ファジィ関係の代数的定式化と表現定理

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九州大学学術情報リポジトリ

Kyushu University Institutional Repository

ファジィ関係の代数的定式化と表現定理

古澤, 仁

九州大学システム情報科学研究科情報理学専攻

https://doi.org/10.11501/3135054

出版情報：Kyushu University, 1997, 博士（理学）, 課程博士バージョン：

権利関係：

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Algebraic Formalisations of Fuzzy Relations and Their Representation Theorems

February 1998

Hitoshi Furusawa

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Abstract

The aim of this thesis is to develop the fuzzy relational calculus. To develop uch calculus in this thesis w study four algebraic formali ation of fuzzy relations which are called fuzzy relation algebras, Zadeh categories, relation algebras and Ded kind categories, and their repres ntation theorems.

The calculus of relations has been investigated ince the middle of the nineteenth century. The modern algebraic study of

(

^binary

)

relations, namely relational calculus was begun by Tarski. Jonsson and Tarski gave the first definition of Boolean relation

.

algebras and studied more generally on relation algebras as Boolean algebras with op- erators. Freyd and Scedrov developed and summariz d categorical relational calculus, which they called allegories. And some various definitions of relation categories exist;

for example, Dedekind categories and Schroder categories were introduced by Olivier and Serrato.

The repre entation problem for Boolean relation algebras was proposed by Tarski as the qu stion of wh th r ev ry Boolean relation algebra is isomorphic to an algebra of ordinary homog neous relations. A negative answer to such problem for Boolean relation algebras was given by Lyndon. But there are many ufficient conditions that guarantee representability for Boolean relation algebras. The first result was provided by Jonsson and Tarski. Schmidt and Strohlein simplified the proof of the representation theorem by using a notion of point relations. They also showed an important role of point relations in computer science. Since the notion is suitable as a relational

(

^algebraic

)

characterisation of points in a set to the author's intuition we follow spirits of Schmidt and Strohlein in this thesis.

In relational calculus one calculates with relation in an element-free style, which makes relational calculus a very useful framework for the study of mathematics, logics and theoretical computer science and also a useful tool for applications. For example nonclassical logics can be formalized by using relation algebras. And much of the theory of nonclassical logics was used in program logics. So relation algebra are fundamental to programs. In fact much of th theory of relation algebras was applied to program s n1antics and prograrn development. The relational calculus can be appli d to not only th theory of progran1s but al o databa , natural language graph set theory

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and so on.

Iathematical treatment of uncertain or incomplete information i an important problen1 in fields of computer scienc such as expert system, natural language pro

cessing knowledge representation, pattern recognition, and databa e . To treat uch informations mathematically, Zadeh introduced the concept of fuzzy sets, and Pawlak introduced the concept of rough sets. It was shown by Cattaneo that the notions of fuzziness and roughness can be formally analyzed in the structure of quasi Brouwer

Zadeh po et. In this thesis we aim at the notion of fuzzy relations. Since Zadeh's invention, the concept of fuzzy sets and relations has been extensively investigated in mathematics, science and engineering. Goguen generalized the concepts to taking values on partially ordered sets. Fuzzy relations and calculus of fuzzy relations are bases of whole theory of fuzzy sets. Also it is well-known that the notion of fuzzy relation is a ba ic one in processing fuzzy information. For example each rule in a fuzzy expert sy tern is formalized mathematically as a fuzzy relation between the sets describing antecedent and consequent. The equations were applied to models in system analysis, patt rn recognition and so on. In the context of applications of the notion of fuzzy relations to computer science, the notion has been studied. In the studies, treatment of fuzzy relations were in an element-wise style. On the other hand, as we discussed above, in relational calculus we calculate with relations in an element-free style, which has given us benefits to apply theories of relations to computer science.

This fact suggests that if we have fuzzy relational calculus in which we calculate with fuzzy relations in an element-free style, we get benefit as in the cas of ordinary r la

tional calculus. This is an original motivation of our research. In fact, fuzzy relation algebras gave a theoretical basis to research of fuzzy difunctional dependency in fuzzy relational databases, and a rewriting system of fuzzy graphs by using single pushou ts based on a study of Zadeh categories.

11

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Acknowledgements

First of all, I would like to express my sincere appreciation to Professor Ya uo Kawahara for his valuable advice, many constructive suggestions and constant encour

agement during the course of this study.

I wish to thank to Professor Setsuo Arikawa and Professor Sachio Hirokawa for their helpful comments and encouragement.

I am grateful to Dr. Yoshihiro Mizoguchi, Dr. Hiroshi Ohtsuka, Satoru Kumamoto and Masao Mori for their valuable suggestions and criticisms.

I thank Dr. Gheorghe �tefanescu and Dr. Wolfram Kahl for their helpful sugges

tions and comments during their stay at Kyushu University. I express my gratitude to Professor Willem Adrian Labuschagne and Professor Dafa Li for their kind sup

port and continual encouragement since our first encounter at Logic Colloquium '95 in Israel.

I also thank to my friends Kusuo Horibe, Yoshihiro Fukumoto, Satoshi Matsumoto and Noriko Sugimoto for their friendship and encouragements.

Finally, I am especially grateful to my parents.

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Chapter 1 Introduction

The importance of relations is almost self-evident. Science is} in a sense}

the discovery of relations between observables. Zadeh has shown the study of relations to be equivalent to the general study of systems (a system is a relation between an input space and an output space).

Goguen}

^J.A.

{Gog67}

The calculus of relations has been investigated since the middle of the nineteenth century. The modern algebraic study of

(

^binary

)

relations, namely relational cal

culus was begun by Tarski

[

^Tar41

]

^. A definition of Boolean relation algebras first appeared in

[

^JT48

]

, and another equivalent definition app ared in

[

^JT52

]

. Jonsson and Tar ki studied more generally on relation algebra a Boolean alg bras with op

erators in

[

JT51, JT52

]

^.

(

^See

[

^Mad91a

]

for more details of the history of the study of Boolean relation algebras.

)

^{Mac Lane}

[

^Mac61

]

and Puppe

[

^Pup62

]

exposed a cat

egorical basis for calculus of additive relations. Freyd and Scedrov

[

^FS90

]

^developed

and ummarized categorical relational calculus, which they called allegories. And some various definitions of relation categories exist; for example, Dedekind categories

[

KF 196 KF97, OS80, OS95

]

and Schroder categories

[

Jon88, OS80

]

were introduced by Olivier and Serrato.

(

Olivi r and S rrato indicated some points of diff r nee and

irnilarity between the two categories in

[

^{0880 OS95}

]

^.

)

1

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In relational cal ulus one calculates with relations in an elernent-free tyle, which makes relational calculus a very useful framework for the study of mathcrnatics, logics

and theoretical computer science and al o a useful tool for applications. For example nonclassical logics can be formalized by using relation algebras [Orl88a, Orl88b, Orl94].

And much of the theory of nonclassical logics was used in program logics. So r lation algebras are fundamental to programs. In fact much of the theory of relation algebras was applied to program semantics and program development [BKSS91, BS91, BZ86, BM97, DOR94, D096, HH86a, HH86b, HH87, Hut93, KM92, 8893]. A typical result which got great benefit from Boolean relation algebras introduced by Hoare and He [HH86a, HH86b HH87]. The relational calculus can be applied to not only the theory of programs but also databases [BDJ 1[93 BBG+94 JOB94, OJB94], natural languages [Bot92a, Bot92b, Sup76, Sup79, Sup 1], graphs [KM94, Miz93, MiK95, 8893], set theory [Kaw95 TG87] and so on. Orlowska described benefit from applying relational cal ulus to a sort of cientific research areas in just one sentence in chapter 6 of [BKS97]

so clearly:

The main advantages of the relational formalisation are uniformity and modularity.

Actually, once problems in these fields are formalized in terms of relational calculus, these problems can be considered by using formulae of relations, that is, we need only calculus of relations in order to solve the problems. This makes easier to research these fields.

Mathematical treatment of uncertain or incomplete information is an important problem in fields of computer science such as expert ystem, natural language pro

cessing, knowledge representation, pattern recognition, and databases. To treat such informations mathematically, Zadeh [Zad65] introduced the concept of fuzzy sets, and Pawlak [Paw82] introduced the concept of rough sets. It was hown by Cattaneo [ Cat96] that th notions of fuzzine and roughnes can be formally analyzed in the structure of quasi Brouwer-Zadeh po et. In this the is we aim at the notion of fuzzy r -

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lations. Since Zadeh s invention

[

^Zad65

]

, the concept of fuzzy sets ha been xten ively investigated in mathematics, cience and engineering. Di Nola and Gerla

[

^{DG 6}

]

^tud

ied on nonstandard fuzzy sets. Goguen

[

^Gog67

]

generalized the concept of fuzzy sets and relations to taking values on partially ordered sets. Fuzzy relations and calculus of fuzzy relations are bases of whole theory of fuzzy sets. Also it is well-known that the notion of fuzzy relations is a basic one in processing fuzzy information. For example, each rule in a fuzzy xpert system is formalized mathematically as a fuzzy relation between the sets describing antecedent and consequent; see 4.1 of

[

^Ped96

]

written by Kandel et al. for details. Fuzzy relational equations were initiated by Sanchez

[

^San76

]

and applied to medical models of diagnosis. Later the equations were applied to mod

els in system analysi

[

Ped85a, Ped85b, Ped87, Ped90a

]

, pattern recognition

[

^Ped90b

]

and so on; see

[

^Ped96

]

for more information of fuzzy modelling. On fuzzy relational equations as a methodology for processing fuzzy information in relational structure are summarized by Pedrycts in

[

^Ped91

]

^. In the context of applications of the notion of fuzzy relations to computer science, the notion has been studied

[

^DK93, ^Nem86

]

^.

In the studies, treatment of fuzzy relations were in an element-wise style. On the other hand, as we discussed above, in relational calculus we calculate with relations in an element-free style, which has given us benefits to apply theories of relations to computer science. This fact suggests that if we have fuzzy relational calculu in which we calculate with fuzzy relations in an l ment-free style, we get benefit as in th case of ordinary relational calculus. This is an original motivation of our research. In fact, fuzzy relation algebras

[

^{KF 95}

]

which are introduced in chapter 3 gave a theoretical ba

sis to research of fuzzy difunctional dependency in fuzzy relational databa e

[

^OJ96

]

^,

and a rewriting system of fuzzy graphs by using single pushouts

[

^MoK97

]

based on a study of Zadeh categori s

[

^KFM96

]

which are introduced in chapter 5.

The ain1 of our r arch is to develop

fuzzy

relational calculu following Tarski's work. Also we con ider representation th orems for fuzzy relation alg bras

[

^{KF 95}

]

^,

3

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Dcd kind categorie

[

KF 196, KF97, OS80, OS95

]

, Zadeh categories

[

^{KF 196}

]

^{and re}

lation algebras

[

^Fur97b

]

. The representation problen1 for Boolean relation algebras was proposed by Tarski as the question of whether every Boolean relation algebra is isomorphic to an algebra of ordinary homogeneous relations in

[

^Tar41

]

. Such prob

lem for Boolean algebras were solved by Stone

[

^Sto36

]

. T he first negative answer to Boolean relation algebras was given by Lyndon

[

^Lyn50

]

. Nat ural constructions for non

repre ntable Boolean relation algebras was provided by Jonsson

[

^Jon59

]

and Lyndon

[

^Lyn61

]

. But there are many sufficient conditions that guarantee repre entability for Boolean relation algebras. T he first result was provided by Jonsson and Tarski

[

^JT52

]

by using a notion of univalent

(

^functional

)

relations. iaddux and Tarski

[

^MT76

]

^gen

eralized the result in

[

^JT52

]

. By using a notion of conjugated quasiprojections

[

^TG87

]

^,

Tarski

[

^Tar53

]

provided such a condition. All above results were generalized by Niad

dux

[

^�l!ad78

]

^{. Maddux}

[

^{1ad91 b}

]

also gave a representation result by using a notion of pair-den e. Concerning algebras generated by symmetric and transitive relations, Jonsson

[

^Jon88

]

and Givant

[

^Giv94

]

each provided such conditions.

(

The readers refer the li t of historical results of such a problem for Boolean relation algebras, which we introduced above, in chapter 2 of

[

^BKS97

]

^.

)

Schmidt and Strohlein

[

^SS85

]

simplified

the proof of the representation theorem by using a notion of point relations. T hey also howed an important rol of point relations in computer cience

[

^SS93

]

. Since it is comfortable for the author to use the notion of point relations concerning application of our relational calculus to computer cience and the notion is suitable as a relational

(

^algebraic

)

characterisation of points in a set to the author's intuition, we follow spirits of Schmidt and Strohlein in this thesi .

T his thesis organized as follows:

In chapter 2, we prepare some notions to be n cessary in the following chapters

\vhi h are definition of fuzzy relation

[

^Zad65

]

£-relation

[

^Gog67

]

and Ded kind cat gories

[

OS80, 0895, KF 1196 KF97

]

and their basic properties.

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In chapter 3, we provide an algebraic formalisation for fuzzy relation . Fuzzy relation treated her are hon1ogeneous ones on a fixed set X ·with values in the unit interval [0 1], that is functions R ^:./y ^xX ^---t [0, 1]. The et of all such fuzzy r lations on X constitutes a fuzzy relation algebra. Unlike Boolean relation algebras, fuzzy relation algebras are not Boolean and so Schroder rule [SS85], an important axiom of Boolean relation algebra , does not work in our case. Instead of Schroder rule we prefer to adopt Dedekind formula, which will be stated as axiom R3 in definition 2.6.

As fuzzy relations are naturally equipped with semi-scalar multiplication by scalars in the unit interval, it is neces ary to add axioms F1 and F3 to the axioms of relation algebras. The axiom F1 requires fuzzy relation algebras to be fuzzy algebras. Th notion of fuzzy algebras is a generalisation of algebras formed by functions with values in the unit interval [0, 1], that is, functions ^a ^: X ^---t [0, 1]. Hence a fuzzy algebra is a complete distributive lattice with semi-scalar multiplication by scalars in the unit int rval. In the fuzzy theory, we call ordinary relations which is fuzzy relations with values in the set

{

0, 1} crisp relations [ Gog67]. Thus crisp relations are special ones in fuzzy relations and form a Boolean subalgebra. The concept of crisp relations is well formalized in terms of semi-scalar multiplication, and then we will introduce axiom A3 for semi-Boolean algebras which claims that crisp relations have their complements (Cf. Proposition 2.1(d)). The final axiom F4, called point axiom, asserts that pair relations formed by point relations play a role of atoms. These are main different points from Boolean case. T he point axiom F4 enables us to construct a function assigning a concrete fuzzy relation on the set of point relations to an abstract relation in a fuzzy relation algebra. Finally we show a representation, an insertion and an isomorphism theorems for fuzzy relation algebras without assumption of Tarski rule [Tar41, SS85].

In chapter 4, we give an algebraic characterisation of cartesian products of fuzzy relation . Jonsson and Tar ki [JT52] characterized carte ian products of Boolean re

lation algebras by u ing a notion of ideal relations. Ideal relations are universal with

5

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respect to the composition with the universal relation \7. A representable Boolean relation algebra which is isomorphic to the algebra of all ordinary hon1ogen ou rela

tions ha no ideal relation except for the zero relation 0 and the universal relation \7.

In order to characterize cartesian products of fuzzy relations algebraically, fuzzy rela

tion algebras required to add two axioms. The former is called an axiom of cartesian products which corresponds to the algebraic characterisation of cartesian products of Bool an relation algebras introduced by Jonsson and Tarski

[

_JT52

)

, and the latter i called a point axiom for cartesian products. Ideal relations in fuzzy relation algebras are required to be crisp in order to keep general properties the same as properties of such relations in Boolean relation alg bras. The second one is defined by using im

proved notion of point relations and the point axiom provid d in section 3.3, since such point relations and the point axiom don't work on cartesian products of fuzzy relation algebras.

In chapter 5 we provide a categorical formalisation for fuzzy relations from a set X to a set Y. Fuzzy relations are set-functions with values in the unit interval

[0, 1),

_that

is, functions R _:X _xY _---t

[0, 1].

_The_setof all such fuzzy relations from X to Y consti

tutes a fuzzy relation category. The category of fuzzy relations is naturally equipped with semi-scalar multiplication by scalars in the unit interval. Also unlike Boolean relation categori s fuzzy relation categories are not Schroder ones

[

_Jon

8] .

Instead of Schroder categories we need to adopt Dedekind categories, named after Olivier and Serrato

[

_OS95

)

, in order to study fuzzy relation categories. First we show a simple repr sentation theorem for Dedekind categories

[

_OS95

]

with a unit object satisfy ing a condition, called the strict point axiom. Secondly we first state the definition of Zadeh categories as a categorical structure formed by fuzzy relations with sup-min compo-

ition. Then some useful properties on point relations, due to Schmidt and Strohlein

[

_SS

-]

and the point axiom will be introduced to prove a representation theorem for morphism in Zadeh categories. In particular, the point axiom enables us to construct

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a function assigning a concrete fuzzy relation between the sets of all point relations to an abstract relation in Zadeh categorie . Finally we show a representation theor m for Zadeh categories satisfying a point axiom without the assumption of exist nee of unit objects. And it is proved that the representation function is a bijection preserving all operations of Zadeh categories. Therefore the representation function turns out to be a functor. However this functor is not generally an isomorphisms of categories since it maps all objects with no points to the empty set. And also we prove that the rep

resentation theorem coincides with the simple representation theorem in section 5.1, when the Zadeh category has a unit object satisfying the strict point axiom.

In chapter 6 we consider relation algebras, which may not be Boolean, and show their representation th orem. Relation algebras in the sense of the chapter are equiv

alent to Dedekind categories [Gog67] (or allegories [FS90]) with just one object. A representation theorem for Dedekind categories are proved in the before chapter with the assumption of existence of a unit object. The discussion have not assumed the existence of a unit object, and £-relations are homogeneous relations on a set X with values on a fixed complete distributive lattice L, that is, functions R _: X _x X _-+ L.

This study is the first step to prove a representation theorem for Dedekind categories without unit obj cts. To prove a representation theorem for relation algebras, we use concepts of scalar relations and point r lations. The concept of scalar relations is an original one, which is defined as a relation included in the identity relation and which commutes with the greatest relation with respect to composition. In the case of £

relations scalar relations can be represent d as scalar matrices as easily understood.

We use the concept of scalar relations to define a new cone pt of crisp relations different from that in above chapter. Also the set of all scalar relations is a complete distribu

tive lattice, which i a sublattice of the relation algebra and scalar relations represent m mb r hip value ^. U ing our concept of alar relation and point relations we de

fin a trict point axiom for relation algebras and then we prove our representation

7

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theorem.

In chapter 7, ,,.e study D dekind categories [0895]. One of the aim in the chapter

to study notions of crispness and scalar relations in Dedekind categorie . A notion of crispness is introduced in chapter 5 under the assumption that Dedekind categories have unit objects which are an abstraction of singleton (or one-point) sets. To capture the notion of crispness without uch a sumption, we use a notion of scalar relations.

The notion of scalar relations in homogeneous relation algebras is introduced in chapter 6. We first define a preoder among object of Dedekind categories which compares the lattice structures on objects in a sense. Next we study notions of scalars and crispness for Dedekind categori s. The scalars on an object form a distributive lattice, which would be seen as the underlying lattice structure. Also the scalars are equivalent to the ideals which w re introduced by Jonsson and Tarski [JT52]. Then we recall the definition of heterogeneous £-relations, and illustrate a few relationships between crispness and lattice structures of scalars. The other aim of the chapter is to prove another representation theorem for Dedekind categories. Such a theorem for Dedekind categories with a unit object satisfying strict point axiom is also proved in chapter 5. In this chapter we show a representation theorem for uniform Dedekind categories sati fying the strict point axiom without the assumption of existence of unit objects, and it is proved that the representation function is a bijection preserving all operations of Dedekind categories.

In chapter 8, we concludes our researches and discu s about several problems for developing the future research.

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Chapter 2 Preliminaries

This chapter gives notions and notational conventions of fuzzy relations, £-relations, and Dedekind categories which are needed in this thesis. l\1ore precise information on these concepts would be obtained from

[

Zad65, KF95, Gog67, Fur97b, 0895, KF95 FS90

]

.

In section

2 . 1 ,

we introduce definition of fuzzy relations

[

Zad65

]

and the basic

properties

[

Zad65 KF95, Gog67

]

. In section 2.2, we provid definition of £-relations

[

Gog67

]

and the basic properties

[

Gog67, Fur97b

]

. In section 2.3, we give definition of Dedekind categories

[

0895

]

, definition of relation algebras

[

Fur97b

]

, and the basic properties

[

0895 KF95, F890, Fur97b

]

.

2.1 Fuzzy Relations

Let

[

0

1]

be the unit interval, that is, the set of all real numbers k with 0 :::; k :::;

1.

In this thesis, a real number k E

[

0,

1]

will be called a scalar and denoted by k, k', k" ^· ^·^· and so on.

(

All real numbers k appearing in the thesi are scalars, that is 0 :::; k

:::; 1.)

The upremum

(

least upper bound

)

and the infimum

(

greatest lower bound

)

of a family

{

k.A.

}

^.A of scalars will be denoted by V >..k>.. and 1\.A.k.A., respectively. In particular, k V k' ⁼max

{

k, k'

}

and k 1\ k' ⁼min

{

k, k'

}

. For two el m nts k k' E

[

0,

1]

the relative

9

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pseudo-complement of k relative to k' will be written as k => k' and defined by

k ^=>k' ⁼

{ ¹

^if k ::; k'

k' otherwise . Now recall some fundamentals on fuzzy relations

[

Zad65

]

.

Definition 2.1 Given possibly different s ts X and Y, a

(

heterogeneous

)

_fuzzy

relation

R

from X toY, written R: _y ^�Y, is a function R: X x Y ---t

[0, 1].

For

(x,

_y

)

_{E X}x Y the value

R(x,

_y

)

_E

[0, 1]

means the degree to which

x

_{andy are}

related under

R.

The set of all

(

heterogeneous

)

fuzzy relations from X to Y will be denoted by F-Rel

(

_X,Y

)

. A fuzzy relation R is contained in a fuzzy relationS, written

R

_�S, if

R(x

y

)

::; S

(x

, y

)

for all

(x,

_y

)

_{E X}x Y. The zero

(

_empty

)

_relation

Oxy

_{and the}_universal_relation

1xy

are fuzzy relations with

Oxy(x,

_y

)

=

0

and

1xy(x,

_y

)

=

1

for all

(x,

y

)

EX x Y, respectively. It i trivial that� is a complete distributive lattice, and

Oxy

_�

R

_�

1xy

for all fuzzy relations R : X ^� y·. We denote the least upper bound and the greatest lower bound of family

{ R

>.. : X ^�Y

}

_>..of fuzzy relations by

U>..R>.. and n>..R>.., respectively. We then have:

and

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for all

(x y)

E X x Y. The compo ite

RS(= So R)

: X ^___, Z of ^a fuzzy relation

R

: �Y ^___,Y followed by a fuzzy relation

S

: Y ^___,Z is defined by

(RS)(x,

z

) = VyEY(R(x, y)

/\

S(y,

z

))

for all

(x,

z

)

E X x Z. This composition of fuzzy relations is called sup-min composition.

The associativity

(RS)T = R(ST)

holds for all fuzzy relations

R, S,

and

T.

The identity relation Ex of a set X is a fuzzy relation such that

E x

(

^{X X}^'

') { ⁼

^{1 if}

^{x = x'}

0 otherwise .

The unit laws Ex

R = R

and

R

Ey

= R

hold for all

R

: X ^___, y·. The converse

(

or transpose

) Ru

: Y ^___, _y of a fuzzy relation

R

: X ^___, Y is defined by

Ru(y, x) = R(x, y)

for all

(y x)

E Y x X. The semi-scalar multiplication

kR( = k

*

R)

: X ^___,Y of a fuzzy relations

R

: X ^___,Y by a scalar

k

is a fuzzy relation such that

(kR)(x y) = kR(x, y)

for all

(x, y)

E X x Y. For fuzzy relations

S

: Y ^___, Z and T : X ^___, Z, the r sidue

T -;- S

: ./y ^___,Y is defined by

(T-;- S)(x y) = VzEz(S(y, z) =* T(x

z

))

for all

( x, y)

E X x Y. A fuzzy relation

R

: X ^___, Y is called nonzero if

R _#-

0 xY.

A fuzzy relation

R

: X ^___, Y is called crisp

[

Gog67

)

if

R(x, y) =

0 or

R(x, y) =

1 for all

(x, y)

E X x Y. It is now obvious

[

Gog67

)

that fuzzy relations together with operations defined above satisfy almost all axioms stated in section 2.3, 5.1

(

3.2

)

, and 3.1. f\Iayb only D3

(

R3

) (

Dedekind formula

)

D4

(

R4

) (

Residue

)

Z2

(

b

) (

F3

(

b

))

, and

A3

(

emi-Boolean alg bra

)

are not obvious· they will be proved in the following:

11

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Proposition 2.1 Let

R R' :

X ^____,.

Y, S

: Y ^____,. Z, and

T

: X ^____,. Z be fuzz_,. relations.

Then the follon·ing holds:

(

^a

) RS n T _� R(S n RUT)

(Dedekind formula).

(b) RS _� T

if and only if

R _� T---;- S

(Residues).

(

^c

) (kR)S

=

(kR)(S n klxz).

(

^d

) R

is crisp if and only if there is a fuzzy relation

R'

such that

lxy

=

R

U

R'

and

Rn R'

=

Oxy.

(

^e

) R

is crisp if and only if

R n klxy

=

kR

for all scalars

k.

Proof.

(

^a

)

^With

Ru(y, x) 1\ T(x, z) :::; (RuT)(y, z),

Dedekind formula follows from

(RS n T)(x z) VyEY(R(x, y) 1\ S(y, z)) 1\ T(x, z) VyEY(R(x, y) 1\ S(y, z) 1\ T(x, z))

for all

( x, z)

^EX ^x

Z.

=

VyEY(R(x y)I\S(y,z)I\Ru(y x)I\T(x,z))

<

VyEY(R(x, y) 1\ S(y, z) 1\ (RuT)(y, z))

VyEY[R(x, y) 1\ (S

n

RuT)(y, z)]

[R(S n RuT)](x, z)

(b)

follows from the following equivalence:

RS _� T

¢::::::i>

Vx\/z: (RS)(x z) :::; T(x, z)

(

^c

)

It is immediate from

¢::::::i>

Vx\/z: VyEY(R(x, y) 1\ S(y, z)) :::; T(x, z)

¢::::::i>

Vx\/z\/y: R(x, y) 1\ S(y, z):::; T(x, z)

¢::::::i>

Vx\/z\/y: R(x, y) :::; S(y, z)

^=?

T(x, z)

¢::::::i>

Vx\/y: R(x, y) :::; 1\zEz(S(y, z)

^=?

T(x, z))

¢::::::i>

Vx\/y: R(x, y) :::; (T---;- S)(x, y)

¢::::::i>

R _� T ---;- S .

[(kR)(S n klyz)](x, z) VyEY(kR(x y) 1\ S(y, z) 1\ k)

- VyEY(kR(x, y) 1\ S(y, z))

- [(kR)S](x, z)

(21)

for all

( x,

z

)

^{E X}^x^Z.

(d) Assu1ne that

R

is crisp and define

R'(x y)

⁼

1- R(x y)

for all

(x, y)

^EXx 1,..

Then it is clear that

R

^U

R'

⁼

1xy

and

R n R'

⁼

Oxy.

The converse is trivial (Cf.

Proof of Proposition 3.2(b)).

(e) It is immediate from

R n k1xy

⁼

kR

¢:=?

Vk

^E[0,

1) : R(x, y)

^1\

k

⁼

kR(x y)

¢:=?

R(x, y)

⁼0 or

R(x, y)

⁼

1

for all

( x, y)

^{E X}x Y. _I

The formula (b) in the last proposition is called "modular law" in [FS90, B 197); in

[Rig48, SS85, SS93) the Dedekind rule is given as

QRnS � (QnSRu)(RnQuS).

Since

these formulae are equivalent, we use the name "Dedekind formula" for the shorter variant in this thesis.

In relational calculus ([FS90, Kaw95, SS93]) a function

R :

^X

----.

^Yis a relation satisfying the uni valency

and the totality

ote that

and

Ru R � ^Ey

^¢:=?

Vy y': Ru R(y, y')

⁼

VxEx(Ru(y, x)

^1\

R(x, y')) ::; Ey(y, y')

¢:=?

Vx, y, y' : y # ^y'

^==>

^{R(x, y)}

^1\

^{R(x, y')}

⁼^{0 ,}

Ex � RRu

¢:=?

Vx: Ru R(x x)

=

VyEx(R(x, y)

^1\

Ru(y x))

⁼

1

¢:=?

Vx: VyExR(x, y)

⁼

1 .

Thus total relations are nonzero if neither X nor Y are empty.

To capture the concept of points in Boolean relation algebras Schmidt and Strohlein [SS85) defined a notion of vector relations. A vector relation (left ideal)

R

^:^X

----.

^Y^is

a fuzz) relation such that

1xxR

⁼

R

1

3

(22)

(

in

[

SS

5]

such a relation is defined as right ideal, i.e. R1 ^{1 T}= R), namely,

1xxR = R � Vx y: R(x, y) = Vx'Ex(1xx(x, x') 1\ R(x', y))

� Vx,y:R(x,y)=Vx'ExR(x' y)

� Vx, x', y: R(x, y) = R(x', y) .

The following lemma motivates the definition of point relations in fuzzy relation algebras.

Lemma 2.1

Let X be a nonempty set. If R ^:

X

^----. Y is a crisp fuzzy relation with Ru R � Ey, Ex� RRu and 1xxR = R, then there is an element y0 E Y such that for all (x, y) EX ^xY, R(x, y) = 1 if y = y0 and R(x, y) =

0

otherwise.

Proof.

Since X is not empty, there is an el ment x0 E X. As R is crisp and total, th re is an element y0 E Y such that R(x0, y0) = 1. But it follows from 1xxR = R that R(x, y0) = 1 for all x EX, and from Ru R � Ey that R(x, y) = R(x, y) 1\ R(x, y0) =

0

for y

#-

^Yo· I

Definition 2.2

A

(homogeneous) fuzzy relation

R on a fixed set X is a function R: X ^XX----+

[0,

1

]

.

The set of all

(

homogeneous

)

fuzzy relations on a fixed set

X

will be denoted by

F

Rel(X).

Every operator of homogeneou fuzzy relations i defined in the am way as in heterogeneous case without attention to domains of relations. Also basic properties we discuss in his section hold in homogeneous case.

(

See

[KF95]

for d tails.

)

The zero relation, the universal relation and the identity relation in

F-Rel(X)

are denoted by

Ox, 1x and Ex, respectively.

2.2 £-Relations

Let L =

(

L �' V /\, 0,

1)

be a fixed complete distributive lattice

(

or. a complete Heyt

ing algebra

) [

Birk4

]

with least element 0 and greatest elem nt

1.

Element. of the

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complete di tributive lattice L will be denoted by l, l', l", ···and so on. The upremum (lea t upper bound) and the infimum (greatest lower bound) of a family

{ l,�}

of ele

ments in L will be denoted by V

>.h

and 1\;._l;._, respectively. For two elements l, l' E L the relative pseudo-complement of l relative to l' will be written as l _=?l'. Now recall some fundamentals on £-relations [Gog67].

Definition

^2.3 Given possibly different s ts X andY, a

(heterogeneous) £-relation

R from X to Y, written R: X ---r Y, is a function R: X _xY ^----7 L.

The set of all (het rogeneous) £-relations from X into Y will be denoted by L

R

el

(X, Y). An £-relation R is contained in an £-relationS, written R � S, if R(x, y) :::; S(x, y)

for all (x,y) EX _xY. The zero (empty) relation Oxy and the universal relation lxy are £-relations with

Oxy(x, y) ⁼0

and

lxy(x,y) ⁼1

for all (x, y) EX _xY, respectively. It is trivial that� is a complete distributive lattice, and Oxy � R � lxy for all £-relations R. \\ e denote the least upper bound and the greatest lower bound of a family { R;._ : X ---r Y

}

>- of L-relations by U ;._R;._ and n;._R;._, re pectively. We th n have:

and

15

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for all

(x, y)

E }{ x Y. The composite

RS(= So R)

: X ^---"7 Z of an L-r lation R : X ^__,y· followed by an £-relation

S

: 17 ^---"7Z is defined by

(RS)(x,

z

)

=

VyE1·(R(x, y) 1\ S(y,

z

))

for all

(x,

z

)

EX x Z. This composition of £-relations is called sup-inf composition.

The associativity

(RS)T

=

R(ST)

holds for all £-relations R,

S

and

T.

The identity relation Ex of a set X is an £-relation such that

E ^x^T

(

^{X X}^'

') {

⁼ ⁰¹ ^if^otherwise

^x

⁼

^x'

The unit laws ExR = Rand REy =

R

hold for all

R

: X ---"7 Y. The converse

(

or transpose

) Ru

: Y ^---"7 ./y of an L-relation

R

: X ^---"7 Y is defined by

Ru(y, x)

=

R(x, y)

for all

(y, x)

E Y x X. For £-relations

S

: Y ^---"7 Z and

T

X ^---"7 Z, the residue

T -;- S

: X ^---"7Y is defined by

(T-;- S)(x, y)

=

1\zEz(S(y,

z

)

=*

T(x,

z

))

for all

(x, y)

EX x Y. An £-relation R: X ---"7 Y is called nonzero if

R #- Oxy.

It is now obvious

[

Gog67

]

that £-relations together with operations defined above atisfy all axioms of Dedekind categories stated in section 2.3. Maybe only D3

(

R3

) (

Dedekind formula

)

and D4

(

R4

) (

Residues

)

are not so obvious; it will be proved in the following:

Proposition 2.2 Let

R

: X __, Y,

S

: Y ---"7 Z and

T

: X ---"7 Z be £-relations. Then the following holds:

(

a

)

R

S

n

T � R(S

n

RUT)

(Dedekind formula).

(b) RS � T

if and only if R

� T-;- S

(Residues).

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Proof.

(

_a

)

_\\Tith

Ru(y,

^x

) 1\ T(x, z)

_�

(RuT)(y, z),

Dedekind formula follows frorn

(RS

n

T)(x

^z

) VyEY(R(x, y) 1\ S(y, z)) 1\ T(x, z) VyEY(R(x, y) 1\ S(y, z) 1\ T(x, z))

for all

( x z)

E

X

x Z.

VyEY(R(x, y) 1\ S(y z) 1\ Ru(y, x) 1\ T(x, z))

<

VyEY(R(x, y) 1\ S(y,

^z

) 1\ (RuT)(y, z))

VyEY[R(x, y) A (S

n

RuT)(y, z)]

[R(S

n

RuT)](x, z)

(b)

RS � T

^�

'ix'iz: (RS)(x, z)

_�

T(x, z)

�

'ix'iz: VyEY(R(x, y) 1\ S(y, z))

_�

T(x, z)

�

'ix'iz'iy: R(x, y) 1\ S(y, z)

_�

T(x, z)

�

'ix'iz'iy: R(x, y)

_�

S(y, z)

^=?

T(x,

^z

)

�

'ix'iy: R(x, y)

_�

1\zEz(S(y, z)

^=?

T(x, z))

�

'ix'iy: R(x, y)

_�

(T 7 S)(x, y)

�

R� T7S

I

In this thesis we will call an £-relation

R

of a set

X

_scalar if there is an l E L

such

R(

^{X X}_'

') {

⁼ ^l ^if

^x

⁼

^x'

0 otherwise .

Now we denote the set of all such scalar £-relations of a set

X S(X).

Then it is clear that the tuple

( S(X) �,

U, n,

Ox, Ex)

is a complete distributive lattice with least element

Ox

and greatest element

Ex,

and also that i isomorphic to L. Moreover scalar

£-relations can be characterized algebraically:

Proposition 2.3

R

is a scalar L-relation of a set

X

if and only if

R

C

Ex

and

Rlxx

=

lxxR.

Proof. Remark that

Rlxx(x,y)

⁼

VzEx(R(x,z)l\lxx(z,y))

⁼

R(x,x)Alxx(x,y)

⁼

R(x x)

for all

x, y

E

X

_if

R � Ex. (

_Similarly

lxxR(x, y)

=

lxx(x, y) 1\ R(y y).)

ow as ume that

R

is a scalar £-relation. Then it is trivial that

R � Ex.

Thu.

Rlxx(x, y)

=

R(x x) 1\ lxx(x, y)

⁼

lxx(x, y) 1\ R(y, y)

⁼

lxxR(x, y).

Next assume

17

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that

R � Ex

^and

Rlxx

⁼

lxxR.

^By

R �Ex

we obtain

R(x, y)

⁼ ⁰^if

x # y.

^Also

R(x x)

⁼

Rlxx(x, y)

⁼

lxxR(x y)

⁼

R(y y)

^{for all}

x, y

^EX. ^Therefore^Ri a calar

£-relation. I

Definition

2.4 A

(homogeneous) L-relation R

on a fixed set X 1s a function

R

:

X

^X^X^---+

^L.

The set of all (homogeneou

) L-

relations on a fixed set X will be denoted by

L

Rel(X).

Every operator of homogeneous L-relations is defined in the same way as in heterogeneous case without attention to domains of L-relations. Also basic properties we di cuss in this ection hold in homogeneous case. (See [Fur97b] for details.) The zero relation, the universal relation and the identity relation in

L-Rel(X)

are denoted by Ox,

lx

and

Ex,

respectively.

2.3 Dedekind Categories

In this section we recall the fundamentals on relation categories, which we will call Dedekind categories following Olivier and Serrato [OS95]. Dedekind categories are

qui valent to division allegories introduced by Freyd and Scedrov [FS90].

Throughout this the is, a morphi m

a

from an object X into an object Y in a Dedekind category (which will be defined below) will be denoted by a half arrow

a :

X

---. Y and the composite of a morphi m a : X ---r Y followed by a morphism

{3

^:

Y

^---.^Zwill be written as

a/3

^:^X^---r ^Z. We denote the identity morphism on an object X by idx.

Definition

2. 5 A

Dedekind category

V is a category satisfying the following:

Dl. [Complete Distributive Lattice]

For all pair of obj cts

X

^andYthe hom-set

D(X Y)

consi ting of all morphisms of X into y· is a complete distributive lattice with

(27)

the least morphism 0 XY and the greatest morphism \7 xY.

D2. [Involution)

There is given a monotone and involutive contravariant functor

�

: V --+ V. That is, for all morphisms

a a'

:

X

^___,.Y,

{3

: Y ^__, Z, the following involutive laws hold:

(a)

(a{3)� = {3�a�,

(b)

(a�)�= a,

(c) If

a � a',

then

aU� a'�.

D3. [Dedekind Formula)

For all morphisms

a

: X ^___,.Y,

{3

: y· ^___,. Z and r : X ^___,.Z the Dedekind formula

a{3

n r

� a({3

n

a�,)

holds.

D4. [Residues]

For all morphi ms

{3

: Y ^__,Z and r :

X

^___,. Z the residue (or division, weakest precondition) r-;-

{3

:

X

^__,Y is a morphism such that

a{3 _�

r if and only if

a _�

r-;-

{3

for all morphi ms

a

: X ^__,y·_

Note that complete distributive lattices are equivalent to complete Brouwerian lattices or complete Heyting algebras. It is clear that both of categories of all sets and fuzzy relation without scalar multiplication and of all ets and L-r lation are Dedekind cat gories.

Example 2.1

Consider a category

Rel0

whose objects are all nonempty sets and in which a hom-set

Rel0(X,

Y) between objects X andY is the set of all (binary) relations on X if

X =

Y and \7 XY

=

Oxy otherwise. That is, a hom-set

Rel0(X

Y) is a singleton set when X and Y are distinct. Then it is ea y to verify that the category

Rel0

is a Dedekind category. The conditions (Dl) and (D2) are trivial, and (D3) and

(D4)

also hold as follows: If

X =

Y

=

Z then (D3) and (D4) are clear. If X

=

Y

#

^Z,

then

{3 =

Oyz, r

=

Oxz and

(-;-{3 =

\7 xx- If

X=/=

Y, then

a=

Oxy and

(-;-{3 =

Oxy.

Throughout the r st of this section, all discussion will assume a fixed Dedekind cat gory V. The greatest morphism \7 xY is called the

universal

morphism and the lea t morphi m Oxy the

zero

morphism. A morphism is

nonzero

if it is not equal to the zero morphism. An objec ^..Y is called

empty

if \1 x x

=

Ox x, and

non empty

otherwi e.

\Ve introduce om basic properties of Dedekind categories.

19

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Proposition 2.4 Let

a a'

^:X ---r y· and

{3, {3'

: Y ---r Z be morphisms in D. Then the following holds:

(

a

)

Ott XY ^_-

0

y X,

n

_v

tt

x1- -^_ n v y X an d I

·

dX

tt

-_ ·d I X.

(

_c

)

_If

_{a� a'}

_and

_{f3 � /3',}

_then

_{af3 �} _a'f3'.

(

_f

) _0xyf3

= Oxz and aOyz ⁼Oxz.

tt tt tt .

_�

( tt )tt tt

Proof.

(

_a

)

_Oxy _� _(Oyx) = Oyx since Oxy

�

Oyx, and \7yx ⁼ Vyx

�

Vxy

· ntt

'""' d ·dtt

- _·dtt

_·d

_- ·dtt (·dtt )tt - (·dtt ·d )tt

-

(·dtt )U

_- _·d

b

_h

SinCe v XY

�

v y X, an I X

-

_I _X_I _{X -} _I _X_I _{X -} _{I X I} _{X -} _{I X}

_-

_I _X _{y t e}

axiom D2.

(b)

(u,\a,\)/3 �

ry <===>

u,\a,\

_�ry -;-

f3

for each ry : X ---r Z by the axiom D4.

¢==:> \j).. :

a,\ �

ry -;-

f3

<===> V).. :

a-\{3 �

ry

<===>

u,\ a,\{3 �

ry

(

_c

)

_If

_{a �} _a'

_and

_{f3 � {3',}

_then

_af3

_�

af3 U a'f3

⁼

(aU a')/3

⁼

a'f3

_�

a'f3 U a'/3'

⁼

a'(/3 U /3')

=

a'f3' by (b).

(

_d

)

It is clear that \7 xx \7 xy

�

\7 xY and idx � \7 xx by the axiom Dl. So \7 xx \7 xY ⁼

\7 xY holds. Similarly it can be shown that \7 xY \7 yy ⁼ \7 xY holds.

(

_e

)

By the axioms Dl and D2

att U 13tt �(a u f3)tt.

Hence

au f3

=

atttt U 13tttt � (att u {3tt)tt

and

(au f3)tt � (att u {3tt)tttt

⁼

att u {3tt.

(

_f

)

follows from the following equivalences:

(29)

and also

Ozy � ^Ozx

⁷

^a�

^{:::=::=}

^Oz}·a � ^Ozx

{:::=::=}

0�, _{z art} _� 0� _z

{:::=::=}

( aOyz )� � 0� _z

{:::=::=}

aOy z � 0 _{x z}

by the axioms Dl, D4, and D2. I

From the statement

(

_c

)

in the last proposition

(n>-a>-),8 � n>-a>.

holds. Also by the statement

(

_d

)

in the last proposition indicates that if \7

xY #- 0 _xY

then both of X and Y are non empty.

Next we provide the axioms Rl-R4 of relation algebras. A relation algebra R, which will be defined b low, is an algebraic structure over a nonempty set R of elements called "relation ' . Originally, relation algebras were formalized by Tarski

[

_Tar41

]

_as

complete Boolean algebras with compo ition and converse. But in this thesis. relation algebras are only complete distributive lattices with composition and converse. In other words, relation algebras

(

in this thesis

)

are complete Dedekind categories

[0895]

_with

just one object.

Definition

2.6 A

relation algebra

R

= (

_R,

_�'

_U,

_n

·

,�, 0,

_\7,_id

)

1s an algebraic structure over a nonempty set R satisfying the following:

Rl.

[Complete Distributive Lattice]

The tupl

(

_R,

_�

_U

_n, 0, \7)

is a complete distributive lattice.

R2.

[Involutive Monoid]

The tuple

(

_R,;

_{,« ,}

_id,

0)

is an involutive monoid with unit element id and zero element

0.

_{That is,}

(

_a

) (a,B)ry = a(,Bry) (

_b

) aid= ida= a, (

_c

) (atl)tl =a, (

_d

) (a,B)� = ,Btlatl,

(

_e

)

_If

_a _� _,8

_{th n}

_att _{� ,ett.}

R3.

[Dedekind Formula) a,B

n

ry � a(,B n a«ry).

R4.

[Residues]

For all relation

,8

and

ry

the residue

(

or division, weakest precondi

tion

) _ry

7

,8

is a relation such that

a,8 �

^ryif and only if

a � ^ry

⁷

^,8

for all relations

0:.

21

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It is clear that algebras

F-Rel(X)

=

(F-Rel(X), S:,

^{U n,}

o,u, Ox, lx Ex)

of fuzzy relations and

L-Rel(X)

=

(L-Rel(X), S:,

^{U, n,}

o,u, Ox, lx, Ex)

of L-relation on a fixed set

X

are relation algebras. Let

R

=

(R, G, U, n,;

,u, 0, V, id

)

be a relation algebra. A relation algebra

R

with V ₌ 0 is trivial and not worth mentioning.

Throughout the rest of the thesis all discussions will assume a fixed relation algebra

R

with V

#

^0. Basic properties of Dedekind categories we discuss in this section hold in also relation algebras without attention to domains.

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Chapter 3 A Representation Theorem for Fuzzy Relation Algebras

The aim of the chapter is to provide an algebraic formalisation for fuzzy relations.

Fuzzy relations treated here are homog neous ones on a fixed set X with values in the unit interval [0, 1], that is, functions R : X _x X � [0, 1]. The set of all such fuzzy relation on X constitutes a fuzzy relation algebra. Unlike Boolean relation algebras fuzzy relation algebras are not Boolean and so Schroder rule [SS85], an important axiom of Boolean relation algebras, does not work in our case. Instead of Schroder rule we prefer to adopt Dedekind formula, which is stated as axiom R3 in definition 2.6.

As fuzzy relations are naturally equipped with semi-scalar multiplication by scalars in th unit in ten al, additional axioms F1 and F3 to the axioms of relation algebras are required. However, th set of all crisp [Gog67] relations are contained in the set of all fuzzy relations and form a Boolean subalgebra. The concept of crisp relations is well formalized in terms of semi-scalar multiplication, and then we introduce axiom A3 for semi-Boolean algebras which claims that crisp relations have their complements (Cf.

Propo ition 2.1(d)). The final axiom F4, call d point axiom, asserts that pair relations formed by point r lations play a role of atoms. These are main different points from Boolean ca e.

Section 3.1 provide a set of th axioms A1-A3 for fuzzy algebras and the ba ic

23

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properties of fuzzy relation algebras. Section 3.2 provides a et of the axioms F1- F3 for fuzzy relation algebras and the ba ic properties of fuzzy relation algebras. In section 3.3 we fir t state the definition of point relations and a point axiom F4. Then some useful properties on point relations ar given for the proof of main results. In particular, the point axiom F4 enables u to construct a function assigning a concrete fuzzy relation on the set of point relations to an abstract relation in a fuzzy relation algebra. In section 3.4, we show a representation, an insertion and an isomorphism theorems for fuzzy relation algebras without assumption of Tarski rule [Tar41, SS85].

This chapter is based on [KF95].

3.1 Fuzzy Algebras

In this section we first introduce a notion of fuzzy algebras as a mathematical structure formed by fuzzy sets, and describe some properties of fuzzy algebras. In short fuzzy algebras are complete distributive lattices with semi-scalar multiplications and a semi

Boolean property. Throughout of the chapter real numbers

k

^E [0, 1] will be called scalars, where [0, 1] is the unit interval, that is, the set of all real number

k

with

O�k�l.

Definition

^3.1 ^A

fuzzy algebra A = (A

�

U, n,

^·^,0,

\7)

i an algebraic structure over a non empty set

A

satisfying the following:

AI.

[Complete Distributive Lattice]

A tuple

(A,�'

^{U n,}0, \7) is a complete dis

tributive lattice with the least element 0 and the greatest element \7.

A2.

[Semi-Scalar Multiplication]

An operation ^· ^: [0, 1] ^x

A---t A

ⁱ a semi-scalar

multiplication of

A.

That is,

(a)

Oa =

⁰

and

^1a

=a,

^(b)

k(k'a) = (kk')a,

(c)

k(U>.a>.) = U>.ka>. and k(n>.a>.) = n>.ka>.

(d)

(1\>.k>.)a = n>.k>.a

( )

If

ka

�

kb

and 0 ^<

k,

then ^a�

b.

(33)

(

The semi-scalar multiplication

k

^·

a

of

a

^E^Aby a scalar

k

^E

[0, 1]

will be written as

ka,

unless confusion occurs.

)

A3.

[Semi-Boolean Algebra]

If

a n k\7 = ka

for all scalars

k,

th n there i an element

b

such that

au b =

^\7and

an b = 0.

Note that complete distributive lattices are equivalent to complete Brouwerian lattices or complete Heyting algebras.

It is obviou that each algebra

(F-Rel(X),

^�'^{U, n,}

*,Ox, 1x)

is a fuzzy algebra. A fuzzy algebra

A= (A �

^U,

n,

^·,

0, \7)

with \7

= 0

i trivial and not worth mention.

Throughout the rest of this section all discussions will be done in a fixed fuzzy algebra

A

with

\7 -f 0.

Proposition

^3.1 Let

a, b

be elements of a fuzzy algebra

A

and

k, k'

scalars. Then the following holds:

(

a

)

If

a � b

] then

ka � kb.

(

b

)

If

k :::; k']

then

ka � k' a.

In particular]

ka

^�

a

and

kO = 0.

(

c

) ^ka

^U

^{k'a = (k}

^V

^k')a (d)

If

k \7 =

\7] then

k = 1.

(

e

)

If

k\7 � k'\7]

then

k:::; k'.

Proof. (

a

)

If

a� b,

then

ka = k(a n b)= ka

ⁿ

kb � kb

by th axiom A2

(

c

)

.

(b)

If

k :::; k',

then

ka = (k

^A

k')a = ka n k' a � k' a

by the axiom A2

(

d

)

. In particular,

ka � 1a = a

by the axiom A2

(

a

)

and

k :::; 1,

and also

0 � kO

^�1

0 = 0

by the axioms A1 and A2

(

a

)

.

(

c

)

Assum

k:::; k'.

Then

ka

^U

k'a = k'a = (k

^V

k')a

by

(

b

)

.

25

ファジィ関係の代数的定式化と表現定理