show that the conjugations in Example 5.3.7 (1) include all congruence relations. It also shows that congruence relations correspond to normal subpolygroups. Theorem 5.3.8. Suppose that P is a polygroup.

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(1) If θ ∈ Conj(P ) and N = θ(e), then N is a subpolygroup of P and θN ⊆ θ. (2) For an equivalence relation θ on P , θ ∈ Con(P ) if and only if θ = θN for some normal subpolygroup N of P . Proof. (1) It is clear that e ∈ θ(e) and θ(e) is closed under −1 . Now, suppose that z ∈ x · y, where x, y ∈ θ(e). Then, eθx and x ∈ z · y −1 so we conclude that e ∈ z 0 · y 0 for some z 0 θz and y 0 θy −1 . But in a polygroup, e ∈ z 0 · y 0 gives z 0 = (y 0 )−1 . So, zθz 0 = (y 0 )−1 θ(y −1 )−1 = yθe which shows that zθe. Thus, θ(e) is a subpolygroup of P . In order to show that θN ⊆ θ, suppose that xθN y, i.e., N xN = N yN . Then, x ∈ N xN = N yN ⊆ θ(e)θ(y)θ(e) = θ(y) since θ(e) is the identity of P//θ. Thus, xθy which completes the proof. (2) It is straightforward to show that θN ∈ Con(P ) whenever N is a normal subpolygroup of P . Now, suppose that θ ∈ Con(P ). In order to show θ ∈ Conj(P ) it suffices to verify the second condition of Definition 3.3.10. Suppose that z 0 θz and z ∈ x·y. Then, x ∈ z ·y −1 θz 0 ·y −1 and θ ∈ Con(P ). It follows that there exists x0 ∈ z 0 ·y −1 such that x0 θx which implies that z 0 ∈ x0 ·y. Hence, θ ∈ Conj(P ). Now, by part (1), θN ⊆ θ, where N = θ(e) is a subpolygroup of P . Suppose that xθy. Then, e ∈ x · x−1 and x · x−1 θy · x−1 . So, there exists z ∈ N with z ∈ y · x−1 . Thus, y ∈ z · x ⊆ N x which gives xθN y. Hence, θ = θN . It remains that to show that N is normal. If y ∈ N x, then xθy which implies that e ∈ x−1 · xθx−1 · y by using the definition. Thus, for some zθe, z ∈ x−1 · y which gives y ∈ x · z ⊆ xN . Therefore, N x ⊆ xN . The other inclusion is similar. So, it follows that N is normal.  By Example 5.3.7 (1), a conjugation relation θN is associated with every subpolygroup N of P , not just the normal ones. The following summarizes a few properties of this embedding. Proposition 5.3.9. Let P is a polygroup. (1) The map N 7→ θN embeds the lattice of subpolygroups of P into Conj(P ) as a join semilattice, note that a join semilattice is a partially ordered set which has a join for any non-empty finite subset. (2) The image of the map in (1) has only θ{e} = δ in common with ConjS (P ). In particular, Con(P ) ∩ ConjS (P ) = {δ}. (3) If G is a group, Con(G) is a sublattice of Conj(G).

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Proof. (1) Suppose that H and K are subpolygroups of P and < H, K > is the subpolygroup generated by H and K. It suffices to show that θH ∨θK = θ . If H and K are comparable, say H ⊆ K, then < H, K >= K and θH ∨ θK = θK = θ clearly holds. Assume that H and K are not comparable. Then, H ⊆ (θH ∨ θK )(e) and K ⊆ (θH ∨ θK )(e). So, < H, K >⊆ (θH ∨ θK )(e). By Theorem 5.3.8 (1), θ ≤ θH ∨ θK . Since the other inclusion is clear, equality holds. (2) If θH ∈ ConjS (P ), then H = {e} by Theorem 5.3.8 and θH = δ. (3) By a standard group theory argument θH ∩ θK = θH∩K . Hence, θH ∩ θK is a conjugation relation which equals θH ∧ θK .  By Theorem 5.3.8 (1) the block θ(e) of a conjugation relation θ is a subpolygroup. A new conjugation relation θ[ϕ] may be obtained from θ by replacing θ(e) block by the blocks of conjugation relation ϕ ∈ Conj(θ(e)). More precisely, Definition 5.3.10. Let P be a polygroup. For θ ∈ Conj(P ) and ϕ ∈ Conj(θ(e)), the ϕ-split of θ is an equivalence relation θ[ϕ] on P defined by  θ(x) if θ(x) 6= θ(e) θ[ϕ](x) = ϕ(x) if θ(x) = θ(e). Proposition 5.3.11. θ[ϕ] ∈ Conj(P ). Moreover, P//(θ[ϕ]) is isomorphic to (θ(e)//ϕ)[P//θ], the polygroup extension of θ(e)//ϕ by [P//θ] (see Section 3.2). Proof. In order to verify the first statement it is enough to show that the product of two θ[ϕ]-blocks is a union of θ[ϕ]-blocks. Along the way we develop a rule for computing the product of two θ[ϕ]-blocks from which the isomorphism is apparent. Let θ[ϕ] = ψ for short. The first two cases are obvious from the definition of θ[ϕ]: (1) ψ(x)ψ(y) = ϕ(x)ϕ(y) if θ(x) = θ(y) = θ(e), (2) ψ(x)ψ(y) = θ(x)θ(y) if θ(x), θ(y) 6= θ(e). When computing these products, replace θ(e) by {ϕ(x) | x ∈ θ(e)}. For the other cases, (3) ψ(x)ψ(y) = θ(y) if θ(x) = θ(e) 6= θ(y), (4) ψ(x)ψ(y) = θ(x) if θ(y) = θ(e) 6= θ(x). In order to verify (3), first note that ϕ(x)ϕ(y) ⊆ θ(e)θ(y) = θ(y). Now, suppose that y 0 θy. Then, y 0 ∈ x · z for some z. So, y 0 ∈ xθ(z) ⊆ θ(e)θ(z) = θ(z). Then, θ(y) ∩ θ(z) 6= ∅ so θ(z) = θ(y) which gives θ(y) ⊆ xθ(y) ⊆

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θ(x)θ(y) as desired. The proof of (4) is similar. Thus, θ[ϕ] is a conjugation relation. The isomorphism is established by comparing (1), (2), (3) and (4) with the definition of the product on the polygroup (θ(e)//ϕ)[P//θ]. . For ϕ ≤ ψ in Conj(P ) let [ϕ, ψ] denote the universal {θ | ϕ ≤ θ ≤ ψ} in Conj(P ). The map ϕ 7→ θ[ϕ] immediately gives: Corollary 5.3.12. Conj(θ(e)) is isomorphic to the interval [θ[δθ(e) ], θ] in Conj(P ). For a subset X of a polygroup such that e ∈ X we let X ∗ = X \ {e}. There is one splitting of a conjugation relation θ that deserves special attention. Namely, for a conjugation relation θ let θS = θ[1Sθ(e) ], where 1Sθ(e) is the unity element in ConjS (θ(e)). In other words, by Proposition 5.3.11, θS is a special conjugation relation obtained from θ by splitting θ(e) into the two classes: {e} and (θ(e))∗ . A few elementary properties of θS are given below. Proposition 5.3.13. Let P be a polygroup. For θ, ϕ ∈ Conj(P ), (1) (2) (3) (4) (5)

θS = θ if θ is special and θS = θ ∩ 1S if θ is not special; θS ≤ θ; θ covers θS in Conj(P ) if θ is not special; ϕ ≤ θ implies ϕS ≤ θS ; θ is determined by θS and θ(e). Namely, θ = θN |θS , a commuting join, where N = θ(e).

Proof. We prove (5). Suppose that xθy. If θ(x) 6= θ(e), then θS (x) = θ(x) so xθN xθS y and if θ(x) = θ(e) (= N ), then xθN yθS y. Therefore, θ ≤ θN |θS .  Information about the structure of Conj(P ) can be obtained from Proposition 5.3.13 (5). For example, if G is an abelian group, every θ ∈ Conj(G) is a join of congruence relation and a special conjugation relation. Lemma 5.3.14. If h is a join retract of a lattice L onto an ideal L (i.e., h : L −→ L satisfies h(x ∨ y) = h(x) ∨ h(y), h(x) ≤ x, h(h(x)) = h(x) for all x, y ∈ L and h(L) is an ideal of L), then h is a homomorphism. Proof. Since h preserves order, h(x ∧ y) ≤ h(x) ∧ h(y). If z ≤ h(x) ∧ h(y), then z ≤ h(x) ≤ x and z ≤ h(y) ≤ y so z ≤ x∧y. Hence, z = h(z) ≤ h(x∧y)

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because z ≤ h(x) ∈ h(L) implies z ∈ h(L) and h fixes elements of h(L). Thus, h(x) ∧ h(y) = h(x ∧ y).  Proposition 5.3.15. The map θ 7→ θS is a lattice homomorphism of Conj(P ) onto ConjS (P ). Proof. Applying Lemma 5.3.14, by Proposition 5.3.13, it suffices to show the map preserves joins. Since ≤ is preserved by Proposition 5.3.13 (4) we only need to show that θ ∨ ϕ)S ≤ θS ∨ ϕS in Conj(P ). Suppose that x(θ ∨ ϕ)S y and x, y 6= e. Then, there exists a sequence x = x0 , . . . , xn = y such that x0 θx1 ϕx2 . . . xi−1 θxi ϕxi+1 . . . xn . If xj θxj+1 and xj , xj+1 6= e, then xj θS xj+1 and similarly for ϕ. Hence, we may assume xi = e and xi−1 , xi+1 6= e for some i. Then, xi−1 ∈ H = θ(e) and xi+1 ∈ K = ϕ(e). If xi+1 ∈ H, then xi−1 θxi+1 θxi+2 and we may drop e = xi from the sequence. Hence we may assume xi+1 6∈ H. Then, e 6∈ xi−1 · xi+1 because if so, xi+1 = x−1 Choose x0i ∈ xi−1 · xi+1 . Then, i−1 ∈ H. x0i ∈ Hxi+1 ⊆ Hxi+1 H ⊆ θ(xi+1 ) and x0i ∈ xi−1 K ⊆ Kxi−1 K ⊆ ϕ(xi−1 ). So, xi−2 ϕxi−1 ϕx0i θxi+1 θxi+2 which means we can shorten the sequence from x0 to xn and eliminate the term xi = e. Repeating the above for all xi = e we obtain x = x00 θx01 ϕ . . . x0m = y where all x0i 6= e. Therefore, (x, y) ∈ θS ∨ ϕS .  For θ, ϕ ∈ Conj(P ) and ϕ ⊆ θ, we define a conjugation relation θ//ϕ on P//ϕ by ϕ(x) (θ//ϕ) ϕ(y) ⇔ xθy, for all x, y ∈ P . The first part of the following theorem gives a lattice version of the first isomorphism theorem from group theory. Theorem 5.3.16. Let ϕ ∈ Conj(P ). (1) The map θ 7→ θ//ϕ is an isomorphism of the interval [ϕ, 1] in Conj(P ) onto Conj(P//ϕ). (2) θ//ϕ is special in Conj(P//ϕ) if and only if θ(e) = ϕ(e). Moreover, the map in (1) is an isomorphism of the interval [ϕ, ϕ] in Conj(P ) onto ConjS (P//ϕ). (3) The map θ 7→ θS is an isomorphism of [ϕ, ϕ] in Conj(P ) onto [ϕS , ϕS ] in ConjS (P ). (4) If ϕ is not special, then the map θ 7→ θS is an isomorphism [ϕ, 1] ∼ = [ϕS , 1S ]. S (5) If N is a subpolygroup of P , then ConjS (N ) ∼ ], a sub= [θN [δ], θN lattice of ConjS (P ).

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Proof. (1) It is a tedious but straightforward argument. (2) We have θ//ϕ is special ⇔ [ϕ(x) (θ//ϕ) ϕ(e) ⇒ ϕ(x) = ϕ(e)] ⇔ [xθe ⇒ xϕe] ⇔ θ(e) ⊆ ϕ(e). But ϕ(e) ⊆ θ(e) always holds since ϕ ⊆ θ. Since θ ∈ [ϕ, ϕ] if and only if ϕ(e) ⊆ θ(e), the restriction of the map in (1) gives the desired isomorphism. (3) If ϕ is special, θS = θ for θ in [ϕ, ϕ] ⊆ ConjS (P ). Assume that ϕ(e) 6= {e}. Since θ 7→ θS is a lattice homomorphism it suffices to show the map is one to one and onto. Suppose that θ1 6= θ2 in [ϕ, ϕ]. Since θ1 (e) = ϕ(e) = θ2 (e), there exists x 6∈ ϕ(e) such that θ1 (x) 6= θ2 (x). By Proposition 5.3.13 (1), θ1S (x) 6= θ2S (x). So, the images are distinct and thus the map is one to one. Now, assume that θ is in [ϕS , ϕS ]. Define θ+ = {ϕ(e); θ(x1 ); . . .}, where θ = {e; (ϕ(e))∗ ; θ(x1 ); . . .}. Since (θ+ )S = θ it suffices to show that θ+ is a conjugation relation. If xθ+ y, then it is clear that x−1 θ+ y −1 since ϕ(e)−1 = ϕ(e) and θ(xi )−1 = θ(x−1 i ) for all i. Thus, the first condition of Definition 3.3.10 holds. In order to verify the second condition, we need to show θ+ (x)θ+ (y) is a union of θ+ -blocks. First, we show ϕ(e)θ(xi ) = θ(xi ). This holds because ϕS ⊆ θ and θ(xi ) 6= ϕ(e) implies that θ(xi ) is a union of ϕ-blocks θ(xi ) = ϕ(xi ) ∪ . . . ∪ ϕ(x0i ) ∪ . . . and ϕ(e)ϕ(x0i ) = ϕ(x0i ) for each component ϕ(x0i ). It remains to see that θ(xi )θ(xj ) is a union of θ+ -blocks. For this it suffices to show (ϕ(e))∗ ⊆ θ(xi )θ(xj ) ⇔ e ∈ θ(xi )θ(xj ). Choose x ∈ (ϕ(e))∗ . Then, x ∈ x0i · x0j for some x0i θxi , x0j θxj . Since θ(x) = ϕS (x) ⊆ ϕS (x0i )ϕS (x0j ) = ϕ(x0i )ϕ(x0j ), eϕx, and ϕ is a conjugation relation, e ∈ (ϕ(x0i )ϕ(x0j ) ⊆ θ(xi )θ(xj ). The proof of the converse is similar. Thus, θ+ ∈ Conj(P ) which completes the proof of (3). (4) It holds by an argument similar to (3). (5) It follows from Corollary 5.3.12 and the observation that θN [ϕ] is special if and only if ϕ is special.  In Theorem 5.3.16, if ϕ ∈ ConjS (P ), then the homomorphism is, in general, not one to one.

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Index

nilpotency, 29, 133 nilpotent, 29 cogroup, 176 weak, 175 color algebra, 163 scheme, 163 regular, 166 commutative strongly, 139 weak, 139 commutator, 27, 124 left, 124 right, 124 complete closure, 60 congruent to, 9 conjugate, 119 convex set prime, 75 coset, 76 double coset algebra, 84 left, 9 right, 9

Hv -group, 139 Hv -homomorphism, 140 automorphism, 140 feebly quasi-canonical, 143 feebly canonical, 143 good homomorphism, 140 inclusion homomorphism, 140 isomorphism, 140 left completely reversible, 143 reversible left, 143 right, 143 right completely reversible, 144 strong homomorphism, 140 weak homomorphism, 140 Hv -semigroup, 139 γ-closure, 70, 101 action kernel, 109 associative general, 6 weak, 139, 148 automorphism generalized, 128

element e-conjugate, 176 infinite order, 10 order, 10 single, 142, 154 equivalent 0-equivalent, 49 e-equivalent, 49

canonical map, 12 core, 97 centralizer, 8 class fundamental, 115 197

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Polygroup Theory and Related Systems

i-equivalent, 49 operationally, 49 fundamental properties, 116 group, 4 abelian, 6 act, 23 alternative, 19 automorphism, 17 automorphism of a graph strongly transitive, 168 commutative, 6 cyclic, 6 direct product, 22 fundamental, 56, 97, 153 general linear, 7 homomorphic, 14 homomorphism, 14 kernel, 14 inner automorphism, 17 integers modulo n, 6 isomorphic, 14 isomorphism, 14 Klein’s four, 22 nilpotent, 28 operator, 23 order, 6 orthogonal, 7 permutation, 19 product, 10 quaternion, 7 quotient, 12 semidirect product, 24 solvable, 25 special linear, 7 special orthogonal, 7 symmetric, 19 homomorphism action, 23 hypergroup, 38 D-hypergroup, 176 KH -hypergroup, 67 P -hypergroup, 38 γn -complete, 62

γn∗ -complete, 65 τ -multisemi-direct hyperproduct, 129 e-reduced, 49 i-reduced, 49 n-complete, 62 o-reduced, 49 canonical, 79 complete, 61 derived, 99 flat, 62 good multihomomorphism, 128 heart, 56 homomorphism, 43 good, 43 inclusion, 43 of type 1, 45 of type 2, 45 of type 3, 45 of type 4, 45 of type 5, 51 of type 6, 51 of type 7, 51 inversible, 61 isomorphic, 43 isomorphism, 43 reduced, 49 regular, 61 total, 38 hypergroupoid, 38 scalar partition, 176 hypermatrix, 113 hyperoperation, 38 hyperring, 112 additive, 113 inclusion homomorphism, 117 multiplication, 113 strong homomorphism, 117 identity, 5, 61, 74 inverse, 5, 61 join space, 74 affine, 75 descriptive, 76 factor, 77

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Index

homomorphism good, 77 kernel, 77 isomorphism, 77 ordered join geometry, 76 projective, 76 spherical, 76 lattice complete, 181 ideal, 181 principal element, 181 principal ideal, 181 join, 181 meet, 181 linear space generated, 74 spanned, 74 normalizer, 8 part γ-part, 59, 101 complete, 58 permutation, 18 cycle, 19 length, 19 transposition, 19 disjoint, 19 generalize, 105 odd, 19 product, 18 subpermutation, 106 universal generalized, 106 points, 74, 80 linear, 74 lines, 80 scalar, 74 poly-monoid, 107 polygroup, 83 P -isomorphism, 110 P -map, 110 τ -direct hyperproduct, 130 action, 107 character, 85 chromatic, 164

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199

conjugacy class polygroups, 84 core, 174 derived from, 176 direct hyperproduct, 86 extension, 86 homomorphism good, 92 inclusion, 92 of type 2, 92 of type 3, 92 of type 4, 93 strong, 92 hypermatrice inclusion representation, 113 representation, 113 intransitive, 111 isomorphic, 110 isomorphism, 93 nilpotent, 132 normalizer, 125 orbit, 110 perfect, 126 permutation, 108 stabilizer, 111 proper, 133 semidirect product, 104 solvable, 126 symmetric, 83 transitive, 111 ultragroup, 175 weak, 148 automorphism, 151 commutative, 156 direct product, 154 inclusion homomorphism, 151 isomorphism, 151 quotient, 152 semidirect hyperproduct, 155 strong homomorphism, 151 weak homomorphism, 151 Prenowitz algebra, 84 product of type zero, 131 quasihypergroup, 38 quotient structure, 46

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Polygroup Theory and Related Systems

relation congruence, 182 equivalence ϕ-split, 184 (full) conjugation, 91, 150, 164 of type 4, 47 fundamenta, 97 fundamental, 56, 115, 117, 140, 153 of type 1, 47 of type 2, 47 of type 3, 47 regular, 53 regular on the left, 53 regular on the right, 53 special, 164 special conjugation, 91 strongly regular, 53 strongly regular on the left, 53 strongly regular on the right, 53 largest conjugation, 180 smallest conjugation, 180 transitive closure, 99 reproduction axiom, 38 ring fundamental, 115 semihypergroup, 38 complete, 60 subsemihypergroup, 40 semihyperring, 113 semilattice join, 183 median, 75 series central, 29 factors of, 175 generalized ascending central, 135 generalized descending central, 133 lower central, 31 normal, 25 solvable, 25 upper central, 31 set partial identities, 42

rough, 39 subgroup, 7 commutator, 27 cyclic, 9 derived, 27 generated by a subset, 9 image, 15 index, 10 inverse image, 15 maximal, 9 normal, 11 trivial, 9 subhypergroup, 40 closed, 41 closed on the left, 40 closed on the right, 40 conjugable, 41 conjugable on the left, 40 conjugable on the right, 40 invertible, 41 invertible on the left, 40 invertible on the right, 40 ultraclosed, 41 ultraclosed on the left, 40 ultraclosed on the right, 40 subpolygroup, 89 derived, 125 generated, 124 normal, 89 Utumi partition, 177

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