Structural Induction on Partial Algebras, II: Introduction to Theory and Application to Partial Algebras [Reprint 2021 ed.] 9783112529041, 9783112529034

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MATHEMATICAL MATHEMATISCHE RESEARCH FORSCHUNG

Structural I n d u c t i o n o n Partial A l g e b r a s

H. Reichel

Band 18 Akademie-Verlag • Berlin

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H. Reichel Structural Induction on Partial Algebras

Mathematical Research • Mathematische Forschung Wissenschaftliche Beiträge herausgegeben von der Akademie der Wissenschaften der DDR Institut für Mathematik

Band 18 Structural Induction on Partial Algebras by

H. Reichel

Structural Induction on Partial Algebras Introduction toTheory and Application of Partial Algebras-Part II by Horst Reichel

Akademie-Verlag • Berlin 1984

Autor: Prof. Dr. Horst Reichel Technische Hochschule "Otto von Guericke" 3010 Magdeburg Boleslav-Bierut-Platz 5

Die Titel dieser Schriftenreihe werden vom Originalmanuskript der Autoren reproduziert.

ISSN 0138 - 3019

Erschienen im Akademie-Verlag, DDR-1086 Berlin, Leipziger Str. 3 (C) Akademie-Verlag Berlin 1984 Lizenznummer: 202 • 100/405/84 Printed in the German Democratic Republic Gesamtherstellung: VEB Kongreß- und Werbedruck, 9273 Oberlungwit Lektor: Dr. Reinhard Höppner Umschlag: Karl Salzbrunn LSV 1025 Bestellnummer: 763 363 0 (2182/18) 02600

P R E F A C E

O u r purpose in writing this book is a n Increasing use of partial algebras in computer science. W e hope to achieve this aim by the development of a special kind of partiality, the so-called equational

partiality,

as a sound mathematical basis for algebraic specifications using p a r tial operations and as one of several possible generalizations of the Universal Algebra to partial algebras. The book presents both the developed theory of equational

partiality

and its applications in algebraic specifications. The most Important concepts with respect to applications are the notions of a canon and of the recursive dependence within canons. The usefulness of partial

al-

gebras is proved by the fact that structural induction alone yields on partial algebras a complete formal calculus of abstract programs

on

abstract parameterized data types. The presentation of the complete theory o f ^ a r t l a l algebras is out of the range of this book. Therefore the author cooperates with Peter Burmeister in writing an "Introduction to Theory and Application of Partial Algebras" consisting of two p a r t s : Part I : "A Model Theoretic Approach to Partial Algebras" by Peter Burmeister , and Part II:"Structural Induction on Partial Algebras" by Horst Reichel. The content of the present book is the result of a long Joint work with Klaus Benecke, Ulrich L . Hupbach, and Heinz Kaphengst, a n d I am very thankful to these colleagues for that collaboration. I would also like to thank Rod Burstall and D o n Sannella for helpful discussions and suggestions during my Visiting Fellowship at' Edinburgh University made possible by a research grant of S E R C . Thanks are due to Wolfgang W e c h l e r for careful proof-reading end useful comments concerning the manuscript. Finally, I want to thank the Akademie-Verlag for friendly cooperation and quick

publication.

Magdeburg, October 1983

H.Reichel

S

C O N T E N T S

INTRODUCTION

9

1 . S P E C I F I C A T I O N OF A B S T R A C T DATA T Y P E S W I T H

PARTIAL

OPERATIONS 1.1. Abstract

13 data

types

- ADT's

1.2. Parameterized abstract 1 . 3 . The d e f i n i t i o n language

of

data

13 types

the-semantics

over a r b i t r a r y

- PADT's

of a

functional

P A D T ' s a s a complex example

2 . EQUATIONAL P A R T I A L I T Y 2.1. Set-theoretical

51 64

notions

65

2.2. Many-sorted algebras 2.3. Equationally

30

partial

67 heterogeneous

algebras

80

2.4. Hierarchy conditions

87

2.5. hep-varieties

97

and h e p - q u a s i - v a r i e t i e s

3 . P A R T I A L A L G E B R A S D E F I N E D BY GENERATORS AND R E L A T I O N S 3.1. Derivability

of

3 . 2 . Free p a r t i a l

algebras

3.3. Partial

quotient

elementary implications

homomorphisms

3.5. Relatively

free p a r t i a l

4 . CANONS -

126 136

algebras

146

I N I T I A L L Y RESTRICTING ALGEBRAIC THEORIES

4 . 1 . Canon model c l a s s e s 4 . 2 . C o m p u t a b i l i t y and d e f i n a b i l i t y abstract

data

types

4 . 3 . Canon morphisms

107 119

algebras

3 . 4 . Theorem of

106

158 159

on

parameterized 173 187

REFERENCES

196

T A B L E OF N O T A T I O N S

199

INDEX

202

7

INTRODUCTION

The main purpose of the book is the initiation of an increasing use of partial algebras in computer science. There are many reasons for the inadequate state of application of partial algebras. Probably, the bad reputation of the theory of partial algebras itself

causes that inad-

equate state. The theory of partial algebras was initiated for mathematical

rea-

sons more or less independently by 0 . Schmidt and his group and by 0. Slominski. In the years from 1965 to 1973 many important

results

were contributed by P . Burmeister, R. Kerkhoff, H. Hoft and V.S. Poythress. But this first and purely mathematical impulse did not initiate a continuously increasing development of the theory of partial algebras. The first results showed that a straightforward generalization of

the

theory of total algebras was not to be supposed. Many different generalizations of the concept of homomorphism, of substructure, and

of

validity of term equations were introduced, and it was not quite clear which of them would be the right ones. At this time there were not sufficiently many examples of partial algebras by means of which one could decide these questions. Meanwhile the extensively developed theory of small categories and to a greater extend the application of universal algebra in computer science act as motives for the development of the theory of partial algebras. In the present Part II of the 'Introduction to Theory Application of Partial Algebras'

we present a special theory of

and par-

tial algebras mainly oriented towards satisfying the requirements

of

applications of universal algebra in computer science and of the theory of small categories. The variety of possible generalizations of

the

theory of total algebras will be thoroughly discussed in Part I

' A

Model Theoretic Approach to Partial Algebras' by P. Burmeister. Next we try to explain the basic idea of the presented theory of 'EQUATIONAL PARTIALITY 1 . One reason for the frequent application of algebra in computer science is ths use of structural induction on free algebraic structures. Very often one has to define a mapping in a finite manner, where

M

is an infinite but finitely

f:M—» P

representable

set. A typical example of this problem is the definition of the semantics of a programming language. In this case the set all syntactically correct programs and

P

M

consists

of

is a set of state transfor-

mations of an abstract intarpreting machine. The finite representation of

M

is given by a phrase structure grammar.

9

A n application of structural Induction to define a function f: M

•

P

may be sketched in the following way: at first one has to

find a finite algebraic theory carrier set of an initial

T

said to be initial if for every fs A

— *

f: M — » •

B P

in such a w a y that

T-algebra

H

, where a

T-algebra

E

M

becomes

T-algebra

A

the is

exactly one homomorphism

exists. In the second step of the inductive definition of we have to define on the set

P

any

T-algebra

P

so

that the desired function is given by the uniquely determined homomorphism

f: M — •

T .

Intuitively a finite representation of an infinite set of a finite algebraic theory algebraic theory a finite set

&

T

T

M

by means

can be understood as follows: A finite

is given by a finite set

of operations and by

of axioms describing which combinations of the funda-

mental operations produce the same value. Among the fundamental operations the nullary operations have the function of atomic generating elements of

M , so that any element of

M

can be represented as

value of a finite combination of non-nullary operations applied

the to

atomic generating elements. If any two such combinations produce the same value then this must be a consequence of the axioms. In the simplest case the theory

T

can be chosen in such a way that the set

of axioms is empty. If we take the simplest non-trivial case, given by one atomic generating element, by one unary generating operation, and by an empty set of axioms, we get exactly Peano's axiomatization of natural numbers. Therefore, mathematical induction on natural numbers is a special case of structural

induction.

In the case of formal languages any production rule of a context free grammar, containing no non-terminal in its right hand side, can be Identified with a constant operation, or an atomic generating

element,

and any other production rule can be interpreted as a generating operation. The existence of different non-terminals corresponds in algebraic terms to different kinds of carrier sets, e.g. we have to use many-sorted or heterogeneous algebras. Combinations of generating

ope-

rations correspond to derivation trees. If the grammar is unambiguous the sets of derivable strings form a free heterogeneous algebra, so that the semantic meaning can be defined by structural induction. The definition of the algebra

B

, which is the range of the semantic

interpre-

tation itself, corresponds in the context of programming languages

to

a description of an interpreter by giving a semantic interpretation to every production rule. More detail in this direction can be found

in

the paper of Knuth (1968) or in a paper of Letitshevski (1968) and in the paper 'Initial Algebraic Semantics* of the ADO group (1975). H o w . ever, the syntactic structure of existing programming languages cannot be

10

completely described In an adequate w a y by context-free grammars.

In

general, most production rules are restricted by so-called context conditions. Algebraically we have to Interpret a production rule with

a

context condition as a partial operation with a necessary and sufficient domain condition. The same phenomenon can be observed in small categories. Small categories have exactly one partial operation, namely the partial composition of morphisms. A n d again, the partial binary operation is equipped with a necessary and sufficient domain condition. The product x«y

exists if and only if the target object of

source object of

x

is equal to the

y .

According to these observations in the theory of equationally partial algebras we equip each operator with a necessary and sufficient domain condition, which is given by a finite set of term equations. A total operator Is then an operator equipped with a trivial domain condition, for instance with the empty set of term equations. W e shall develop the theory of equationally partial structures in such a way that the notions and results of total algebras will be reproduced if every operator has a trivial domain condition. W e shall prove that a certain syntactic hierarchy condition for the domain conditions signatures (or of heterogeneous operator schemes) is necessary and sufficient for a nice and simple theory of equational partiality. W i t h approach we can describe the Inductive power of production rules of real programming The application of structural

this

context-restricted

languages.

Induction on partial algebras will

be demonstrated in Chapter one by specifications of several abstract data types and of parameterized abstract data types containing

partial

operations. W e present these applications of structural induction

on

partial algebras without any preceding theoretical explanations in order to show that those specifications are not significantly harder then specifications using total operations only. In the following three chapters we develop thoroughly the algebraic foundations of structural induction, of parameterized abstract data types with partial operations, and of functional enrichments of parameterized abstract data

types.

In Chapter two we develop the algebraic theory of equational partiality. The basic notions are hierarchical equationally partial signatures (hep-signatures), hep-signatures with elementary

implica-

tions as axioms, and finally hep-quasi-varieties as classes of models of hep-signatures with elementary implications. It will be shown that

11

due to the hierarchy condition of the eyetem of domain conditions the weakest concept of homomorphism between partial algebras 19 sufficient for the development of a sound theory of equationally partial algebras that for instance completely reflects the structural behaviour of small categories. In this respect Chapter two mainly contributes to the Universal Algebra of partial algebras. Chapter three develops the algebraic foundations of structural induction on partial algebras. It is mainly based on the existence of partial algebras that are freely generated by sets of generators and sets of defining relations on the generating elements. This concept, well-known in the theory of groups, is the algebraic key for a' powerful and flexible approach to structural induction which works as well for total algebras as for partial algebras. This concept can also be used for constructions of partial quotient algebras. The theorem of homomorphisms in Section 3.4 represents the most significant difference in the structural behaviour of total algebras and of equationally partial algebras . In, Chapter four we deal with classes of equationally partial algebras that are exlusively motivated by semantic problems of parameterized abstract data types and by functional enrichments of parameterized abstract data types. These model classes are defined by so-called CANON's , or by initially restricting algebraic theories. On the basis of this concept we define computable enrichments and definable enrichments of canons

and prove some necessary conditions for definable and

for computable enrichments of canons. In the last section we show how canons and canon morphisms can be used for the definition of the semantics of constructions that compose complex

specifications from simpler

ones. Finally we consider constructions like instantiations of parameterized abstract data types that preserve computable enrichments of canons.

12

CHAPTER

1

SPECIFICATIONS OF ABSTRACT DATA TYPES WITH PARTIAL OPERATIONS 1.1

Abstract Data Types

-

ADT's

In connection with investigations of modularity concepts of large software projects the understanding of the data type concept was essentially modified. W h e r e a s in programming languages like A L G O L 60, PL/1, and PASCAL a data type is considered to be a set, at least since

the

early seventies it seems to be generally accepted that data types are not just sets, but sets equipped with operations. For instance, if

we

deal with strings the following operations can be considered to be

rel-

evant: r

equal(String,String)

— B o o l

append(String, Character)

— S t r i n g

interlock(String,String)

— S t r i n g

contained(Character.String) length(String) empty

— •

—•>

Bool

Nat

— S t r i n g

Here we see that four different kinds of objects occur which are .characterized by the type names 'Bool', 'String 1 , 'Nat', and

'Characters',

respectively. This situation can algebraically be represented by 4-sorted algebra A

Bool

=

A

• ANat

=

"

(set

Characters = < a ' b ' c ^Characters

'

ar,

a

consisting of four carrier sets , namely natural numbers), •

and

A

string =

8et

of

d which consists of six fundamental

equalA: A s t r ± n g X A s t r l n g

—

appendA: A g t r i n g X A c h a r g c t e r s interlock: A s t r ± n g X A S t r i n g

a11

strings over

operations:

A0ool — -

Agtring Agtring

A

contained

X AStplng

lengthA: A s t r i n g

—

emptyA: {0}

AString

Depending on

: Acharacterg

on

— *

A

ABool

Nat

the choice of the actual alphabet, i.e. depending

the interpretation of the type name 'Characters', we would

different concrete data

get

types.

13

The example demonstrates that a concrete data type can be identified with a many-sorted algebra. But this understanding is not sufficient t«^ put into practice the suggestion of Hoare (1972) : "In the development of programs by stepwise refinement the programmer is encouraged to postpone the decision on representation of his data until after he has designed his algorithm and has expressed it as an abstract program operating on abstract data." The problem consists in finding a precise definition of an abstract data type and of an abstract program operating on an abstract data type. The fundamental idea in abstract data type specifications is the mathematical presentation of an abstract data type by a class of behaviourally equivalent many-sorted algebras. In the literature on abstract data types there are several different definitions of behavioural equivalence. Here we use the mathematically most simple version of equivalence which considers many-sorted algebras as behaviourally equivalent if they are isomorphic. In Chapter four

we will make precise the no-

tion of an abstract program operating on an abstract data

type.

However, there are many very different interpretations of the preceding scheme

(») which have nothing in common with the intended model

of strings over an alphabet. Although schemes of the form

(*)

are

used in the programming language ADA as specification part of a package it has no semantics. In

ADA the semantics of a package is given by the

implementation, or by the body of the package. This approach does

not

represent the behaviour of an abstract data type by the description of the properties of the available manipulation functions, but by giving a representant of the behavioural class defined in terms of

other ADA

packages and of the basic types and basic functions of A D A . A

disadvan-

tage of this approach can be illustrated by the following example. If the natural numbers w o u l d be represented by the special model of Roman numbers, i.e. I,11,111,IV,V,VI,...

, everybody sees the difficulties in

discovering the distrlbutlvlty law of addition and multiplication of natural numbers. Therefore we prefer axiomatic tools in descriptions of the external behaviour of abstract data types. Since our aim is the demonstration of the usefulness and

feasibility

of structural induction on partial algebras we will not explain the fundamentals of algebraic specifications of abstract data types. Readers who are not familiar with the basic ideas should consult the paper 'Introduction to Data Algebra* by S.N. Zllles in LNCS 86 (198o) or 'A Look at Algebraic Specifications' by the same author (see S . N . Z l l les (1981)) .

14

Coining back to the preceding example one can easily check that the intended interpretation satisfies the following properties expressed by the equations; length(empty) • 0 length(append(w,x))« length(w) + 1 interlock(w,empty) = w interlock(w,append(v,x)) where

w,v

" append(interlock(w,v),x)

are variables for strings and

x

is a variable for

characters. The next

interpretation of the scheme

(*) shows that the system

(mm) of equations is not strong enough to exclude all those interpretations which are not isomorphic to the intended interpretation. Let the interpretation B

B

" length B (f) = f(x) Q

equal (f.g) »

B

'

'»"ctions

with

B

strings. Evidently,

Nat ° ANat "

Characters

set

The interpretation

0

•

h(x) - f(x) • g(x), x 6 B c h a r a c t e r s f(x) f 0

iff

for x e B_. iff f(x) - 0 Characters overall xe B c h a r a c t e r s

T

iff

f(x)-8(x)

forali

F

iff

f(x)+g(x)

for some

x eBcharacters xe B c h a r a c t e r 8

represents multi-sets and does not

A

and

B

B

but not by

,

represent

are not isomorphic since there is

equation which is satisfied by append(append(w,x),y)

,

an

A , namely

= append(append(w,y),x).

This example indicates that we need an additional tool for excluding undesirable

interpretations.

This additional exclusion of interpretations will be done by the concept of

equations, if of (««) G

of axioms. Informally we can say

"initial satisfaction

an interpretation C

C

of

(*)

initially satisfies the set

that

(*»)

of

satisfies only those equations which are consequences

. W h e r e an equation

tj = t 2

of equations if every algebra

D

is called a consequence of a set satisfying

each equation

in

G

15

also satisfies

"

* Interpretation

B

shows that the equation

append(append(w,x),y) = append(append(w,y),x) is not a consequence of

(*•), so that

the set of equations given by

B

does not initially satisfy

(«•).

A s next we will make the concept of initial satisfaction

more pre-

c i s e . Readers wanting a more detailed explanation should look

into the

paper 'An Initial Algebra Approach to the Specification, Correctness, and Implementation of Abstract Data Types' of the ADO group ( see AD3

(1978)). In terms of universal algebra one can say that an algebra

scheme

(«)

initially satisfies

scheme

(•)

satisfying

fJ C

• D .

(••)

(««)» iff for every

C

algebra

of D

of

there is exactly one homomorphism

Obviously any two algebras are isomorphic if they ini-

tially satisfy

(**) . This shows that initial satisfaction

fits for

specifications of isomorphism classes or in other words for definitions of abstract data types. But, it was shown by Bergstra and Meyer (1983) that there are very simple abstract data types that are not specifiable by finite sets of equations. They show that the set of all finite subsets of natural numbers with the cardinality function may serve as such a counterexample. A finite specification of that data type requires conditional equations of the form «1 "

r

e

l

n "

r

n

— *

« - r .

A s next we introduce the syntactic form for writing

specifications

of abstract data types by means of a simplified version of strings. STRING

is definition sorts

Nat, String, Characters

oprn

zero

> Nat

succ(Nat)

*• Nat

a,b,...,y,z

— •

Characters

append(String .Characters) length(String) axioms

w,v:Strlng,

• String

• Nat

x:Characters

length(empty) = zero length(append(w,x)) = succ(length(w)) interlock(w,empty) = w interlock(w,append(v,x))=append(interlock(w,v),x) end STRING

16

The key-word definition indicates initial satisfaction for the following axioms. After the key-word sorts the new sort names follow which correspond to the specified data types. After the key-word oprn the sequence of fundamental operators follow, where in brackets, following the operator name, the number, the ordering and the required sorts of the arguments of that operator are declared by an n-tuple of sort names. The sort name after the arrow declares the result sort. This notation is intentionally similiar to function declarations in programming languages. This implies that in the preceding specification 'zero —*• Nat* denotes a constant of type 'Nat',i.e. an operator without any arguments that produces an object of type 'Nat', and that append(String, Characters)

•

String

denotes an operator which needs two arguments, the first is of sort 'String' and the second of sort 'Characters', and the resulting value is again of sort 'String'. a,b,...,y,z —»Characters declares a set of constant operators with the names a,b,...,y,z all of type 'Characters'. After the key-word axioms the declaration of the variables follows that are used in the following axioms. In the preceding specification each axiom is an equation. A verification of the specification STRING would require the proof initiality of the interpretation which is suggested by the chosen names of sorts and of operators.

of

The next example illustrates the use of conditional equations by a specification of the set of natural numbers, of the set of all finite subsets of natural numbers equipped with the cardinality function. CARDINALITY is definition sorts oprn

Nat, Sets, Bool zero — • Nat succ(Nat)

— *

true, false eq(Nat.Nat) 0 • Sets Sets U {Nat}

Nat *• Bool »• Bool Sets

card(Sets) »• Nat N a t e Sets • Bool axioms x,y: Nat, m: Sets eq(x,x) = true eq(x.y) = eq(y.x) eq(zero,succ(x)) = false oq(x.y) = eq(succ(x),succ(y))

17

(m U {x}) U {x}

-

(m U {x}) U Rseq

Nat

Rseq

*• Bool

add(x:Nat,r:Rseq iff x £ r) axioms

x,y: Nat,

Rseq

r: Rseq

x $ nil if x ^ r

then

y £ a d d ( x , r ) • eq(x,y)

end EXAMPLE1 In the specification

EXAMPLE1 the specification

NAT

is used as

prerequisite. In this w a y complex specifications can be written by stepwise enrichments of presupposed

specifications.

Of course, this example could also be specified with a total adding operator. In this case we w o u l d have to add a new axiom that guarantees that the addition of a natural number does not change the sequence if

x£ r

then

n

contained in the sequence

r

r , i.e.

add(x,r) » r

A n o t h e r w a y of making the adding operator to a total one would be the use of a so-called error value, as it was suggested by Goguen The resulting specification EXAMPLE2

is

NAT

(197^,

is then:

with definition

sorts

Rseq

oprn

nil error

•

Rseq >

Rseq

19

add(Nat,Rseq) axioms

x,y: Nat,

Reeq

r: Rseq

x IN

x • 0 and equal.

with

while(l) = while(2) = ... = while(x) = ...

Since there is no axiom that implies the equality of

while(l)

. with

any natural number, the initiality of the enrichment by definition in WHILE*

causes the creation of 8 new object of sort

a new object contradicts meter of

WHILE*

M .

The creation of

the persistency so that not each actual para-

can be extended to a model of

We hope that the similarity of the axioms in

WHILE*. WHILE

with the ordi-

nary explanation of the semantics of a while-statement has been observed. We interpret this similarity as a justification of the introduced calculus of functional enrichments of PADT's. Before we will deal with the actualization of formal parameters of an initial canon we specify a further example for which the class of actual parameters is of greater interest than in the previous specifications . The object of the specification is some kind of an inhomogeneous storage. For this inhomogeneous storage besides the sets of addresses and values a set of types and some additional functions may be given that associate with each address and with each value an object of sort 'TyDe'. In addition the equality of addresses may be given. This system of sets and functions is considered as a parameter of the inhomogeneous storage. O n the basis of that parameter we define the set of all states of the storage and we define

store- , read- , and

operations. For these operations we assume that a value stored by an address

a

only if

a

and

x

x

remove-

can be

are both of the same type.

Evidently, one can only read a value by an address, if before under that address any value has been stored. These requirements imply

that

both operations become partial ones. The remove-operation will be spe-. cified as a total operation that will not change the state of the

43

storage if we try to remove an information by an address under which before no value has been stored. STORAGE

¿s

BOOL

with

requirement

sorts

Address, Value, Type

oprn

a-tvpe(Address)

Type

v-type (Value)

equal(Address.Address) axioms

'

• Type •>

Bool

a.aj.ag! Address

equal(a,a) = true if

e q u a l ( a 1 ( a 2 ) = true

then

aj = a^

with definition sorts

Storage

oprn'

initiate

»

Storage

written(Address,Storage)

»

Bool

store(Storage,a :Address,x:Value iff

a-type(a) = v-type(x) )

—r-Storage

read(a sAddress ,s tS'orage iff

written(a.s) = true )

remove(Address»Storage) axioms

Address,

Value

Storage

s: Storage,

x,y: Value

written(a.initiate) = false if writtenfaj,s) = false, a - t y p e ( a 2 ) = v-type(x) writtenfa^storeis.Bg.x))

then

= equal(a1#a2)

if a-type(a)=v-type(x), a-type(a)=v-type( y)

then

store(store(s,a,x) ,e,y) = store(s,a,y) if a-type(aj)=v-type(x), a - t y p e ( a 2 ) = v-type (y)

f_l _f_i

then

store(store(s,a^,x),a2,y)=store(store(s,a2,y),a1#x) if a-type(a) = v-type(x)

then

remove(a,store(s,a,x)) = s

fi

if written(a.s) = false then

remove(a.s) = s

if a-type(a)=v-type(x)

read(store(s,a,x)) = x

then

end STORAGE In this specification explicit

domain conditions within the declara-

tion of the partial operators are used as distinct from the initial canon

WHILE

. Therefore we payed attention that no additional domain

conditions are caused by the axioms. By means of the actualization of the parameter part one can derive different kinds of storages. It is important that actual parameters themselves may be PADT's .

44

PADT's. This possibility enables stepwise refinements of

In the following the abstrect storage will be actualized by setting the disjoint union of the sets of integers, truth values, and strings over an arbitrary alphabet for the parameter sort

'Value'. The parame-

ter sort 'Address' will be actualized by a set of strings overs a p o s sibly different alphabet, and we will finally set the two-element {static,dynamic} functions

for the parameter sort

'a-type'

and

'v-type'

set

'Type'. Therefore the attribute

associate with each address and

each value one of these both attributes. This actualization uses another parameterized abstract data type as actual parameter so that by further actualizations both the set of characters of the strings that form the addresses and the set of characters that form a part of t£e values may by actualized. To achieve the sketched actualization we specify at fi rst the actual parameter. PARAMETER1

¿s

NAT

with

requirement

sorts

Character

oprn

c-eq(Character,Character)

axioms

-

Bool

x,y: Character

c-eq(x,x) = true if

c-eq(x,y)=true

then

x=y

with definition sorts oprn

String nil

»

String

append(String.Character)

»

s-eq(String,String) axioms

x,y: Character,

String

Bool u,v: String

s-eq(u,u) = true s-eq(u,v) = s-eq(v,u) s-eq(u,append(u,x)) = false s-eq(append(u,x),append(v,y))

= (s-eq(u,v)

c-eq(x.y))

with definition sorts oprn

Union, Kind static, dynamic

•

b-inj(Bool) n-inj (Nat.)

axioms

Kind

Union •

Union

s-inj(String)

»

Union

u-attr(Union)

•

Kind

x: Bool, y: Nat, v: String

u-attr(b-inj(x)) = static u-attr(n-inj(y)) = static u-attr(s-inj(v)) = dynamic end PARAMETERl

45

For an actualization of the parameter sort 'Character' in the initial canon

PARAMETER1

Initial canon PARAMETER2

_is

BOOL

sorts oprn

we define a special set of characters by

the

PARAMETER2. with definition

A a,b,c

»

A

2-eq(A,A) axioms

•

Bool

x: A

2-eq(x,x) = true 2-eg(a,b) » 2-eq(a,c) = 2-eq(b,a) = 2-eq(b,c) = 2-eq(c.a) = * 2-eq(c,b) » false end PARAMETER2 A f t e r the specification of a as actual parameters afterwards

PAOT

and an

ADT

that will be used

we introduce the last operation

for

the construction of an initial canon. This construction in called 'Mpdlficatlon'. The general syntactic scheme of a modification is the following: is

adlf led [ by

(+)

end such (i) ( Ü )

(ill)

for

s:

for

s;

for

6*

for

6"

(t2 J

according

X

that: S*..

, ,s are sort names of the initial canon

6*,.. ..6 1

and """

are operators of the initial canon S_ "j

for

sorts

operators of (iv)

1'

' t 2 :

u

2

u

i —

•

T h e e

r0

—

v

, where B

o • u , g: u

—e—» u

be

s u l t of the composition is 2

*

The looping construction requires three arguments, p: u f: "

ui#

, can

o »

Bool,

and corresponds to the well-known con-

struction 'while

p

do

f

else

g' : u

o » u

.

51

The arguments of the looping construction have to satisfy the following conditions. The input tuples of all three arguments are of equal type, given by

u , the output sequence of

given by

'Bool', and the output sequences of

to the common input sequence of

p

is the sequence of length one f

and

g

are equal

p.f.g •

The branching construction corresponds to another well-known construction of programming languages, namely to

where

'if p

then

p: u

o

g

else

- Bool,

f

g: u

—o

» v,

guments. Again it is required that

f: u

p,f,g

o > v

are the ar-

have equal input sequences,

and that the sequence of length one 'Bool' is the output sequence of p .But now we require only that

f and

g

have equal output

which may be different from the common input sequence of

sequences p ,f , g .

The source tupling is applicable to any finite sequence of tuplefunctions tl,

u

— e —

t2: u

—

v

tns u

2

—

with a common input sequence. The result is the tuple f

n):

u

v

where the output sequence v

the output sequences

1

'

,,,v

v

r

function

2 '"''v n

i 2'*'vn

v

—

v

1 v

i» 2'"

: u

' B

Bool

Operators =

, target =

A

B

),

B = target ,

{ t r u e , f a l s e } , A g o r t s « = sorts( B

A A A A A and nil , app , length , concat , and val empty sequence of sort names of

B

)*,

are interpreted as the

, as the appending of a sort name

to a given sequence of sort names, as the length of sequences of sort names, as the concatenation of sequences of sort names, and as the selection of

the i-th component of a sequence,

respectively.

57

The interpretation of the remaining sorts

F

and

F*

remaining operators is the kernel of the definition of We start with the definitions of the sets

i

Ap

A

the

Apm

:

A

(B,_t: w

o • u)

| such that

B

is an enrichment by

definition B ,on andAB 1_t: is a w-term oftuple J w

o

and

and of

semantics.

A F « = { ( B , t* J w

o •

UjjCUjiX.. ,#u r )

enrichment by definition of M _t : w B

o »

uj#...#ur

with

o » u

, and

B

with

| such that

w,u e sorts(B )* B

is an

, and

is a w-term tuple sequence on w,u1,...,ur€

sorts(B ) } .

The interpretations of the operators 'in','out','in*' are naturally defined by A

a

in (B,Jr: w

o «• u) = w ,

A

o u t ( B , _t: w

o »

u) = u

,

A

( i n * ) ( B , tf J w — e - » Since the sort

F*

. .#u r ) = w .

represents only an auxiliary construction of

special sequences of w-term tuples the interpretation of the operators 'first' and 'add' is naturally given by f i r s t A ( B , t: w where

w

o •

u

o »

u) =

t*: w

o -

u)

is interpreted as w-term tuple sequence of length

one, and a d d A ( ( B 1 , tj: w - e — with ® 2 ' - 2 :

= such that

u ^ . . .#u p ), ( B 2 , t s w - » — > w

° *

length(_t*) = length(t*)+1,

1 £ i ^ r = length(t*),

u)) =

and

,t|(i) = I*(i)

for

t*(r+l) = t .

The most interesting part of the definition of the semantics is the interpretation of the operators

"base",'compose','prj'loop','branch*,

'away', and 'tuple' since these operators represent the constructions of

tuple-functions.

Semantics of

'base":

For each operator B

58

we define

6: w

o - S

in the signature of the initial canon

base A (5) = ( B , 6 : w

S)

where the operator is considered as a term over B , and that term is considered as a w-term tuple of one component. Semantics of

'compose':

The elements

(Bj,^:

— e - ^ - U j ) , ( B j , t 2 : v»2

o - u 2 ) of A p are A in the domain of the partial composition ,compose , if and only if

Uj = Wg . In that case we set compose A ( ( B j , t^ : w x

o >

u^)

a ( B ^ with B 2 . t: Wj with

t(i) = [l 2 (i)T (t^) [.^(^(Ii)-'

w

for u

l

t,2(i) :

o »

u2(i)

o •

2

u2)) =

u2)

1 *i^length(u2),

where

( i )

denotes the term that results from simultaneous substitution of

o ,»

^ 2 (i).: u^

^(j)

for each

j

o »

u2(i)

by the

for the J-th variable in with

1

J, ^ length (Uj) .

We demonstrate this composition of w-term tuples by a simple example Let

= (g.c.d: Nat X Nat — N a t ,

Jt2 = (less: Nat X Nat — • l.c.m: N a t X N a t — * be term tuples on

Nat)

and

with

_t2

Nat,

Nat)

NAT with BOOL. The composition of

yields a term tuple triple

plus: Nat X Nat — p -

Bool, times: Nat X Nat — p -

NatXNat

Bool X Nat X Nat

( y ^ y g . y j ) £ {true, false } X »1 X U

^

that produces

for each pair (x,y) E H

a X

»

in such a w a y that yj = g.c.d(x.y) t 3 : w w) A be given arguments of the partial operation 'loop ' contained in the domain of that partial operation. Then the intended semantics the substitution of the parameter sort by the sequence p(M)

o » w

o »

M

requires

in the intial canon

WHILE

w e sorts( B )* , and of the parameter operators

Bool , f(M)

o •

w, _tj : w

o > w

M, g(M)

o » M

by

tj : w

o -

Bool,

respectively.

This kind of modification requires a slight generalization of the modification of initial canons as defined up to now. On the left-hand side of a modification list we permit now sequences of sort names and w-term tuples of the modified canon. With this generalization we are able to define l o o p A ( ( B l t tjsw — B o o l ) , ( B 2 , _t2:w - e - » w) . ( B j , _t 3 :w — e - » w) ) = = (WHILE modified by ( B x wi.th B w

for

2

with B ^ )

according

M

for

P

t 2 for

f

for

g

end , w h i l e : M

o »

M )

If the semantics of the loop-construction should be given by the following flowchart, then this change in the semantics of the loopconstruction can easily be realized by the use of a corresponding tial canon

LOOP

defined in the

ini-

following.

61

The kernel of the changed definition of the semantics is the specification of the initial canon LOOP

is

BOOL

LOOP.

with requirement

sorts

M

oprn

p(M)

o •

f (M)

—

g(M)

Bool M M

with definition oprn

loop(M)

axioms

o - M

x: M

if

p(f(x)) = false then

iL

P(f(x)) = true, loop(g(f(x))) - loop(g(f(x)))

loop(x) = f(x)

loop(x) = loop(g(f(x)))

then

fi

end LOOP Using this specification we define l o o p A ( ( B l f tjsw

Bool) , ( B 2 , tz-.\N - « — w ) , ( B 3 , t 3 :w - e — w) ) =

= (LOOP modified by ( B t with E ^ with B , ) according w for M for

lz Semantics of

P

for

f

for

g

end , loop: w

o » w )

'branch':

The semantics of the branch-construction will be defined in the same way as the semantics of the loop-construction. Evidently, we have

to

use another initial canon. We start with the specification of the initial canon

BRANCH.

BRANCH

BOOL

¿s

with requirement

sorts

P,M

oprn

p(P)

o m Bool

f(P)

M

g(P) - e —

M

with definition oprn

branch(P)

axioms

end

62

—e—-

M

x: P

if

P(x) = true, f(x) = f(x) then

If

p(x) = false, g(x) = g(x) then

BRANCH

branch(x) = f(x) branch(x) = g(x)

Using this specification w e define b r a n c h A ( (IBj, B l t tjiw ^ :w — e — Bool) Bool). , ( B g i, .tgiw — e — u ) , ( B 3 , _ t 3 : w ( BRANCH modified by ( B t with B ^ with B ^ ) w u

¿2 ¿3

for

P

for

M

for for

P f

for

9

end , branch: P

—*-»-u))

according

o »

M )

The preceding semantics of the branching does not assume the applicability of both a l t e r n a t i v e s . Therefore, a further conceivable tics of the branching could be given by the following

is

BOOL

sorts

requirement

P,M

oprn

with

with

seman-

specification

p(P)

o »

Bool

f(P)

M

g(P)

M

definition

oprn

branch*(P)

axioms

o »

M

x: P

if p(x)=false, f(x)=f(x), g(x)=g(x) then if p(x)=true, f(x)=f(x), g(x)=g(x) then

branch*(x)=g(x) branch*(x)=f(x)

end The initial canon

BRANCH*

presumes the applicability of both alterna-

tives and of the branching condition as sufficient domain condition of

,

branch*(P)

ment in

• o » M ' , and due to the initiality of the last

BRANCH*

The homomorphism algebra

A

is

enrich-

this condition is also a necessary domain condition. f: 1 ( 2 )

» A

uniquely

not surjective, i.e. not

is the meaning of an abstract

any

defined by the semantic ( B , t: w

o -

u)

tuple-function.

Finally w e repeat that it is not the aim of this section to introduce a functional language and to contribute to the research work on functional programming, rather than to demonstrate the applicability of structural induction even if the generating constructions are not defined for all arguments, and to use for this demonstration a non-triviaL example.

63

CHAPTER EQUATIONAL

2 PARTIALITY

Whereas in the first chapter it is shown that there are strong motivations for the development of a theory of equationally partial manysorted algebras, the following chapter develops the algebraic basis of the preceding use of structural induction on partial algebras. A c c o r ding to that aim the following two chapters are a contribution to the universal algebra of partial algebraic

structures.

The existence of different sorts of elements in all examples of Chapter one is algebraically reflected by the U9e of many-sorted algebras (or heterogeneous algebras as they are called by Birkhoff

and

Lipson (1970)). However, the use of many-sorted algebras causes some notational difficulties. These notational problems could be considerably reduced if we were to use the mathematical language of category theory, but this language is a very abstract one and the uses of abstraction to achieve generality are not the same as in other branches of math-ematics and computer science. The theory of categories is a very useful and powerful tool in the hands of a well-trained

mathematician.

Because most mathematicians and specialists in computer science are not familiar with the basic concepts of category theory, we will not use this tool here. Readers who are interested in category theory should consult the famous book

' Categories for the working mathematician'

by

MacLane (1971) or a recent paper of Goguen and Burstall (1983) . In the following we will use the traditional language of sets,

functions,

operations, relations, and so on. We deal mostly with S-index and functions between S-indexed families of sets, where

S-

families

represents

the set of different sorts of objects. To overcome the notational problems of many-sorted algebras we introduce in the first section the underlying set-theoretical

notions

and concepts. In the second section we Introduce the concepts of total many-sorted algebras, of partial many-sorted algebras, of homomorphisms and of so-called existence equations. The third section is devoted to defining the concept of equational partiality and equationally partial algebras. The fourth section deals with hierarchy conditions for both systems of domain conditions of operators and sets of axioms. It is proved that hierarchical systems of domain conditions are necessary and sufficient for the definition of substructures by the requirement that the inclusion forms a homomorphism. This implies that for this

64

theory only one concept of homomorphlsm is necessary to yield a' nice theory of structure. On the basis of the results of the fourth section we introduce in the last section of this chapter hierarchical equationally partial varieties, briefly hep-varieties, and hep-quasl-varieties. The class of small categories turns out to be a typical example of a hep-variety.

2.1

Set-theoretical Notions

A s usual w e denote eeta by capital letters, elements and functions by small letters. For finite sets the elements which belong to the set can also be given within brecee. Let

S

be a n y finite set. A n S—Indexed family

sets will briefly be called and family an A

f - (fg: A s — B

S - f u n c t i o n or and

any

B

are

seS

9

|

S-set.

seS)

A • (A

| s€S)

Correspondingly, an

of

S-indexed

of functions will briefly be

called

S - m a p p i n g . If it is evident from context that

S - s e t s , we denote an S-function by

the set

A

and thNe mapping

S

the 8-component of the

S-set

A

f : A 8

and of the

fi A «• B

9

8

S-mapplng

*

both

B .

For

will be called f: A

B

respectively. Let main x

fs A A

of

• B

g: B

denoted by

f: A » C

a n y element of the do-

fgs A

> B

will be denoted by

xf£ B .

If addi-

is a further mapping, the composition will be • C , sometimes also by

application of the composition to any xfgec

xeA

» B . The element that is assigned to the element

by the function

tionally

be a n y function and

ft A

o r by

xeA

fog: A

- C .

The

is then denoted by

(xf)geC

if we want to stress the successive application of

f

and

g . This

notation is compatible both with the diagrammatic and the applicative order of the composition. W e hope that readers will not be confused irritated by the use of

xf

instead of

f(x) .

Very often w e have to deal with sets of variables taking values an

S-set

or

in

A » (A g | s e S ) . In this case the particular s-component in

w h i c h each variable we define an

xeX

can take on values must be fixed. Accordingly

S-sorted system of variables as a mapping

Now, any

xeX

v: X

S

can take on values in

in an

S-set

A • (A8 | s e S )

v: X

» S .

o n l y . A n assignment of is therefore given by a

65

function

ouch that

x£6Axv

for each

xeX.

Ay

denotes the set of all assign-

ments of the S-sorted system of variables

vs X

» S

In the

S-set

A - (A9 | s e s) . Let v: X

any

V

A

The mapping

B

v —

v

fv: A v

assignment

* Bv

af^eB^

» S

S-function and

S-function to a function

x(afv) - ( * £ ) f x v 6 B x v

by setting

the

be any

S-sorted system of variables, then we can extend in a

natural w a y the

vj X

• B g | 8 6 S)

f - (f 8 s A a » S

S-set

f: A

every

xiX, a e A

y

transforms any assignment

of the same

in the

S-function

for

.

££

A

to

v

an

S-sorted system of variables

B • (Bs | s e S )

by pointwise application of

- B .

For the sake of simpler formulae we allow the subscript to be omitted from the functions

f :A 8

• B

8

,f : A

8

V

s

and

- B

V

v

V

respectively« if this subscript can easily be derived from context. M n«W

'-{0,1,2,...} we set

of variables string

w •

s

denotes the set of natural numbers, and for

[n] - { l , 2 , . . . , n } w : [n]

• S

is2***sn

where

and

[0] -

can obviously be identified with the s^ • lw

string will be denoted as usual by systems of variables of the form

for every

A s [0]

of all finite strings of elements of

For a n y string

ws [n]

"

w : [n'J

S

ie[n].The S . s"

empty

denotes the set

S , e.g. the set of all S-sorted

of all non-empty strings of elements of

A » (A g | s e S )

any

A finite sorted system

»

S . S+

denotes the set

S .

an assignment

£€A

in an

w

S-set

will also be denoted by

£ » (aj,a 2 ,... ,a n )

with

» la

for every

ie[n]

as ordered n-tuples are usually denoted. According to that convention A^

can be Identified with the Cartesian product , i.e. A

w " Alw

X

A

2W

x

•••

x

A

nw '

W e recall that for the empty string sists of ons element for every

S-set

7 : [o]

» S

A » (A^ | s € S )

the set

A-,

clusion of the empty set into the union of the s-components of the

66

con-

which is the in-

S-set* If ft A

Therefore we can set w : [n]

»

M

"

S

fi A ^ — A binary

an

{0}* A •

| seS)

is an

S-set,

is a n y function, and if w e use the notation

M

A ^ , then the value associated to

w i l l be denoted by

S-relation

S - i n d e x e d family

every

a»

la a string,

to denote elements of by

A

s e S . An

S-equlvalence in

q

in an

{> •

A

S-set

A • (As | s e S )

such that

in an

S-set

if the binary relation

valence relation for every and symmetric for every

s e S , i . e . If

^ A

8 £

is given by

«

X A8

for

Is refered to ae A

x

8

*8

an

i» an equi-

is reflexive, transitive

s€S .

Finally we will point out an important effect. be a sorted

(alt...,an)

( a l t . . . , a )f .

| s €S)

S-relation* (>

(a 4 ,...,a ) i n

system of variables and

that at least one variable exists no assignment of

x€X

v: X

Let

A • (A^ | s e S )

exists with • S

in

A

v: X an

»- S

S-set so

A x v - 0 . Thsn such that

there

Ay « 0

.

This effect causes the theory of many-sorted algebras to differ seriously from the theory of homogeneous algebras, especially with

respect

to the 'equational theory' of many-sorted algebras.

2.2

Many-sorted A l g e b r a s

Before we deal with partial algebras we Introduce the basic notions of total many-sorted Definition 2.2.1 £

A

algebras. s i g n a t u r e

- (S,

f o r any

2-algebra

e£S,

S-inclusion

B£S, Obviously,

«very

i o r

i s an

B . An

A

S-subset

S - c a r r i e r of a 6 :w

as

underlying

A ^ B

of

the

2-subalgebra

if

a€A B — w t h e a p p l i c a t i o n of 6 to £ y i e l d s an element of A , I.e. A ie B c l o s e d w i t h r e s p e c t to t h e a p p l i c a t i o n of fundamental o p e r a t i o n s of t o arguments t a k e n from

, where

S - s e t of a

with

» s . and f o r e v e r y

A.

F o r t h e sake of completeness we summarize h e r e some well-known

facts

o f t h e t h e o r y o f t o t a l many-sorted a l g e b r a e w i t h o u t p r o o f . P r o o f s more d e t a i l s can be found i n the book of Lugowski (1976) o r i n paper of B i r k h o f f (2.2.6)

Let

set-image

and L i p s o n

A,BeALGt(2)« Af? B

of

the

gebra o f

B ,

i m a g e

, denoted by

According to ( 2 . 2 . 6 ) r e p r e s e n t a t i o n s of (2.2.7)

F o r any

equivalence

(1970). B

®r

anV

S-set

which w i l l

S-homomorphism

A

i s an

t h e { B , E , c j - s e t of o r t h o g r a p h i c a l l y

A 8 £

S-indexed f a m i l y

X A^ , a € S ,

r e l a t i o n , 6 t w —• s ,

kerf

Be Q

for every

i € [ n ] , where

£-homomorphism

» (kerf J

seS)

ker f 8 - { ( * , y ) i A

s

i s a congruence r e l a t i o n i n

74

h o m o m o r p h i c

correct

f o r m a l e x p r e s s i o n s i s a c a r r i e r of a s u b a l g e b r a o f

relatione

F o r any

the

I

holds f o r every

(2.2.8)

B

2-subal-

.

c o n g r u e n c e (1^,1^) 6

f: A

S - c a r r i e r of a

be c e l l e d t h e Af

and

the

^ •

is called

in

A

fs A

B

p8 x.^gA^

with'

• S . the

of

a

If

and f o r e v e r y w: f n ]

seS)

0.

S-indexed

• family

with

X A8 A .

| * f e - yf 8 }

. seS . I

(2.2.9) Theorem of Howomorphlanis Let be given £-homomorphlsms f: B — A , gs 8 » C , so that Bs » C g is eurjective for every s e S , briefly g: B - C surjective. Then there exists a 2-homoraorphlem h: C • A with f « g°h If and only if k s r g ^ Q k e r f ^ for every sort name s e S .

1« •

The homomorphlsm theorem can be applied to the problem described above if we take B • T ( 2 , v ) , for A the semantic algebra, for C the homomorphic image T ( 2 » v ) i of ij T ( S , v ) » 0 ,i.e. the eubalgebra of orthographically correct representations of formal expressions, and if we assume that f: T ( £ , v ) — • > A is the extension of any assignment A , and that g: T(.2?#v) » i® the restriction of i: T ( S « v ) • 0 to its homomorphic image. The condition that equal representations imply equal meaning guarantees then by (2.2.9) the existence of a homomorphiem h: T ( 2 , v ) i A which defines the meaning for every orthographically correct representation . The logical structure of a compiler can be reconstructed from this algebraic definition of semantics by h: T(^J,v)i » A . The aim of syntacic analysis is to decide whether a given string is an orthographically correct representation of a formal expression and if so to construct the corresponding internal representation. Sometimes even the postfix polish notation is used as the internal form of formal expressions. The second part of a compiler, the semantic interpretation, corresponds to a description of the semantic algebra in terms of the machine or assembler language of the target computer. The main point of this description is the definition of the fundamental operations of A by suitable programs. Now that this example is complete we shall make a closer study of the relations between T ( ^ , v ) and partial S - algebras for a general signature S and any S-sorted system of variables vt X » S . Definition 2.2.10. For any signature S p a r t i a l

» (S, : {xg - k^) - { g.k} op(defcomp) »{source, target} op(def id ) - 0 . Definition 2.4.1. An ep-slgnature 0 « ( S » ( d e f 5 I Be. £2)) is said to be h i e r a r c h i c a l If every sequence of not necessarily palrwise distinct operators 50. with

»

' "*

€ op(def 6 i )

for every

there is a natural number

n

16*1

with

terminates, or In other words,

op(def£> n ) • 0 .

Hierarchical ep-signstures will be called for short.

h e p - s i g n a t u r e s •

Clearly, for the ep-signature 6 Q we can build an infinite sequence for every operator fiTe£?e .namely 6,6,6,... becauee {? € op(def 6 ), o so that is not a hierarchical ep-eignature. For the ep-signature we can construct the infinite sequence f> 9• f> 9> •••< f> g> • • • because fe op(def g ) hierarchlcal. As mentioned above

and

g eop(def f ) , so that

8^

is also not

IS hierarchical since any appropriate ss-

quence of operators has at most the length two for op(def id ) • op(def source ) » op(def target ) • 0 and op(def comp ) « (source, target} .

87

The Importance of the concept of hep-signatures becomes evident

by

virtue of the following theorem. Theorem 2.4,2. 8-algebras (i)

6

For any ep-slgnature

A, B

6 » (Z,(d»f6 I 6^X2))

and

the following conditions are equivalent:

Is hierarchical;

(ii)

Every bijective homomorphlsm

f: A

- B

(111)

Every injectlve homomorphlsm

f: A — »

is an isomorphism;

B

reflects solutions

» B

reflects solutions

of existence equatione; (iv)

Every injectlve homomorphlsm

fs A

of domain conditions. Before proving Theorem 2.4.2 that a homomorphlsm

f: A

we will make more precise the notion

• B

reflects solutions of existence equa-

tions and of sets of existence equations respectively. Let

(v: t^Stg )

and

(v:G)

be any existence equation and any set

of existence equations respectively. A homomorphlsm flects solutions of

(v: tjStg ) or of

(vsG)

f: A

» B

re-

if for every assignment

«€ Av !

f

v

£ B

(v:

tlIt2)

£fv68(vsG)

iM

iMplie8

P110«

¿€A(v:

or

tlSt2)

1cA(V:G) •

Proof of Theorem 2.4.2: First we prove that

(iv)

implies

(il). Therefors let

f: A — »

B

be a bijective homomorphlsm. Then there exlste the inverse S-mapping f" 1 - ( f"1, B e — We have to show that j) £ B j ^ g

and

f" 1

As |

.68).

is a homomorphlsm. Let be

£ • b f" 1 e A^ . Because of

according to (iv) that £ • — * w l t A d e f 6 • momorphlsm f: A — » B yields b 6 B f _81

—

- —b fW_ 1 6 A

1 bff8ff 8 8a

—

This proves that

B

6T € Si

88

l n J®ctlvity of the ho-

» A

= a— 6 A f 8 - -b f "WX 6 A f o 8. ie a homomorphlsm.

Second we show Indirectly that For that purpose we define

w — » 8 ,

because

= -b 6 0 . a f 5B — W f :

Se,Q,6:

a f^ • b € B d e f S

(1) Q

Is a consequence of

(ii) .

®8 the set of ell those operators

for which there Is a non-terminating sequence

with

5 1 + 1 e op(dof

)

If w e assume that

6

and

F*

F* - F s - { 0 }

determined total dom 6

= Fw

e*

for every

seS.

F

is the uniquely

e-algebra, e.g. for every

6 €-£2

The fundamental operations of dom b

= 0 .

-algebras in Ssction 2.3 we define two 8-algebras

with

F

i - 0,1,2

is not hierarchical, then

Analogously to the F

for

f f" = 1

F*

with

6 tw — »

e .

are given by

for every

6 « Q \ Q . , where

for every

6€

6 :w

—

s

.

The S-function e . ( e 8 s F* with

e8 - I d ^ j

» Fa | e£S )

for every

se S

is a bijective homomorphism

F

e* * — — F . however it is not an isomorphism. This contradiction to (ii) can only be reeolved if - 0 , which means that 0 is hierarchical. It is not hard to show that If

def 6

• ( w : 0 ) , where

e-algebra, so that for -feAdef& -

f

w

e

B

-€A(w:

"

A

Now

w *

for

( w s tjStg ) t St

)

for

S.€\

**e

ever

Y

Y

holds provided •

with

a89U,ne

ever

(iv)

6 : w

s, a f^ €

then

(lil)

Brfef g

Bdef 5

- 8

(w:

1

St

2

for every

obviously

def 6T + ( w : 0 ) , a g A w t

is true.

and

) ^ def 6 , so that by

(iii)

(w: t j S t g J e d e f Ç . This means

which is what we have to show. Next we turn to the implication (iv) — ( i i i ) . By a simple induction one can prove that (iv) implies the following slightly generalized condition (iv)* Every injective homomorphism of

def t

for any term

f: A

» B

The proof goes like the proof of Corollary 2.3.5. with f' A

vs X

• S

» B ^ —

f

"v 6

This implies

reflects solutions

t e T ( £ J , u ) 8 , u: Y

be any exietence equation,

» S . Let

a gAy

(vs t j § t 2 ) and

any injective homomorphism with B

(v: - '

v

tl

St2)

€ B

d

.

f v

' £f

v

eB

d - f

and

a f y t®

- a f y x\

89

Condition t®.

t|

(iv)* together with the definition of the term functions yield

S ^ d e f V Since

f: A

• » B

—

€ A

def tg'

*A , • eeS

t

•V a V •

is the eort name with

The lnjectivlty of tj£

l(a»v) " ^ i V

is a homomorphlsm we obtain by

f

where

8nd

f :

(2.3.4)

•

tlft2 ST(2,v)8 .

— • » Bs

• t 2 £ , so that

'

yielde

a tj • a t*

or in other words

* £ A ( v : tjSt,, ) ' We conclude the proof of Theorem 2.4.2 by showing that (1) implies (iv) . This is the main part of the proof. The presupposed hierarchy of 6 allows us to prove (iv) by the following principle of induction: Let be a poset (partially ordered set) In which svery descending chain is finite. Let P contains all minimal elements of | 6"' £ 5 , 6 * + 6 } if { 6" '

in the carrier of an equoid

• Q^ , q 2 « A

» Q2

be an isomorphism h: Qj m Q2 with morphlems are unique up to isomorphism.

A

natural

exist, then there

q2 - q ^ h

must

, so natural homo-

This notion of a natural homomorphiam Is derived from categorical structure theory, eee MacLane (1971), and it coincides with the notion of the natural homomorphlsm nat (3 t A

» A/^

of a congruence relation in a total algebra associating with every its congruence class [aj^ - a (nat (> ) . element a e A Here we mention only that a natural homomorphlsm between equoids is in general not a surjective S-function. The problems of existence and of construction of quotient equoids will be treated in Section 3.3. The problame arise from the fact that by set-theoretic factorizations new solutions of domain conditions may be created in such a way that the value of the fundamental operation for this new eolution cannot be reduced to any repreeentatlve of a congruence claas. To illustrate thie we return to the e C A T - e q u o i d A as defined above. If we chooee the S-relation ^ given by ?ObJ " {( 2 .3)} •

?Mor - { ( e 2 ' e 3 > >

then what we just described happens.

2.5

hep-Varletles and hep-Quasi-Varietles

Until now we have only generalized the concept of a signature, and we have not yet any meane available to deecrlbe such properties of operators of sets of operators as associativity, commutatlvlty, or dlstributlvlty. As mentioned in Section 2.2 no kind of validity of term

equations

is known which is sufficient for describing the basic properties of the set of operations for small categorlea. Therefore we need other means.

97

which have not a l o w e r e x p r e s s i v e n e s s than e x i s t e n c e Oust a s

f o r s i g n a t u r e s we u s e t h e t h e o r y o f

to discover

suitable

small categories,

means. L e t

which says

A

A

comp (comp ( x , y ) , z ) holds,

provided

are a l l

c a n be f o r m a l i z e d i n

the

Definition An

2.5.1.

is

G

- comp ( x . c o m p compA(y,z), defined.

following

of

(v:

The e l e m e n t a r y A

G

g i v e n by a n

of

exl8tence

G

• H ) K

~

A

(vs

s ' x s )

be a n y

variables

signature.

of

v: X — —

S ,

by an e x i s t e n c e

the,elementary

equations

(v:G)

( v s G —-•» ' 1 * * 2 *

tjStg )

G

, the

the

equation

implication, p r e m i s e

h o l d s

in a

partial

' by

• tjStg )

.

• H ) S - s o r t e d system of

.

We draw a t t e n t i o n

to

iff

the f a c t

We i l l u s t r a t e

where a

•

A

(ws0)

A

A Aw

-

- S

and by

A A

, denoted

two

( V J H )

by

.

•

i m p l i c a t i o n s c a n c a u s e demands

by means o f

the hep-signature

SCAT

implication0

with

satisfies

£

( V S G )

comp(comp(l,2),3)

ws[3j —-»{ObJ.Mor} SCAT-equoid

vs X

Z-algebra

that

this

t o w h i c h we a d d t h e e l e m e n t a r y (ws 0

variables

(vs G ) , (vs H ) .

holds I n a p a r t i a l

, ( V S G

applicability.

98

equation

implication.

existence equations

A

If

conditional

i m p l i c a t i o n

s e t s of (vs

, and

i m p l i c a t i o n

be d e n o t e d b r i e f l y

(vs Is

equation

If

A H> ( v : An

the

compA(compA(x,y),z)

2? - ( S , * s Q

Implication

(v:G)

This w i l l

x,y,z

of

(y,z))

T h i s k i n d of

S - e o r t e d system of

set

the elementary

A

A

guide

law

way.

c o n c l u s i o n

t h e

-algebra

the a s s o c i a t i v i t y

the elementary i m p l i c a t i o n ,

and b y a f i n i t e of

have. aa a

• tjStg )

g i v e n by a f i n i t e

d o m a i n

equations

categories

f o r a n y morphisms A

e l e m e n t a r y (vs

S

Let

us c o n s i d e r

that

compA(x,y),

compA(x,conipA(y,z))

snail

lwthis

8

comp(l,comp(2,3))) 2w

-

3w -

Mor.

elementary i m p l i c a t i o n ,

then

A(wscomp(comp(l,2),3)-comp(1,comp(2,3)))

of

holds, so that every assignment a e A Is a solution of the sxlstence " e "" equation (w: comp(comp(l,2),3)»comp(l,comp(2,3))) . This means that A A comp (x,y) is defined for every (x,y)e A M o r X A M o r . Thus, coop is a total binary multiplication. Consequently, we have used the wrong implication to give the associativity law of small categories. The class of small catsgories is given by those fying the following set < ^ C A T iXCAT -

of elementary implications:

(v: f — - source(ld(e))S e ) ,(vs 0 — (v: ff

6 C A T - e q u o i d s satis-

target(id(e))8 e ) .

• comp(ld(source(x)) ,x) « x ) ,

(vs {target(x) = source(y)}-—Psource(x)>sourc(comp(x,y)) ), (v: { t a r g e t ( x ) a s o u r c s ( y ) } — » target(y)-target(comp(x.y)) ) , (v: £ target(x )«>80urce(y), target(y)«source(z) j • comp(comp(x,y),z)"Comp(x,comp(y,z)) ) where

v: {e,x,y,z} — { O b j ,Mor}

with

ev

• ObJ

and

x v • y v • z v = Mor . The total unary operators source(Mor) — • Obj , target(Mor) — Obj associate with every morphlsm x € A M o r i t 8 8 0 _ c ° H a c ' source object 80urce A (x)e , and target object target A (x)e A 0 |,j • The unary total operator id(Obj) — » Mor associates with evsry object e£A it8 80 called ha8 Obj " identity morphlsm i d A ( e ) 6 A M o r » according to the first two elementary implications, the given object e as its sourcs and target object. For any morphlsm x ^ A M o r t'10 ™orphisms id A (source A (x)) 6 A M ( j r and le^Jtarge^fx)) £ A M { j r are called the left identity and right identity of x respectively. Their properties are given by the third and fourth elementary implications. The equatlonally partial binary operator comp(x: Mor.y: Mor Iff target(x)»80urce(y) ) associates with any ordsrsd pair of morphisms morphlsm

A

comp (x,y)6 A M o p

the source object of A

condition. If

(x,y) e ^ q , .x ^Mor

provided ths target object of

y , i.e.

comp (x,y)

» Mor

iff (x,y)

x

8

equals

is a solution of the domain

is defined, then its sourcs object equals

ths source object of the first factor, according to ths fifth slemenA tary Implication, and target objects of comp (x,y) and the second factor are equal, according to the sixth elementary Implication. The last elementary implication is the correct form of the associativity law of the partial binary composition. A typical example of a email category can be obtained if one takee A

Obj *

M

a n y

eet

8et

®»

^or "

the

8at

of

811

functions fs X — »

Y

99

with

X , Y € M . The fundamental operations of the email category are

then given by i d ^ X ) - Id x « X — •

X

A

for every

eource (fi X — » Y ) - X , every comp

A

(f: X

x e M » AobJ ,

A

target (fs X — •

Y)

- Y

for

• Y )6A M o f . , and

is the composition of functions«

It is not the aim of this book to give an introduction to the theory of cetegorles, although the author completely agrees with Burstall and Goguen who consider this theory as very useful for computer scientists, see Goguen, Burstall (1983). As the example of small categoriee shows, interesting classee of equoids can be described by a hep-signature 8 with a set (v: target(id(source(x))) = aource(x)) the implication (v: 0 — - target(id(source(x))) £ source(x) ) should be a consequence of a suitable set of minimal axlom8. Indeed, this Implication is a consequence of { I f we replace e by source(x) . In the same way one can see that (v: 0 — » is a consequence of

target(x) = source(id(target(x))) ) {a^} » where

def(v: comp(x v id(target(x)))«x ) • (v: target(x) 2 « aource(id(target(x))) ). Finally def(v: comp(x,comp(y,z))»comp(comp(x,y),z) ) »(v: { target(x)«source(comp(x,y) , target(comp(x,y)) • 2 source(z)} ) holds, so that (vs { target(x)'eource(y) , target(y)§eource(z)} — — » ^target(x)>source(comp(y,z)), terget(comp(x,y))» -source(z)} ) has to be a consequence of the remaining axioms. It is not hard to see that this implication is a consequence of {a 5 ,Sg} . Therefore, the partial order Is given by Sj < a 4 , a 2 < a 3 , a g < a 7 , a & < a 7 , and the condition of a stable est of axioms is satisfied. The importance of the concept of stabls sete of axioms stems from the following theorem.

102

Theorem 2,5.3. Let

0

be any ep-algnature, ej

and

a 6 domtj/ldomt*

because

A

If we assume

s£S

A

stjf at2

a«}'. + 4 * 5 %

with

tj.tg £ T ( 2 , v ) s . Then

A h— (v: G — •

def(tj2t 2 ) ) .

, the lnjectlvity of

• % t ; - atjf8 + at^fe - £fvt®

For

a £ A , _v (v:G)

B

yields

•

Since f: A » B is not only an lnjective homomorphism, this Inequality causes

On the other hand

f: A — »

B t— (v: G — »

t

S-function but also a

.

j " t 2 ) holds because

B )— (X .

this leads to

afv6B(v.6)

tjitjj)

•

But this contradicts + a v !

•

and hence A

— *1

A

-*.X2

Is true for every

S € A ( v . g ) . This mesne

A (—• (v: G >

«

—

^

*

and the proof le finished.

I

The following corollary shows how the concept of stable sets of axioms is related to the well-behaviour of homomorphiems between (©«-(vt

^ tS

t

i)

2

and

.

From (X t— ( v * v B s G* — *

H"t)andat-

(X t— ( W v " s G* — •

tjT-

f o l l o w s by means of Consequently,

for every

G" S G

with

vVfvtG

1 1

)

2

—

tjtS

t2 t ) .

T)

EDIII. (vs

f i n i t e S - s o r t e d subsystem finite

t

(vs H * t

,

tgT)

v * s X*

v« F*( # , G , v ) / = 9 for every s £ S , and denote the elements of F ( # , G , v ) .i.e. the equivalence classes, by (3.2.2.b) [t.Gj-{(r,G)

| Ci f- (vs G

n-tuples of elements of ing way: (3.2.2.C)

F(£?,G,v)

•tSr)}. will be abbreviated in the follow-

[I.GJ - ([lt,G],[2t.G],...,[nt,G]) €F(«-,G,v) w where

w s f n j — S

and

,t€T(2J»v)w«

Additionally to the S-set F(££,G,v) we define an -indexed family of fundamental operations in the following way: If 6-fP.fiT: w — s. w: [n] — * • S then the domain of the fundamental operation 6F(£?,G,v) Is defined by

(3.2.2.d)

d o m f f F ( « ' G ' v ) . {ft,G}| t £ T ( S , v ) w a

120

h - (v: G —

and g

i5

T(2.v)

}

|

6F(#»G.V)

And- the application of (3.2.2.e)

[lfG]6F(tf.G.v)

„

i8

defined by

[l5T(tt.v)tGj

#

The independence of the definitions(3.2.2.d) and (3.2.2.e) from the choice of representativee results immediately from the generalized substitution rule (3.1.5). Before proving that

F(¿?,G,v) is an

# - a l g e b r a freely generated by

( v : G ) we give another important property of Theorem 3.2.3. Let equations, where

(wsH ) Y

y€Y, (X t —

. given by

is a solution of (vs G

P r o o f . The case

ws Y

» S

be any eet of existence

may be Infinite. A n assignment

f £F(^,G,v)w every

with

F(C?,G,v).

•

yf»

(ws H )

[yt.G] in

,

teTiS.v)^

for

F(tf,G,v) if and only if

Ht: ) .

( w : H ) » ( w : 0 ) is trivial because

a

(vs G

rtS

t t )} ,

. [tt.G]

t € T( £ .u) , u : Z

S .

Consequently, we obtain a

(v: G

•

tj

t § tjT ) ,

(vs G

t 2 t S t 2 T ) , and

[tji.G] = [ t 2 7 . G ] for

f >[t,G]£F(ff1G,v)(||!H) #

For every

t - (v: G

tj t 1 t 2 T )

(w: tjitg ) £ ( w s H ). iZ h -

(vs G — •

, and so

This is identical with

hT ) .

121

For the proof of the converse direction we assume (X

I— (v: G

H t )

or a H- (v: G — • for every

t^t 2 t 2 1 )

(w: tj'tg ) e(w: H ) .

As s result of this assumption and Corollary 3.1.4 (v: G —

a y -

t 1 l 2 tjj: ) ,

t j . 6 t2 t )

h- (v: G —

(X

follow, and so we obtain 6

[t.Gj € d o m t ^ ' ^ n d o m t ^ * '

' ^

and

[i.e]eF(a.e.v)(lItiit2, for every

(ws tj8t 2 ) £ (w:H ), i.e.

f - [!.G]eF(tf,G.v)(wsH) .

I

Theorem 3.2.4. For every set of elementary implications set of existence equations F(C?»G,v) is an

(v: G ) , v: X

¿? -algebra freely generated by

Proof. First of all we prove that fore let (w: H and

•

t

i"t2 )

_f 6. F( (X ,G,v) ^ (J ( - (vi G

where

¿eT(2,v)w

In addition,

F(£?,G,v)

wlth

Y

#

and every

» S , the partial algebra

"" S

(v: G ) .

ie an be

an

y

(X -algebra. Thereaxlom

j . According to Theorem 3.2.3

of

this means

• H t ) with

_f = [l» G ] •

¿it— (w: H — »

^ holds as a consequence of (3.1.3),

so that Corollary 3.1.8 yields (X\—

(ws G — •

tj^i t 2 l ) .

A further application of (3.2.3) leads to f - [t.ej6F(a.G.v)(witi2t2) Consequently,

F(£?,G,v)

•

satisfies every axiom of

(X .

Next we show that the assignment e£F(#,G,v)v

.given by

is a solution of ( v s G ) For

ieT(2J,v)v

with

e • [i,G] and

122

in

x e - [x,G]

for every

x£X,

F(£?,G,v) .

xi" x G « gT .

for every

xeX

the following holds

This implies Ci I— (vs G — • G ) so that Theorem 3.2.3 yields eg F(C?,G.v) (v}G J . Finally we assume that A is any solution. According to (3.2.2.b)

£?-algsbra and

£eA(vtG)

an

Y

(X h— (v: G — tSt ) holds for every [t ,G] e F( (X ,G,v) , and by Theorem 3.1.9 we obtain A

(vsG) g

A

(vst.St) *

Hence, for every define the homomorphism

A a edon t , and so we can

It follows that

a* j F(#,G.v) —

A

by [ t , G ] a # - a t A for every [t,G] e F( £f,G,v) . It 4remains to show that this defines rsally a homomorphism with £av • — • For every

x6X,

xea* Now let and

6€

- (xs) a* , with

- [x,G] a* - a x A

6 : w

- xa .

• e , w: [ n]

» S

, be any operator

[t.G]€dom6F . On account of (3.2.2.d) thia implies ct »— (vs G — •

t e r T ( z ' v ) S ter T (S.v)

and therefore (Lt.Gle^^'6'^)»^

-

[t6T(S.v)>Gj8#

.

.(tffTiS.vijA.

- (a(lt)A a ( n t ) A ) 5 A - (ta*)ffA. £ This proves that a : F(££,G,v) — * A is a homomorphism with the required property. Since the S-set X* » (X* | s € S ) given by X* - {[x,G] | xv-s } is a generating S-set of F(#.G,v) , the assignment extended to a homomorphism in at most one way.

aeAv

can be •

By means of the prsceding results it is easy to prove the completeness of ¿?-derlvabllity. Theorem 3.2.5. For any two sets of exlstsnce equations (X I- (vs G

(vs G ) , (vs H ),

H)

123

A

holds If and only If Proof.

(V.G) —

Since Theorem 3.1#9

A

(v:H)

18

true

for

ever

Y

#-algebra

A.

represents the necessity, we have only to

prove the sufficiency. A

/ (V

.ui I V in )

18

true

* o r every

(X-algebra, then we obtain for

F ( # , G . v ) the relation

For

1 6T(2IV)v

with

xi » x

for every

x£X ,

[i.G]eF(tf,G,v)(v:H) holds, if and only if

(X I— (vs G Since

i -

Id

x(2

[i.G]

Hi ) .

v)

an (w: {target(x)2 8 ource(y) , z 8 comp(x.y)}) . If we consider the set of existence equations (w: Gj) = ( w:{target(x)§source(y) , source(x)8source(z) , target(x)Starget(z)} ) then

F(£fp A T ,G 3 ,w) can be represented by the following diagram:

The small category, represented by the following diagram • source(y) target(x)

source(x)

target(y) source(z)

comp(comp(x,y),z)

I^jt

tar

9»t(z)

can be described by (w: G 4 ) • (ws{target{x)2source(y) , target(y)-80urce(z)}). Finally,

F(£?* AT ,0,w)

describes the following small category:

source(x)o-

•otarget(x)

source(y)o-

•otarget(.y)

80urce(z)o-

»otarget(z)

We see that free

¿?* AT -algebras ,i.e.

¿i"*AT-algebrae freely gene-

rated by an empty set of existence equations est,

They coincide

(v:0),ere of no inter-

with small categories of isolated morphlems.

125

3*3

Partial quotient algebras

At the end of Section 2.4 we eald that the conetructlon of quotient algebras by means of congruence relatione cannot be generalized In a straightforward way from total algebras to equolde. We demonetrete this assertion by small catsgorles.

to

For this rsason we conelder again the email category A isomorphic " i t h v s i x , y i — » { O b j . M o r } and x v » y v - Mor.

The two-sorted relation e

e

?ObJ " { ( » > l

(>

eeA

with

objJ

U

t (target(x).source(y))}

and

?Mor - { < " . • > I n , e A M o r ) is a congruence relation. I.e. q Is a two-sorted equivalence relation and the carrier of a subequoid of A X A . However, the algebraic structurs of A cannot be carried over canonically to the two-sorted quotient set A/g , given by ( A /«?>„ or - {

C * v m

9

}

( A / R ) o b J - {[source(x)]

[target(x)] ^ • [source(y)]^,

[targetiy)]^} . Only the unary total operations source A : A ^ can be carried over to

—

AQbj A/p

,

target A : A M o r —

A

obj

by

source^^f^x]^) • [source(x)l^ , target A / ?([x]^) - [target(x)]^ - [eource(y)J ? - source A / ?([y]^) target A / ?([y] ? ) - [target(y)] ? . But now there is a non-trivial solution of the domain condition of the binary composition, namely ([x]^,[y]^) , end there is no congruence class that could represent the value of the composition compA/?([xl?,[^) . This fact could cause the conjecture that no quotient S*AT-algebra A/(> exists. But, this conjecture is caused only by lack of success of en elementwise construction. On the other hand the homomorphism f: A — » B as dsfined In Section 2.4 Is natural for .

126

In the following wo show that for every many-sorted relation q , for every aet of elementary implications (X , and for every (Z -algebra A a homomorphlem ft A • B between & -algebras exists which is natural for ^ . To begin with, we return to the preceding example of the homonorphlsm f: A m B natural for (> as defined in Section 2.4 on the pages 94 and 95. We recall that A is isomorphic to F ( , v ) and B is isomorphic to F

(#£ A T t{ t a r g« t (*) S 8 °"rcs(y)},v) •

We can aee that up to isomorphism B is constructed from A by adding the ordered pairs of £ , considered as existence equetione, to ths set of existence equations which describes A . In this construction no missing elements annoy, becauss [comp(x,y),(v:{target(x)Ssource(y)})J exists in Fitf^T.itargstfxjSsourcefyjj-.v) due to # * A T b— (v: { tsrget(x)2source(y)} — c o m p ( x , y ) S c o m p ( x , y ) ). In order to investigate the general construction of quotient & -algebrae, we show first that every (X -algebra A is isomorphic to an algebra F(££,G,v) where ( v : G ) can be derived canonlcally from A . Let (X be any set of elementary Implications and A any gebra of type 2 » ( S , ( * : Q — » S* X S ) . Then we define:

(X -al-

X A - {(a.s) | a e s , a i A j ; V

A

:X

S

A

a

WITH

wlth

• A e (Va)

(8'8)VA • a

8

( » )

•

8

a

f o r 8V8l for ever

y

"y

(a,s) I *e t ( S . v a ) 9 , a £ S , a - e A t A } .

is a solution of

(v A , G A ) in

A , and the unique extension

to e homomorphlem •A

: f

(^'gA'vA) —

A

is an isomorphism, i.e. (3.3.1)

127

This representation shows that we do not restrict the generality if we deal only with ¿^-algebras of ths form F(£?,G,v). Obviously, there are many different sets of existence equations which deecribe ons and the sane C( -algebra. In general it is recursively undecidable whether F(£T,G,v) and F(£f,H,w) are isomorphic, even in the case that ( v : 6 ) , ( w : H ) are finite sets of existence equations on finite S-sorted systems of variables« Now we can start with the construction of a natural homomorphism f : F(tf.G.v) — F(tf.H.w) for every S-sorted relation (> in F ( # , G , v ) . For every binary S-relatlon in F(£?,G,v) we define the so-called associated set of exletence equations (v:G ) by (3.3.2) (vsG p ) - { (vs t 1 S t 2 ) | ( [tj.G] , [t 2 ,G]) ses} . Theorem 3.3.3. For every the homomorphism fsF(ii.G.v) —

S-sorted binary relation

£

in

F(£?,G,v)

F( £?,GUG ? ,v)

with [t ,G] f • [t.GUG^] is natural for Proof.

for every

[t, G] e F( £?,G,v)

^ .

First we show that the previously defined S-function f:F(tf,G,v) —

F(6t,GUG?,v)

is a homomorphism. Let every

ee f(#»g»v)(v.q) c£ -algebra

B

th

e universal solution of

and every solution

exactly one homomorphism b*:F(#,G,v) - —

B

with

ebj

b£B, — tv

;

(v:G) , I.e. for there exists

» b.

The assignment • _1 e F( (X ,G U G^ t v) v with x_l» [x,GU G J for every x e X is a solution of (v: G ) in F( (X ,G U G ,v) because of G £ G U G^ and Theorem 3.2.3. The given S-function fsF(#,G,v)

»

F((7,GUG?,v)

is then the unique extension of the solution to a homomorphism. It remains to prove that f:F(tf,G,v) • F(6?,GU G^,v) satisfies condition (2) of Definition 2.4.8.

128

l6F(t?,GUG^,v)|v.Gj

If

B

with

is any

-algebra and

[tj,G] g g

g: F ( # , G , v ) — »

- [t 2 ,G] g 9

for svsry

thsn the assignment

e g v e Bv

g : F( ¿f ,G, v) — »

is ths unique extension of

morphism. phism

B

Since

_e g y £

B

is a solution of

(V'GUG

h: F(£?,GUG^,v)

•

f-h : F( (I.G.v)

—

B

B

t

)

lere

^

with

is a homomorphism

( [ t ^ G ] , [t2,G] ) £

.8 e S ,

(v:GUG^)

in

e gy £ Bv

B

and

to a homo-

exists exactly one homomor-

lhy- sgy

.

By that

B

is a homomorphism with e(f h ) y If

(efy) h y = 1 h y -

h': F ( i 2 , G U G ^ t v ) — •

B

egy

and

g - f-h

.

is a further homomorphism

with

f-h*«g

thsn

IK

-(*

f

v>

h

;

-

£(fvh;) a

£(f'h")v " 1 9 v

- l h

v

and so h - h' .

•

Theorem 3.3.3 shows that ths preceding construction of the natural homomorphism

fs-A

• B

between

for constructions of quotisnt

( ^ ' ^ - a l g e b r a s is representstive

(X-algebras.

A s nsxt w s are concerned with a second useful construction, the so-callsd

p u s h o u t - c o n s t r u c t i o n

theory of categories. Lst f: A between

B ,

forms a (i) (ii)

g: A — »

C

C( - a l g e b r a s with common domain f*s C — •

Q .

g*: B — •

p u s h o u t f-g* = g . f

#

of

hs Q — •

Q'

Q

f:A

• B , g: A

f's C — »

Q ' , g": B — •

with

g * h - g* f: A

Q uniquelyupto

and •

f*.h - f

if

Q'

with

B , g: A — »

. C

determine the

isomorphism.

Before we w i l l give a general construction of we show

» C

thsre exists exactly one homomorphism

Evidently, the homomorphisms 'pushout-#-algebra

A . A couple of homomorphisms

;

for every coupls f*g' = g.f•

, taken from the

be givsn homomorphisms

pushout-¿{-algebras

that the pushout gsnsralizes the construction of sums

coproducts of

£?-algebras. This construction

or

corresponds to the con-

struction of ths frss sum of groups in ths cass of goneral

(Z-algebras.

129

The construction of sums can be derived from the more general pushout construction by means of so-called initial Definition 3.3.4. Let ft -algebra •X

A

'ZC e ALG(£?)

be any class of

i n i t i a l

is called

¿^-algebras. OC -algebras.

with respect to

An

the class

. or initial Of-algebra , If (i) (li)

A £ K. j For every B 6 b: A — » 8 .

there exists exactly one homomorphlsm I

Corollary 3.3.5. For every set of elementary implications Ct -algebra

F(#,0,a)

Proof. Let

B

la initial with respect to

ft

the

A L G ((X).

be any ¿T-algebra. Then there exists exactly one aesign-

ment of the empty etring

» S

0

in the carrier of the ¿^-algebra

B , namely the inclusion of ths empty set into ment is the only solution of exactly one homomorphlsm We point out that

(¿>0)

in

b:

B . This unique assign-

B , so that by Theorem 3.2.4 — »

B

exists.

I

F(ft.,0,?l) may be the empty algebra or that some

of its carrier sets may be empty. For instance,

is the empty category. Now let

a:

— A ,

b: F(ft,ft,A) — •

B

be homomor-

phisms between ¿^-algebras. According to (3.3.5) these homomorphisms are uniquely determined by g: 3 — *

C

A , B € ALG(£?) . I f

are homomorphisms between

main, then in any case

the compositions

coincide with the unique homomorphlsm A n element-free description of phisms

I: A — »

A+B

,

of homomorphisms f: A — » homomorphlsm

hsA + B

A +B

C , g: B — • -

C

a»f

with

and

b«g are equal and

c: F((X.,0,'h)

the sum

J: B — »

f: A — » C ,

¿f-algebras with common codo-

A+B

» C .

is given by homomor-

such that for each pair C

there exists exactly one

i*h • f

and

J«h • g .

If we now consider the puehout of the uniquely determined homomorphisms a:

— » A ,

JL

a': B — • > Q , ere isomorphic.

b:

—*>

B

given by the homomorphisms

M

b :A — »

Q

, than

the

¿T-algebras

Q

and

A+B

Therefore the construction of the sum of £? -algebras

is a special case of the conetruction of a pushout of homomorphisms bstween (Z-algebras.

130

Now we describe the general pushout construction. For this purpose we introduce a notatlonal convention. If fs Fitf.Gj.Vj) — is a homomorphism denotes any

and

with Then

[*»Gj]

an

Y element of

F(«,G2,v2) .

9: Ftit.Gj.Vj) —

F(£*,G 3 ,V 3 )

S , v2s Y —

f*s F(0:.G 3 ,v 3 )

S

form a pushout for 2

+V

3

S

f

y u z

, Vj» Z — -

• F( 0;.G.v 2 >v 3 )

9*: F ( ^ . G 2 . V 2 ) —

V

tf

Let be given homomorphisms

fi F f t f . G ^ V j ) —

Vj : X — •

FtCX^G^.v^) , then

f » ® ! ] * " [* *»G2l •

terra with

Corollary 3.3.6.

F(tf,G 2 .v 2 )

and

Y r\ Z - ff .

and

F( £?,G,v 2 + V 3 )

and

g , where

s

~ ~

S

with

x(v 2 +v 3 ) - x v 2 x(v 2 +v 3 ) • x v 3

if

xeY

if

and

x€Z,

and where G - G2 U G 3 U {(v2+v3: x f S * g ) | The homomorphisms

f'

and

g'

xex}.

are defined by

[t,G2]g

#

- ft,G]

for every

[t.G 2 ]

F(6*,G 2 ,y 2 )

[t.G3]f

#

- [t.G]

for every

[t,G 3 ]

F(tffG3.v3).

£

and

M

Proof. f r and g* are homomorphlsme according to the same reasoning as in the proof of Theorem 3.3.3. So it remains to show that f.g* a gof* and that Let

M

fr

and

g

M

form a pushout for

¿ € Fi^.Gj.Vj)^

If we can prove

,q j

l(f.g*V

V

1

- l(g f*)

holds for every

and

g .

denote the universal solution. V

1

then

according to condition (2) of Definition [x f,G] • [xg,Sj

f

f.g # - g f*. follows

3.2.1

x £ X , since

( v 2 + v 3 : x f I x g ) C (V 2 +Vjs G)

for every

x€X.

This yields *(I(f'9*)) • (xl)(f.g # ) - ( [ x . G j f ) g*. and so

[xf,G2]g#-

- [x f.G] - [xg,G] - x(l(g.f*)),

131

l(f.g*) - l(g«f*)

and

f.g* - gif* .

Now let g': F(0t,G 2 ,v 2 ) — •

B

f's F(tt,G 3 ,v 3 ) —

B

be homomorphisms with B£ALG(£?:) and Then there Is exactly one assignment —€

B

v +v 2

with 3

f»g* • g»f* .

yj>- y(ig') z b = z(J.f')

for every for every

yeY and zi Z ,

where F

1 e F( (X .G^.Vg) (y^jG^) •

^'G3'V3>(V3:G3)

are the universal solutions, respectively. Because of

G 2 £ g , Gj & G , v 2 ^

interpret

G^

and

G3

v

2+v3 •

and

v

3 ^ v2+v3

as sete of existence equations on

*e v

2

+v

can

3 '

8n
t^ does not depend on

yeY

B J> t 2 does not depend

the following holde:

bt®

- (¿g") t B - ( l t ^ ' ^ ' ^

1

) g- - [tj.Gjg« - (x f) g' -

- x(f.g') - x ( g . f ) - [t 2 .G 3 ] f' - b t\ . This means "

6 B

(v 2 +V 3 SG) '

Consequently, there is sxactly one homomorphlstn h: F ( « . G , v 2 + v 3 )

B

with

k h ^

}

-b

.

where k£F(i*tG.v2+v3)(v2+V3:G) denotes the universal solution. If

k + v 2 , l< |v 3 ,

b*v2,

b to the S-sorted subsystems it follows

132

denote the restrictions of v2

and

v3

of

v2+v3;

Jt

and

respectively,

• k

Vj ,

lg'.kv2,

if'» b W

J (

lg'.

b*v2

and additionally J, f * h - ( k i v 3 ) h - ( k h ) v 3 .

biVj.^g'

l g ' h - ( k l v 2 ) h - (k h) V j . b t V j . i f ' , and eo f*»h - f

and

g*»h - g' .

I

Since the pushout construction is not 90 familiar outside the theory of catsgories

we illustrate this construction by some examples.

Furthermore, this construction will be used extensively in Chapter 4 for the construction of complex specifications of abstract data typea of simpler o n e s . First we remark that the pushout construction encloses the of substructures' . Evidently, the sst-theoretic union of bras is in general not the carrier of an A U B

of

(X-subalgebras

stand the smallest

A

and

B

-subalgebra of

Therefore, the union of

- s u b a l g e b r a . By the union

of an C

'union

eource(x 3 ),

target(x 2 )Starget(x 3 ),

target(x 1 )§source(x2)} ) >

and

for i=1 2 3. *i 9 * x i « » The small category can be graphically described as follows:

According to Section 3.4 the homomorphism g: A - C is natural for its kernel. Since B and C are not isomorphic and kerf • k e r g , the homomorphism f: A » B cannot be natural for its kernel. Let f: A » B be any homomorphism between small categories qj : A — • > Qj may be natural for kerf , so that a functor h

l!

* B

with

and

f » qjohj

exists. Qj

is Isomorphic to the homomorphic image

(A f)

If and only if

hj: Qj » B is injective. One can prove generally that Qj and (A f) are not isomorphic iff there are morphisms x,y,z A M with A A A a x ji source (x), y + source (y), z > source (z), target (x) ¡i source(y), and If

z ^ , o r - c o m p B ( x ^ o r . y f„ o r ) • hj: Qj

» B is not injective there is a non-trivial homomorphism

q 2 s Qj — • which is natural for

Q2 kerhj

and there is a homomorphism

137

hg i Q 2

• B

hj • q 2 o h 2 *

with

One can prove additionally that ( h 2>ObJ " ( Q 2 ) O b j

B

Obj

la always Injective, and ao In there are no morphiems x,y,z having the propertie8 described above with respect to hj: Qj — B . Thi8 implies that h 2 : Q 2 — » B becomes Injective for both aorte Obj , Mor and that the homomorphlc image (A f) becomes isomorphic to Og . These considerations show that the homomorphlc image (A f) of a homomorphism between small categorise can be reconstructed by one or by two factorizations Qj or Q 2 , respectively. This existence of Iterated quotients motivates the following definition. Definition 3.4.1. Let 6 be any hep-signature and (X any set of elementary Implications. A finite or countable sequence

( (fi. I

l e I

)•

of homomorphisms between (X -equolds is called a c h a i n of i t e r a t e d q u o t i e n t s of f s A — » B , where A,B€ALG(C?), if (1)

f • fj

(2)

q ^ Q4

(3)

qA

and

fj » q-j»'^!

» ^i+i

i9

Is natural for

not

an

ker f t

f°r every

iel ;

isomorphism for every for every

i€I

iei .

If I « [n], then the natural number n la called the l e n g t h of the chain of iterated quotlente of f: A — » B . If alternatively I » 14 , then we say that the chain of iterated quotients has infinite length, • With this notion the theorem of homomorphism of total algebras can be formulated as follows: Evsry chain of iterated quotients of a homomorphism between total algeSrae is of length one. The preceding considerations show that every chain of iterated quotients of a homomorphism between small categories is at moat of length two (see Reichel (1969)). According to the following theorem the length of chains of itsrated quotients is not bounded in the general case. Theorem 3.4.2. There are chains of iterated quotients of homomorphisms between equolds of infinite length. Proof : Theorem 3.4.2 will be proved by constructing an infinite chain of iterated quotients. According to that let us consider the

138

hep-signature 6

la

aorte oprn

end

N ,P n — » N a(N) — • N o(P) — . » N r(x sN Iff x 8 - n ) — •

and let ua take the 8-equoids The 9-equoid A -Is given by A,, - {(O.x) | x e W } n

A

,

A

and

B .

Ap - 0 .

- (0,0) ,

(O.x) ® A • (O.x+l) for every A A , dom an » dom r - 0 . The 8-equoid B„ • n

P

0

B

B

x e 3N . and

may be defined ae followsi

» X

H

,

Bp -

U ,

- (0,0) ,

(x,y) a 8 • (0,0) xi

for every

Q

(x,y)e B N ,

• (x+1,0) for every x € B p , B B dom r • B N and (x,y) r • x for every

One can see easily that one homomor'phlsm f: A

•

A

Is an Initial

(x,y)e B N .

6-equoid, so that exactly

B

exists, which is given by (O.x) f N fp: 0

—

-(0,0)

for every

(0,x)£AN,

Bp .

Therefore,the kernel of

f: A

(kerf)N

- AN X An

(ker f) p

- ApXAp .

» B ie given by

and

By virtue of the construction of natural homomorphisms. according to Theorem 3.3.3, we obtain the first quotient ( V n

q^: A

" t (0.0>.(1.0),(l.l).... # (l t x)....

» Q 1 by setting: | x€*l},

(Qjip - { 0 } . n< Q l> - (0,0) , (0.0) s { Q l } - (0,0) , (1.x) s ( Q l ) - (l.x+l) On

(Q

for every

x

,

l l ' . (1,0) ,

139

don>r(Ql'-

(0,0)

and

(0,0) r ^ l * - 0 .

The u n i q u e l y d e t e r m i n e d homomorphiem f 2 s Qj — ^

B

with

f • fj

-

i s g i v e n by (J.O) ( f 2 ) N

-

(J.O)

for

(l,x) (f2)N

-

(0,0)

for every

0(f2)p

k

(J.x) J»

(A

x € K

with

-

o

r

k

2

~

x ^ 1 ,

and

by

(k,0),(k,l)

(Ak)p - { o , l , . . . , k - l } n

f

( ' ^ k »

(Afc)N - { ( 0 , 0 ) , ( 1 , 0 )

}

.

- 0 .

I n d u c t i v e l y we d e f i n e

(A

J-O.l

(k,x),...

|

xe»},

,

(0,0) *(0,0)

e(V -

(k,x+l)

k ' • (J+1,0)

dom r ( A k }

if

for

x-0 if

and

J£k-1

k-J ,

J-0,1

x£W k-1 ,

- {(0,0),(l,0),....(k-l,0)}

( J »0)

-

J

for

,

,

J-0,1,.. .,k-l .

The u n i q u e l y determined homomorphism f k . Qk —

B

I s g i v e n by (J.O) ( f k ) N -

(J.O)

for

(k,x)(fk)N-

(0,0)

for every

xe»

J (fk)P

for

J-0,1,...,k-l

,

-

J

J-0,1,...,k

, with

x t 0 ,

and the homomorphism Q

Q

k —'"

which i s n a t u r a l f o r

k+1 ker f k

i s d e f i n e d by

(J.0)(qk)N

-(J.O)

for

(k,x) (qk)N

-

for every

J (qk)p

-

J

(0.0) for

J-0,l,...,k,

J-0,1,....k-l

x € W with

x / 0 ,

.

T h i s e x p l i c i t l y d e f i n e d i n f i n i t e c h a i n of i t e r a t e d q u o t i e n t s of a homomorphism f : A - 3 p r o v e s Theorem 3 . 4 . 2 .

140

N o w w e a r e l o o k i n g Into s i g n i f i c a n t relatione b e t w e e n the m a x i m a l c h a i n of I t e r a t e d q u o t i e n t s of a h o n o m o r p h l e m honomorphlc image

f: A

• B

F o r this r e a s o n w e c o n s i d e r the h o m o m o r p h l c image B

of Proof 3 . 4 . 2 .

(Af)„ - {(x.O) | x € W } , (Af)

the

(A f) .

m o m o r p h l s m fs A — •

so that

and

(A f)

p - H ,

is a n i n f i n i t e s u b e q u o i d , w h e r e a s the

image c o n s i s t s of o n e e l e m e n t , n a m e l y w i t h i n c r e a s i n g index

k

the

of the h o -

W e eee that

kth

set-theoretic

(0,0) . F u r t h e r m o r e w e see

i t e r a t e d quotient

and m o r e s i m i l a r to the h o m o m o r p h l c i m a g e . H e n c e , the q u e s t i o n a r i s e s . C a n the h o m o m o r p h l c image

( A f)

Q^

that

becomee

more

following

of a h o m o m o r p h i s m

be

c o n s i d e r e d a s the limit of the m a x i m a l c h a i n of i t e r a t e d q u o t i e n t s of f: A — »

B ?

In both c a s e s of total a l g e b r a s a n d small c a t e g o r i e s , w h e r e the m a x i mal

c h a i n s of i t s r a t e d q u o t i e n t s a r e of finite l e n g t h , this

question

c a n bs a n s w e r e d p o e i t i v e l y , b e c a u s e w e c a n e v i d e n t l y d e f i n e that

the

last quotient of a finite c h a i n of i t s r a t e d q u o t i e n t s is the limit of this c h e i n . But w h a t is the limit of a n i n f i n i t e c h a i n of i t e r a t e d q u o t i e n t s ? T h e following d e f i n i t i o n of the limit of a c o u n t a b l e c h a i n of P h i 8

"8

f

A

f

l L. A

1 ^ ^

f

2 S*. A

2 ^ ^

3

f

f

„ 1

L.

3 ^ ^

„

•••

mor-

n_

n

• • •

Is d e r i v e d from the t h e o r y of c a t e g o r i e s , w h e r e It c o r r e s p o n d s to the c o l i m l t o t the c h a i n . D e f i n i t i o n 3 . 4 . 3 . Let implications, ( f

i ' Ai

A

a c o u n t a b l e c h a i n of l i m i t (1)

be a h e p - s i g n a t u r e ,

a set of

elementary

i+1

I

1

¿?-equolds.

An

¿?-equoid

L

is c a l l e d

a

of this c h a i n , if there a r e c a n o n i c a l q

(2)

0

and

i!

A

i

L

^

morphisms

*ith

for e v e r y f a m i l y of ^

9

i

s A

i —

8

)

B

w l t h

with

f

iocli+i

f o r

every

i 6 >1 , e n d

homomorphisms

there is e x a c t l y o n e h: L — •

"

9

i "

f

i'9 i + l

f o r

ever

Y

homomorphism

gA - q »h

for e v e r y

1 e J4 .

|

141

Intuitively, condition (1) requires that L is an upper bound and condition (2) implies that L is a least upper bound of the chain. Additionally, condition (2) guaranteee that a limit of a chain is unique up to isomorphism. If we allow in Definition 3.4.3

finite chains

( f

A i8 A i — i+1 I i e M ) and modify conditions (1) and (2) for finite index sets, then every finite chain f 1L. A f2o f„ A n . 1 * 2 ^ ^ ' ' • ^ ^ n+1 has a limit, namely L • An+1 where the canonical homomorphisms q^: A t — L for it[n+lj A

are defined by q

n+l " V

q

i "

f

Id

An+1 '

f

l

+

of

r for

1° "'i+l

f

"

n • "n-1 " f n-l' f n •

n-l°fn ' i ' e -

1 6

W

•

As next we are concerned with the existence of limits of infinite chains . Corollary 3.4.4. Let

6

be a hep-eignature and

(X any set of elemen-

tary Implications. For every infinite chain ( flS A

±

—

Ai+1 I l e w )

of homomorphisms between (X -equoids there exists a limit which is unique up to Isomorphism.

L£ALG(e,#)

Proof. According to general-theorems on colimits in categories (see MacLane (1971) ) Corollary 3.4.4 would be an immediate consequence of Theorem 3.3.3 and Corollary 3.3.6. But we will give here an explicit construction o-f L . If S the set

is ths set of sort names of L* - {(x,J) | J i W ,

For every ordered pair morphism f

142

i.j' A i — -

A

b

v

eeS

x €(AJ)e} .

(i,j)£AIXl ) G/K where K is the commutator of G . With respect to partial algebrae we can also speak of a small category freely generated by a directed graph. This construction can be implemented by the 80-called category of finite paths over a directed graph (see Hasse.Michler (1966) ). From Definition 3.5.5 it follows immediately that a T^-equold F freely generated by a Tj-equoid B according s Tj » T2 is determined uniquely up to isomorphism. By means of the calculus introduced in Section 3.2 we can prove that for every theory-morphism *

Bf*

B = B £# ©

and

there is e x a c t l y one f«g

#

homomorphism

—

B

M

with

M

(f ® g ) £

«

according

f . T h i s p r o v e s that

: Tg

A

is f r e e l y g e n e r a t e d b y

A£

» T .

A s next w e p r o v e c o n d i t i o n (1) of D e f i n i t i o n 4 . 3 . 1 . Let be g i v e n BeALG(T2,A2). £;

(T0>

Then

such that

C £ ALGITJ, A J )

B e c a u s e of B © C

6 ALG(T,

C £