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A new algebraic approach to L-fuzzy relations convenient to study crispness
Information Sciences 139 (2001) 233±252
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A new algebraic approach to L-fuzzy relations convenient to study crispness Michael Winter Department of Computer Science, University of the Federal Armed Forces Munich, 85577 Neubiberg, Germany Received 9 January 2000; received in revised form 17 March 2001; accepted 25 June 2001
Abstract The aim of this paper is to develop a suitable calculus of L-fuzzy relations. We show that for this purpose the theory of Dedekind categories is too weak. Therefore, we introduce the notion of a Goguen category as a suitable algebraic de®nition and show several properties of this kind of a relational category. Ó 2001 Elsevier Science Inc. All rights reserved.
1. Introduction The calculus of binary relations has been investigated since the middle of the 19th century. It plays an important role in the development of logic and algebra. Furthermore, since the mid-1970s it has become clear that this calculus is a fundamental conceptual and methodological tool in computer science just as much as logic. While computer science applications are evolving rapidly in several areas as in communication, programming, software, data or knowledge engineering, exact sciences are needed to understand existing methods. One important application is the treatment of uncertain or incomplete information. To handle such kind of information, Zadeh [12] introduced the concept of fuzzy sets. Later on, Goguen [3] generalized this concept to L-fuzzy sets and relations for an arbitrary complete distributive lattice L instead of the unit interval 0; 1 of the real numbers.
E-mail address: [email protected] (M. Winter). 0020-0255/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 0 - 0 2 5 5 ( 0 1 ) 0 0 1 6 7 - 0
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In this paper we aim to give a suitable algebraic de®nition for L-fuzzy relations. It is well known that a reasonable category of L-relations constitutes a Dedekind category introduced in [6]. Unfortunately, this theory is too weak to express some standard properties of L-relations. In particular, we show that there is no formula in the theory of Dedekind categories expressing the fact that a given L-relation is 0±1 crisp, i.e., all entries are either the least element 0 or the greatest element 1 of L. We introduce the notion of a Goguen category as a suitable algebraic definition for L-relations. First of all, crispness in this theory coincides with 0±1 crispness of L-relations. Furthermore, we show that under an assumption of the underlying lattice, i.e., the lattice of scalar relations, the notion of s-crispness [2,5] and crispness are equivalent. We assume that the reader is familiar with the basic concepts of category theory.
2. Dedekind categories Throughout this paper, we use the following notation. To indicate that a morphism R of a category R has source A and target B we write R : A ! B. The collection of all morphisms R : A ! B is denoted by RA; B and the composition of a morphism R : A ! B followed by a morphism S : B ! C by R; S. The identity morphism on A is denoted by IA . In this section we recall some fundamentals on Dedekind categories [6,7]. These categories are called locally complete division allegories in [1]. De®nition 1. A Dedekind category R is a category satisfying the following:
1. For all objects A and B the collection RA; B is a complete distributive lattice. Meet, join, the induced ordering and the least and the greatest elements are denoted by u; t; v; AB ; AB , respectively. 2. There is a monotone operation ^ (called conversion) such that for all rela^ tions Q : A ! B and R : B ! C the following holds: Q; R R^ ; Q^ and ^ ^ Q Q. 3. For all relations Q : A ! B; R : B ! C and S : A ! C the modular law Q; R u S v Q; R u Q^ ; S holds. 4. For all relations R : B ! C and S : A ! C there is a relation S=R : A ! B (called the left residual of S and R) such that for all Q : A ! B the following holds: Q; R v S () Q v S=R. A Dedekind category R such that every collection RA; B is a Boolean algebra is called a Schr oder category. We denote the complement of a relation R : A ! B, if it exists, by R. If, in addition, RA; B is atomic for all A and B, R
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is called a heterogeneous relation algebra. According to the equivalence of the so-called Tarski-rule )
R 6
AB
CA ; R;
BD
CD
for all objects C and D
to a generalized version of the notion of simplicity known from universal algebra, we call a Dedekind category (or a Schr oder category or a heterogeneous relation algebra) simple i the Tarski-rule is valid. Furthermore, we de®ne the ^ right residual by Q n R : R^ =Q^ . In the following lemma we have collected some basic properties of relations in a Dedekind category. We will use these properties throughout the paper without mentioning. Proofs may be found in [1,8±11]. Lemma 2. Let R be a Dedekind category, let A; B; C be objects of R, and for i 2 I, let Q1 ; Q2 ; Qi : A ! B, R1 ; R2 ; Ri : B ! C and S; S1 ; S2 : A ! C be relations. Then
BC
AB
BA
AB ;
AB ; R1
BB
and I^ A IA ,
AB ;
BA
;
AB
AB ,
AC ,
AA ;
3. Q1 ;
^ AB
BA ;
2.
^ AB
1.
^
4. wi2I Ri wi2I R^ i , F F ^ ^ 5. i2I Ri i2I Ri , 6. if Q1 v Q2 and R1 v R2 then Q1 ; R1 v Q2 ; R2 , 7. Q1 ; R1 u S v Q1 ; uS; R^ 1 ; R1 , F F 8. Q1 ; i2I Ri i2I Q1 ; Ri , 9. Q1 ; wi2I Ri v wi2I Q1 ; Ri , 10. Q1 v Q1 ; Q^ 1 ; Q1 , 11. S=IC S and IA n S S, 12. Q2 v Q1 ; R2 v R1 and S1 v S2 imply S1 =R1 v S2 =R2 and Q1 n S1 v Q2 n S2 , 13. S=R1 ; R1 v S and Q1 ; Q1 n S v S, F F 14. wi2I S=Ri S= i2I Ri and wi2I Qi n S i2I Qi n S. An important class of relations is given by mappings. De®nition 3. Let Q : A ! B be a relation. Then we call 1. Q univalent i Q^ ; Q v IB , 2. Q total i IA v Q; Q^ or equivalently i Q; 3. Q a map i Q is univalent and total,
BC
AC
for all objects C,
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4. Q injective i Q^ is univalent, 5. Q surjective i Q^ is total, 6. Q an isomorphism i Q and Q^ are mappings. Notice that if Q is an isomorphism Q^ ; Q IB and Q; Q^ IA hold. In the following lemma we collect some fundamental facts concerning this class of relations. Again, proofs may be found in [1,8±11]. Lemma 4. Let U : A ! B be univalent, let F : D ! A and G : E ! B be mappings, let H : C ! C be total and injective, and for i 2 I, let Q : A ! C and Ri ; S : B ! C be mappings. Then 1. U ; wi2I Ri wi2I U ; Ri , 2. F ; Q=S F ; Q=S, 3. Q=S; G^ Q= G; S, 4. Q=S Q; H = S; H . Later on, we will show that there is no formula expressing the fact that an Lrelation is 0±1 crisp. For this purpose, we have to de®ne a language and its meaning. We require a set of object variables and a set of typed relation variables, i.e., every relation variable is of the form r : a ! b where a and b are object variables. Now, terms and formulae are de®ned as follows: De®nition 5. The set of terms of type a ! b and the set of formulae are de®ned inductively as follows:
5. 6. 7. 8. 9. 10. 11. 12.
Any relation variable r : a ! b is a term of type a ! b. If a is an object variable then Ia is a term of type a ! a. If a and b are object variables then ab and ab are terms F of type a ! b. If u is a formula and r : a ! b a relation variable then fr : a ! b j ug and wfr : a ! b j ug are terms of type a ! b. If t is a term of type a ! b then t^ is a term of type b ! a. If t1 and t2 are terms of type a ! b resp. b ! c then t1 ; t2 is a term of type a ! c. If t1 and t2 are terms of type a ! c resp. b ! c then t1 =t2 is a term of type a ! b. If t1 and t2 are terms of type a ! b then t1 t2 is a formula. If u1 and u2 are formulas then u1 ^ u2 is a formula. If u is a formula then :u is a formula. If u is a formula and r : a ! b is a relation variable then 8r : a ! bu is a formula. If u is a formula and a is an object variable then 8au is a formula.
1. 2. 3. 4.
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Given a Dedekind category R, an environment r is a function mapping each object variable a to an object A of R and each relation variable r : a ! b to a relation R : r a ! r b. The update rA=a, respectively rR=r : a ! b, of r at the object variable a, respectively at the relation variable r : a ! b, with the object A, respectively with the relation R : r a ! r b, is de®ned by r b iff a 6 b; rA=a b : A iff a b; 8 iff a c; < Ar d rA=a r : c ! d : iff a d; r cA : r r : c ! d iff a 6 c ^ a 6 d; rR=r : a ! b c : r c;
rR=r : a ! b s : c ! d :
r s : c ! d R
iff r : a ! b 6 s : c ! d; iff r : a ! b s : c ! d:
As usual we denote a sequence of updates rA=aB=bR=r on an environment r by rA=a; B=b; R=r. Furthermore, given a class F of mappings FA : A ! A for every object A of R and an environment r we denote with F r the environment de®ned by 1. F r a r a for all object variables a, ^ 2. F r r : a ! b : Fr a ; r r : a ! b; Fr b r : a ! b.
for
all
relation
variables
^ Notice that F rR=r : a ! b F rFr a ; R; Fr b =r : a ! b. Furthermore, if F is a class of isomorphisms, i.e., every FA is an isomorphism, the function r 7! F r is a bijective function on the class of environments.
De®nition 6. The value VR t r of a term t of type a ! b and the validity R r u of a formula u in a Dedekind category R under an environment r are de®ned inductively as follows: 1. VR r : a ! b r : r r : a ! b, 2. VR Ia r : Ir a , 4. VR
F
:
ab r
r ar b ,
ab r
3. VR
:
r ar b ,
5. VR fr : a ! b j ug r :
F
fR j R rR=r:a!b ug,
6. VR wfr : a ! b j ug r : wfR j R rR=r:a!b ug, ^
7. VR t^ r : VR t r , 8. VR t1 ; t2 r : VR t1 r; VR t2 r,
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9. VR t1 =t2 r : VR t1 r=VR t2 r, 10. R r t1 t2 i VR t1 r VR t2 r, 11. R r u1 ^ u2 i R r u1 and R r u2 , 12. R r :u i R 2r u, 13. R r 8r : a ! bu i R rR=r:a!b u for all R : r a ! r b, 14. R r 8au i R rA=a u for all objects A. Later on, the following lemma plays an important role in the proof of Theorem 1. Lemma 7. Let t be a term of type a ! b, let u be a formula, let R be a Dedekind category, let r be an environment, and let F be a class of isomorphisms. Then the following hold: ^ 1. VR t r Fr a ; VR t F r; Fr b . 2. R r u iff R F r u.
Proof. The assertions are shown simultaneously by structural induction. · t r : a ! b ^ ^ VR t r r r : a ! b Fr a ; Fr a ; r r : a ! b; Fr b ; Fr b ^ ^ Fr a ; F r r : a ! b; Fr b Fr a ; VR t F r; Fr b :
^ ^ ; Ir a ; Fr a Fr a ; VR t F r; Fr a . · t Ia VR t r Ir a Fr a · t Fab or t ab g Analogously to t Ia . · t fr : a ! b j ug G G VR t r fR j R rR=r:a!b ug fR j R F rR=r:a!b ug G ^ =r:a!b ug fR j R F rFr a ;R;Fr b G ^ Fr a ; fS j R F rS=r:a!b ug; Fr b ^ Fr a ; VR t F r; Fr b :
· t wfr : a ! b j ug Analogously to t of isomorphisms. · t t1^
F fr : a ! b j ug since F is a class
^ VR t r VR t1 r^ Fr b ; VR t1 F r; Fr a ^ ^
^ ^ Fr a ; VR t1 F r ; Fr b Fr a ; VR t F r; Fr b :
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· t t1 ; t2 VR t r VR t1 r; VR t2 r ^ ^ ; VR t1 F r; Fr c ; Fr c ; VR t2 F r; Fr b Fr a ^ Fr a ; VR t1 F r; VR t2 F r; Fr b ^ Fr a ; VR t F r; Fr b :
· t t1 =t2 . Using Lemma 4 (2)±(4) we get VR t r VR t1 r=VR t2 r ^ ^ ; VR t1 F r; Fr c = Fr b ; VR t2 F r; Fr c Fr a ^ Fr a ; VR t1 F r=VR t2 F r; Fr b ^ Fr a ; VR t F r; Fr b :
· u t1 t2 R r u () VR t1 r VR t2 r ^ ^ () Fr a ; VR t1 F r; Fr b Fr a ; VR t2 F r; Fr b () VR t1 F r VR t2 F r () R F r u: · u u1 ^ u2 R r u () R r u1 and R r u2 () R F r u1 and R F r u2 () R F r u: u1 () R F r u. · u :u1 R r u () R 2 u1 () R 2 r F r · u 8r : a ! bu1 R r u () R rR=r:a!b u1 for all R : r a ! r b () R F rR=r:a!b u1 for all R : r a ! r b ^ =r:a!b u () R F rFr a ;R;Fr b 1 for all R : r a ! r b () R F rS=r:a!b u1 for all S : r a ! r b () R F r u: · u 8au1 R r u () R rA=a u1 for all objects A () R F rA=a u1 for all objects A () R F rA=a u1 for all objects A () R F r u:
3. Matrix-algebras, fuzzy- and L-relations Given a Dedekind category R, an algebra of matrices with coecients from R may be de®ned.
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De®nition 8. Let R be a Dedekind category. The algebra R of matrices with coecients from R is de®ned by 1. The class of objects of R is the collection of all functions from an arbitrary set I into the class of objects ObjR of R. 2. For every pair f : I ! ObjR ; g : J ! ObjR of objects from R , a morphism R : f ! g is a function from I J into the class of all morphisms MorR of R such that R i; j : f i ! g j holds. 3. For R : f ! g and S : g ! h, composition is de®ned by G R; S i; k : R i; j; S j; k: j2J
4. For R : f ! g, conversion is de®ned by R^ j; i : R i; j^ . 5. For R; S : f ! g, join and meet are de®ned by R t S i; j : R i; j t S i; j;
R u S i; j : R i; j u S i; j:
6. The identity, zero and universal elements are de®ned by : i1 6 i2 ; f i1 f i2 fg i; j : f ig j ; If i1 ; i2 : If i1 : i1 i2 ; fg i; j : f ig j : Obviously, a morphism in R may be seen as a (in general non®nite) matrix indexed by objects from R. If R has only one object the objects f : I ! ObjR of R may be identi®ed with the set I. Lemma 9. R is a Dedekind category. Proof (sketch). The critical point is the de®nition of the left residual in R . As shown in Lemma 2 the in®nite variants of meet-subdistributivity and joindistributivity hold in R. A straightforward computation shows that these properties are also valid in R . Now, it is possible to de®ne the residual of S and R as the join of all elements Q satisfying Q; R v S. Let L L; _; ^; 0; 1 be a complete distributive lattice with least element 0 and greatest element 1. The induced ordering is denoted by 6 . Notice that we have pseudo-complements in L, i.e., for all elements k and l there is an element k ! l such that k ^ m 6 l () m 6 k ! l. L may also be considered as an oneobject Dedekind category with identity 1 and composition ^ (the residual is given by the pseudo-complement). Consequently, L is a Dedekind category, called the full category of L-relations. This category (or every subcategory closed under all operations) seems to be a good model for generalized fuzzy relations, also called L-fuzzy relations in [3]. An L-relation R is called 0±1 crisp i R i; j 0 or R i; j 1 holds for all i and j.
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Let 0; 1 R be the unit interval, i.e., the set of all real numbers r such that 0 6 r 6 1 holds. Then 0; 1; inf; sup; 0; 1 is a complete distributive lattice. Hence, we get the full category of fuzzy relations 0; 1 as a special case.
4. Crispness in Dedekind categories In some sense a relation of a Dedekind category may be seen as an L-relation. The lattice L may be equivalently characterized by the ideal relations, i.e., a relation J : A ! B satisfying AA ; J ; BB J , or by the scalar relations.
De®nition 10. A relation a : A ! A is called a scalar on A i a v IA and 0 AA ; a a; AA . Furthermore, a is called linear i for every scalar a : A ! A, 0 0 a u a AA implies a AA . Notice that the notion of ideal was introduced by J onsson and Tarski [4] and the notion of scalar was introduced by Kawahara and Furusawa [5]. Given a complete distributive lattice L, the ideal elements J and the scalars a are exactly these relations such that k iff x y; J x; y k for all x and y; a x; y 0 iff x 6 y
for an element k 2 L. Notice that the collection of ideal elements on A is isomorphic to the collection of scalars on A via the mappings / J : J u IA and / 1 a : a; AA . Furthermore, for all S : A ! B, the relation AA ; S; BA is an ideal element and IA u AA ; S; BA is a scalar. If S 6 AB then both relations are nonzero since otherwise we have BB
IA u
AA ; S;
BA
;
AB
AA ;
AB
AB :
AA ; S;
Sv
Lemma 11. Let a; b be scalars on A, and let R : A ! B. Then 1. 2. 3. 4.
a; R R u a; a; b a u b, a; a a, IA u AA ; a;
AB ,
AA
a.
All proofs not given in this section (including the proof of the lemma above) may be found in [2,5]. Using the scalars, there are two notions of crispness in an arbitrary Dedekind category.
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De®nition 12. A relation R : A ! B is called (l-crisp) s-crisp i a; Q v R implies Q v R for all (linear) nonzero scalars a and all relations Q : A ! B.
Obviously, AB is s-crisp. Furthermore, a little computation shows that the class of s-crisp relations is closed under meet. But, unfortunately, AB may be not s-crisp. Dedekind categories in which AB is s-crisp are characterized by the following lemma: Lemma 13. In a Dedekind category the following statements are equivalent:
1. All nonzero scalars are linear. 2. AB is s-crisp. 3. For every R : A ! B, its pseudo-complement R ! 4. Every complemented R : A ! B is s-crisp.
AB
is s-crisp.
Considering L-relations the following hold. Lemma 14. In L all s-crisp relations are 0±1 crisp. The converse is, in general, not true. A property of the underlying lattice L equivalent for L-relations to Lemma 13 (1) is required. Lemma 15. Let L be a complete distributive lattice. Then the following properties are equivalent: 1. In L all 0±1 crisp relations are s-crisp. 2. k ^ l 0 implies k 0 or l 0 for all elements k; l 2 L. The following lemma shows that s-crispness grasp 0±1 crispness i the properties listed in Lemma 13 are valid. Lemma 16. Let A be an object of a Dedekind category. If there is a nonzero and nonlinear scalar a on A then there is no s-crisp relation R : A ! B except AB for all objects B.
Proof. Suppose R : A ! B is s-crisp and a; a0 are nonzero scalars such that a u a0 AA holds. Then a; a0 ; AB a u a0 ; AB AB v R. We conclude a0 ; AB v R and further AB v R since R is s-crisp. We have seen that the notion of s-crispness does not grasp 0±1 crispness of L-relations. Also, the notion of l-crispness does not work. If L is a Boolean lattice the identities are the only linear scalars in L , and hence, all relations are l-crisp. But the following hold:
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Lemma 17. In L all 0±1 crisp relations are l-crisp. Unfortunately, we are able to show that the theory of Dedekind categories is too weak to express the notion of 0±1 crispness of L-relations. Theorem 18. There is no formula u in the language of Dedekind categories such that for all lattices L and L-relations R : A ! B, L rA=a;B=b;R=r u for all environments r () R is 0±1 crisp: Proof. Consider the Boolean algebra B4 : P fa; bg, i.e., the powerset of the set fa; bg. Let X fxg and Y fx; yg be the sets. Then the relation FY :
fag fbg
fbg fag
is an isomorphism in B 4 Y ; Y . Now, take F as the class of isomorphisms given by FY on Y and IZ on all other sets Z. Suppose there is such a formula u. Then B 4 rX =a;Y =b;R=r u for all environments r where R is the 0±1 crisp relation R : fa; bg ; . Lemma 7 shows that B 4 F rX =a;Y =b;R=r u. Since F rX =a; Y =b; R=r r FX ; R; FY^ R; FY^ fag fbg fa; bg ; ; fag fbg ; fbg fag we conclude B 4 F rX =a;Y =b;S=r u where S is the relation above. Since the function r7!F r is bijective, B 4 rX =a;Y =b;S=r u for all environments r. From the de®nition of u we conclude that the relation S is 0±1 crisp, a contradiction.
Nevertheless, we are able to map every relation R : A ! B to the least s-crisp relation Rs : wfQ : A ! B j R v Q and Q is s-crispg containing R. There is another possible way to characterize Rs . Consider the function G U R : a n R: a6
AA a scalar
This function is monotone, and R IA n R v a n R, since a is a subidentity. F This gives us R v a6 a n R U R. Altogether we conclude that for every R AA there is a least ®xpoint lR of U greater than R.
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Theorem 19. Let R be a Dedekind category. Then Rs lR for all relations R.
Proof. v: It is sucient to show that lR is s-crisp. Suppose there is a nonzero scalar a andF a relation Q such that a; Q v lR . Then we conclude Q v a n lR v a6 a n lR U lR lR . AA w: It is sucient to show that Rs is a ®xpoint of U. For every nonzero scalar s s a we have a; a n Rs v Rs and Rs is s-crisp. This implies F a n R v sR since s s U R a6 a n R v R . The other inclusion Rs v U Rs is trivial. AA
5. Goguen categories In the last section we have shown that we need an additional concept to de®ne a suitable algebraic theory of L-fuzzy relations. Our approach introduces two operations one which maps each relation to the greatest 0±1 crisp relation it contains and one which maps each relation to the least 0±1 crisp relation which includes it. We now give the abstract de®nition. De®nition 20. A Goguen category is a Dedekind category together with two operations " and # satisfying the following:
1. R" ; R# : A ! B for all R : A ! B. 2. " ; # is a Galois correspondence, i.e., R" v S () R v S # for all R; S : A ! B. ^ " 3. R^ ; S # R" ; S # for all R : B ! A and S : B ! C. 4. If a 6 AA is a nonzero scalar then a" IA . 5. For any set fXa : A ! B j a : A ! A scalarg such that (a) Xa Xa" for all a : A ! A. (b) w Xa XF M for all sets M of scalars in GA; A, a2M
and all R : A ! B the following equivalence holds: G a; Xa () a n R# v Xa for all a : A ! A: Rv a:A!A a scalar
Furthermore, a Goguen category G is called Boolean based i for all R : A ! B the relation R" is complemented, i.e., R" t R" ! AB AB . G is called linear i all nonzero scalars are linear and nonlinear i G is not linear. The obvious de®nitions of
"
and
#
for L-relations give the standard model.
Theorem 21. Let L L; _; ^; 0; 1 be a complete distributive lattice. Then L together with the operations
M. Winter / Information Sciences 139 (2001) 233±252
"
R x; y :
1 0
iff R x; y 6 0; iff R x; y 0;
#
R x; y :
1 0
245
iff R x; y 1; iff R x; y 6 1
is a Boolean based Goguen category. Furthermore, for a relation R in L we have R" R R# iff R 0±1 crisp. Proof. The assertion R" R R# i R 0±1 crisp follows immediately from the de®nitions of " and # . Since every scalar a : A ! A is of the form a x; y
k 0
iff x y; iff x 6 y
for an element k 2 L, we will identify throughout this proof the scalar a with the element k and denote it by a. 1. Axiom 1 is true by de®nition. 2. Suppose R" v S and R" x; y 6 0. Then by the de®nition of " we have 1 R" x; y 6 S x; y and hence R x; y 6 1 S # x; y. The other implication is shown analogously. 3. De®ne two operations " ; # : L ! L on the lattice L by a" 0 if a 0 and a" 1 otherwise, and a# 1 if aW 1 and a#W 0 otherwise. The operations satisfy a ^ b# " a" ^ b# and a2M a" a2M a" for all subsets M of L. Then we conclude _
# "
R^ ; S x; z
_
!" R^ x; y ^ S # y; z
y "
#
R y; x ^ S y; z
y
_
_
R y; x ^ S # y; z
"
y "^
^
R x; y ^ S # y; z R" ; S # :
y
4. By the de®nition of " we conclude for every nonzero scalar a : A ! A 1 iff x y a" x; y IA x; y: 0 iff x 6 y 5. Suppose fXa : A ! B j a : A ! A scalarg is a set of 0±1 crisp relations with the required properties. ()) Suppose a n R# x; y 1. Then the 0±1 crisp relation Q de®ned by 1 iff x0 x and y 0 y; 0 0 Q x ; y : 0 else is included in a n R# . We conclude Q v a n R# () Q v a n R () a; Q v R () a a; Q x; y 6 R x; y:
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F By the assumption we get a 6 b b; Xb x; y. Let M be the set of all elements b such that Xb x; y 1. Then G _ _ a 6 b; Xb x; y b ^ Xb x; y b ^ Xb x; y b
b
_
b
_
b2M
M:
b2M
W 0 W Now, let M 0 : fa ^ b j b 2 Mg. We have M a ^ M a and 1 Xb x; y 6 Xaub x; y for all b 2 M since Xaub u Xb X aubtb Xb . Finally, we conclude x; y XW 0 x; y Xa x; y: 1 w Xb x; y XF b2M 0
b2M 0
b
M
(() Suppose R x; y a. Using the relation Q de®ned above, we conclude # # assumption we get Q v a n R and hence 1 6 a n R x; y. From the F 1 Xa x; y. This gives us R x; y a a; Xa x; y 6 b b; Xb x; y. According to our intuition we de®ne crispness in an arbitrary Goguen category as follows: De®nition 22. A relation R : A ! B of a Goguen category is called crisp i R" R. The crisp fragment G" of G is de®ned as the collection of all crisp relations of G. Notice that the fact that " ; # is a Galois correspondence implies 1. 2. 3.
R v R"# and R w R#" , " # #"# RF R"#" and F R " R , " # i2I Ri i2I Ri and wi2I Ri wi2I R#i .
In the following lemma we have collected some basic properties of Goguen categories. Lemma 23. Let G be a Goguen category. Then the following hold:
1. I"A IA . 2. R#" R# . 3. R"# R" . 4. " is a closure and # a co-closure operation, i.e., R v R" , R"" R" and R# v R, R## R# . 5. R R" () R# R. " 6. AB AB and "AB AB . ^ " 7. R^ ; S " R" ; S " . ^ # ^" "^ 8. R R and R^ R# .
M. Winter / Information Sciences 139 (2001) 233±252 "
247
"
9. R; S " R" ; S " and R" ; S R" ; S " . 10. For all nonzero ideal relations J, J "
AB .
Proof. 1. If IA 6 AA then this is a special case of Axiom 4. Otherwise, R IA ; R AA ; R AA follows for all relations R : A ! B and we conclude I"A AA IA . "^ # " # # # 2. Using 1 and Axiom 3 we compute R#" I^ A ; R IA ; R IA ; R R . " "#" "# 3. R R R by 2. 4. First, we conclude R v R"# R" and R" R"#" R"" using 2 and 3. Analogously, we get R w R#" R# and R# R#"# R## . 5. Follows from R v R# () R" v R and 4. # # 6. By 4 AB v AB and AB v "AB , and hence AB AB and "AB AB . Using 5 we get the assertion. ^ ^ " " 7. We immediately conclude R^ ; S " R^ ; S "# R" ; S "# R" ; S " . ^ ^ ^ " " " 8. Using 7 we get R^ R^ ; IA R^ ; I"A R" ; I"A R" ; IA R" . Furthermore, the computation ^
#
"
X v R^ () X " v R^ () X " v R () X ^ v R () X ^ v R# () X v R#
^
gives us the second assertion. 9. We immediately conclude ^
"
^^
"^
"
R; S " R^ ; S " R^ ; S " R"
; S " R" ; S "
and ^^
"^
R" ; S" R" ; S" ^
"
R" ; S^
^
^
"
^
S ^ ; R" " S ^ ; R^ "
"^
^
S " ; R^ R^ ; S " R" ; S " :
J " a;
AB
" a;
" " AB
a" ;
BA
" AB
such that J a;
IA ;
AB
AB
:
AB
and a 6
10. Suppose J 6 AB , and let a : IA u J ; holds. We conclude
AA
Notice that the last lemma shows that R is crisp i R# R. Lemma 24. Let G be a Goguen category, and let fXa j a : A ! A scalarg be a set of crisp relations satisfying wa2M Xa XF M for all sets M of scalars in GA; A. Then for all R : A ! B the following hold:
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M. Winter / Information Sciences 139 (2001) 233±252
G
R
#
a; Xa () a n R Xa for all a : A ! A:
a:A!A a scalar
Proof. ()) It is sucient to show that Xa v a n R# for all a. This property follows from G a; Xa R ) Xa v a n R () Xa" v a n R () Xa v a n R# : a; Xa v a
(() It is sucient to show that
F a
a; Xa v R. This property follows from
#
a; Xa a; a n R v a; a n R v R:
#
As indicated in the proof of Theorem 21 in L the relation a n R is # characterized by a n R x; y 1 () a 6 R x; y. In fuzzy theory this relation is called the a-cut of R. We are now able to prove the so-called a-cut Theorem of fuzzy theory in an arbitrary Goguen category.
Theorem 25 (a-cut Theorem). Let G be a Goguen category. Then for all R : A ! B the following hold: F # 1. R a:A!A a; a n R , a scalar F # 2. R" a6 a n R . AA a scalar
Proof. 1. By Lemma 24 the assertion follows from !# !# G # G # w a n R w a n R anR M nR : a2M
a2M
a2M
2. We immediately compute !" " G G G # # # " R a; a n R a; a n R a" ; a n R a
a #
a n R :
a
G a6
AA
The last theorem shows another interesting relation between Rs and R" since R is the least ®xpoint above R of U de®ned in Section 4. s
Lemma 26. Let G be a Goguen category, let Q; R : A ! B be relations, and let a 6 AA be a scalar. Then the following hold:
M. Winter / Information Sciences 139 (2001) 233±252 #
a n R" R" . " Q" u R" Q u R" . " IA u AA ; R; BA IA u AA ; R" ; BA . If AB 6 AB then AB ; BC AC . If R 6 AB then CA ; R" ; BD CD for all objects C and D. R" v R s . " If a is linear then R" a; R .
1. 2. 3. 4. 5. 6. 7.
249
Proof. 1. The assertion follows from #
G
#
#
a n R" R"" R" :
R" R"# IA n R" v a n R" v
a6
AA
a6
AA
"
#
a6
G
a n Q u a n R
AA
G
#
"
a n Q u R" Q u R" :
a6
2. Using 1 and the a-cut Theorem we get G G # # # Q " u R" a n Q u R" a n Q u a n R"
a6
AA
AA
3. It suces to consider the following computation: IA u
AA ; R;
BA
"
I"A u IA u
AA ; R;
BA
" AA ; R;
IA u
AA ; R;
IA u
AA ; R
"
;
"
AA ; R;
BA
" IA u
" AA ; R;
BA
"
BA
" " BA BA
"
I"A u IA u
AA ; R
"
;
" BA
:
a6
AA
4. As AB 6 AB the composite AB ; BA is a nonzero ideal on A and so AB ; BA " AA by Lemma 23 (10). On the other hand " " AB ; BA "AB ; BA "AB ; "BA AB ; BA . 5. Since R 6 AB the composite CA ; R; BD is a nonzero ideal. It follows " " ; R; ; R ; . CD CA BD CA BD 6. From the monotonicity of U we conclude G G R" a n R# v a n R U R v lR Rs a6
AA
since R v lR implies U R v U lR lR . " 7. It is sucient to show R" v a; R . First, a u b; b n R a; b; b n R # v a; R, which proves b n R v a u b n a; R and hence b n R v a u bn # a; R . Since a u b 6 AA i b 6 AA and the set of all scalars fa u b j b : A ! Ag is a subset of all scalars in GA; A we conclude
M. Winter / Information Sciences 139 (2001) 233±252
G
#
b n R v
R"
b6
a u b n a; R
#
AA
#
"
b n a; R a; R :
v
b6
AA
G
G
250
b6
AA
Notice that property 4 of the last lemma implies that for all A, the sets Sc A of scalar on A are isomorphic via the mapping a 7! BA ; a; AB uIB . For this reason we identify these sets. We call this set the underlying lattice of G and denote it by LG . From properties 6 and 7 of the last lemma we are able to conclude a theorem corresponding to Lemmas 14 and 17 for L-relations in an arbitrary Goguen category. Theorem 27. Let G be a Goguen category. Then the following hold: 1. All s-crisp relations are crisp. 2. All crisp relations are l-crisp. Proof. 1. Suppose R is s-crisp. Then we conclude R" v Rs R v R" . 2. Suppose k; Q v R for a linear scalar k and a crisp relation R. Then we con" clude Q v k; Q v R" R. Furthermore, we have a result similar to Lemma 15 concerning the characterization of these Goguen categories such that the notions of s-crispness and crispness coincide. Theorem 28. Let G be a Goguen category. Then the following statements are equivalent: 1. G is linear. 2. All crisp relations are s-crisp, i.e., R" Rs .
Proof. 1 ) 2: By Theorem 27 since the class of l-crisp and the class of scrisp relations are equal. 2 ) 1: Let a be a nonzero scalar and a0 a scalar such that a u a0 AA holds. Then a; a0 a u a0 AA . We conclude a0 AA since AA is crisp, and hence s-crisp. We have collected closure properties of the class of crisp relations in the following lemma:
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Lemma 29. Let G be a Goguen category, and for i 2 I, let Qi ; Q : A ! B, R : A ! C and S : B ! C be crisp relations. Then the following hold: F 1. i2I Qi and wi2I Qi are crisp. 2. Q^ is crisp. 3. Q; S is crisp. 4. R=S and Q n R are crisp. 5. If Q u T and Q t T are crisp then T is crisp. 6. If Q is complemented then Q is crisp. Proof. F F F " # 1. i2I Qi i2I Q"i i2I Qi and wi2I Qi wi2I Q#i wi2I Qi . ^ " 2. Q^ Q" Q^ . " " 3. Q; S Q; S " Q" ; S " Q; S. " 4. It is sucient to show R=S v R=S. This follows from "
"
"
"
R=S ; R R=S ; R" R=S; R" R=S; R v S " S: "
5. First, Q u T Q" u T Q" u T " Q u T " . Furthermore, Q t T Q t T " Q" t T " Q t T " . Since GA; B is a distributive lattice, we conclude T " T . 6. By 4. since AB Q u Q and AB Q t Q are crisp. The last lemma gives us the following corollary: Corollary 30. If G is a (Boolean based) Goguen category then G" is a simple (Schr oder) Dedekind category.
Acknowledgements I am grateful to Hitoshi Furusawa, Yasuo Kawahara and Gunther Schmidt for reading, commenting and contributing to this paper. Last but not least, I wish to thank the anonymous referees for their helpful comments.
References [1] P. Freyd, A. Scedrov, Categories, Allegories, North-Holland, Amsterdam, 1990. [2] H. Furusawa, Algebraic formalisations of fuzzy relations and their representation theorems, Ph.D. Thesis, Kyushu University, 1998. [3] J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967) 145±157. [4] B. J onsson, A. Tarski, Boolean algebras with operators, part I, Amer. J. Math. 73 (1951) 891± 939;
252
[5] [6] [7] [8] [9] [10] [11] [12]
M. Winter / Information Sciences 139 (2001) 233±252 B. J onsson, A. Tarski, Boolean algebras with operators, part II, Amer. J. Math. 74 (1952) 127±162. Y. Kawahara, H. Furusawa, Crispness and representation theorems in Dedekind categories, DOI-TR 143, Kyushu University, 1997. J.P. Olivier, D. Serrato, Categories de Dedekind. Morphismes dans les Categories de Schr oder, C. R. Acad. Sci. Paris 290 (1980) 939±941. J.P. Olivier, D. Serrato, Squares and rectangles in relational categories ± three cases: semilattice, distributive lattice and boolean non-unitary, Fuzzy sets and systems 72 (1995) 167± 178. G. Schmidt, T. Str ohlein, Relationen und Graphen, Springer, 1989 (English version: Relations and Graphs. Discrete Mathematics for Computer Scientists, EATCS Monographs on Theoret. Comput. Sci., Springer, 1993). G. Schmidt, C. Hattensperger, M. Winter, Heterogeneous relation algebras, in: C. Brink, W. Kahl, G. Schmidt (Eds.), Relational Methods in Computer Science, Advances in Computer Science, Springer, Vienna, 1997. L.H. Chin, A. Tarski, Distributive and Modular Laws in the Arithmetic of Relation Algebras, University of California Press, Berkeley, CA, 1951. M. Winter, Strukturtheorie heterogener Relationenalgebren mit Anwendung auf Nichtdetermismus in Programmiersprachen, Dissertationsverlag NG Kopierladen GmbH, M unchen, 1998. L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338±353.
www.elsevier.com/locate/ins
A new algebraic approach to L-fuzzy relations convenient to study crispness Michael Winter Department of Computer Science, University of the Federal Armed Forces Munich, 85577 Neubiberg, Germany Received 9 January 2000; received in revised form 17 March 2001; accepted 25 June 2001
Abstract The aim of this paper is to develop a suitable calculus of L-fuzzy relations. We show that for this purpose the theory of Dedekind categories is too weak. Therefore, we introduce the notion of a Goguen category as a suitable algebraic de®nition and show several properties of this kind of a relational category. Ó 2001 Elsevier Science Inc. All rights reserved.
1. Introduction The calculus of binary relations has been investigated since the middle of the 19th century. It plays an important role in the development of logic and algebra. Furthermore, since the mid-1970s it has become clear that this calculus is a fundamental conceptual and methodological tool in computer science just as much as logic. While computer science applications are evolving rapidly in several areas as in communication, programming, software, data or knowledge engineering, exact sciences are needed to understand existing methods. One important application is the treatment of uncertain or incomplete information. To handle such kind of information, Zadeh [12] introduced the concept of fuzzy sets. Later on, Goguen [3] generalized this concept to L-fuzzy sets and relations for an arbitrary complete distributive lattice L instead of the unit interval 0; 1 of the real numbers.
E-mail address: [email protected] (M. Winter). 0020-0255/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 0 - 0 2 5 5 ( 0 1 ) 0 0 1 6 7 - 0
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In this paper we aim to give a suitable algebraic de®nition for L-fuzzy relations. It is well known that a reasonable category of L-relations constitutes a Dedekind category introduced in [6]. Unfortunately, this theory is too weak to express some standard properties of L-relations. In particular, we show that there is no formula in the theory of Dedekind categories expressing the fact that a given L-relation is 0±1 crisp, i.e., all entries are either the least element 0 or the greatest element 1 of L. We introduce the notion of a Goguen category as a suitable algebraic definition for L-relations. First of all, crispness in this theory coincides with 0±1 crispness of L-relations. Furthermore, we show that under an assumption of the underlying lattice, i.e., the lattice of scalar relations, the notion of s-crispness [2,5] and crispness are equivalent. We assume that the reader is familiar with the basic concepts of category theory.
2. Dedekind categories Throughout this paper, we use the following notation. To indicate that a morphism R of a category R has source A and target B we write R : A ! B. The collection of all morphisms R : A ! B is denoted by RA; B and the composition of a morphism R : A ! B followed by a morphism S : B ! C by R; S. The identity morphism on A is denoted by IA . In this section we recall some fundamentals on Dedekind categories [6,7]. These categories are called locally complete division allegories in [1]. De®nition 1. A Dedekind category R is a category satisfying the following:
1. For all objects A and B the collection RA; B is a complete distributive lattice. Meet, join, the induced ordering and the least and the greatest elements are denoted by u; t; v; AB ; AB , respectively. 2. There is a monotone operation ^ (called conversion) such that for all rela^ tions Q : A ! B and R : B ! C the following holds: Q; R R^ ; Q^ and ^ ^ Q Q. 3. For all relations Q : A ! B; R : B ! C and S : A ! C the modular law Q; R u S v Q; R u Q^ ; S holds. 4. For all relations R : B ! C and S : A ! C there is a relation S=R : A ! B (called the left residual of S and R) such that for all Q : A ! B the following holds: Q; R v S () Q v S=R. A Dedekind category R such that every collection RA; B is a Boolean algebra is called a Schr oder category. We denote the complement of a relation R : A ! B, if it exists, by R. If, in addition, RA; B is atomic for all A and B, R
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235
is called a heterogeneous relation algebra. According to the equivalence of the so-called Tarski-rule )
R 6
AB
CA ; R;
BD
CD
for all objects C and D
to a generalized version of the notion of simplicity known from universal algebra, we call a Dedekind category (or a Schr oder category or a heterogeneous relation algebra) simple i the Tarski-rule is valid. Furthermore, we de®ne the ^ right residual by Q n R : R^ =Q^ . In the following lemma we have collected some basic properties of relations in a Dedekind category. We will use these properties throughout the paper without mentioning. Proofs may be found in [1,8±11]. Lemma 2. Let R be a Dedekind category, let A; B; C be objects of R, and for i 2 I, let Q1 ; Q2 ; Qi : A ! B, R1 ; R2 ; Ri : B ! C and S; S1 ; S2 : A ! C be relations. Then
BC
AB
BA
AB ;
AB ; R1
BB
and I^ A IA ,
AB ;
BA
;
AB
AB ,
AC ,
AA ;
3. Q1 ;
^ AB
BA ;
2.
^ AB
1.
^
4. wi2I Ri wi2I R^ i , F F ^ ^ 5. i2I Ri i2I Ri , 6. if Q1 v Q2 and R1 v R2 then Q1 ; R1 v Q2 ; R2 , 7. Q1 ; R1 u S v Q1 ; uS; R^ 1 ; R1 , F F 8. Q1 ; i2I Ri i2I Q1 ; Ri , 9. Q1 ; wi2I Ri v wi2I Q1 ; Ri , 10. Q1 v Q1 ; Q^ 1 ; Q1 , 11. S=IC S and IA n S S, 12. Q2 v Q1 ; R2 v R1 and S1 v S2 imply S1 =R1 v S2 =R2 and Q1 n S1 v Q2 n S2 , 13. S=R1 ; R1 v S and Q1 ; Q1 n S v S, F F 14. wi2I S=Ri S= i2I Ri and wi2I Qi n S i2I Qi n S. An important class of relations is given by mappings. De®nition 3. Let Q : A ! B be a relation. Then we call 1. Q univalent i Q^ ; Q v IB , 2. Q total i IA v Q; Q^ or equivalently i Q; 3. Q a map i Q is univalent and total,
BC
AC
for all objects C,
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4. Q injective i Q^ is univalent, 5. Q surjective i Q^ is total, 6. Q an isomorphism i Q and Q^ are mappings. Notice that if Q is an isomorphism Q^ ; Q IB and Q; Q^ IA hold. In the following lemma we collect some fundamental facts concerning this class of relations. Again, proofs may be found in [1,8±11]. Lemma 4. Let U : A ! B be univalent, let F : D ! A and G : E ! B be mappings, let H : C ! C be total and injective, and for i 2 I, let Q : A ! C and Ri ; S : B ! C be mappings. Then 1. U ; wi2I Ri wi2I U ; Ri , 2. F ; Q=S F ; Q=S, 3. Q=S; G^ Q= G; S, 4. Q=S Q; H = S; H . Later on, we will show that there is no formula expressing the fact that an Lrelation is 0±1 crisp. For this purpose, we have to de®ne a language and its meaning. We require a set of object variables and a set of typed relation variables, i.e., every relation variable is of the form r : a ! b where a and b are object variables. Now, terms and formulae are de®ned as follows: De®nition 5. The set of terms of type a ! b and the set of formulae are de®ned inductively as follows:
5. 6. 7. 8. 9. 10. 11. 12.
Any relation variable r : a ! b is a term of type a ! b. If a is an object variable then Ia is a term of type a ! a. If a and b are object variables then ab and ab are terms F of type a ! b. If u is a formula and r : a ! b a relation variable then fr : a ! b j ug and wfr : a ! b j ug are terms of type a ! b. If t is a term of type a ! b then t^ is a term of type b ! a. If t1 and t2 are terms of type a ! b resp. b ! c then t1 ; t2 is a term of type a ! c. If t1 and t2 are terms of type a ! c resp. b ! c then t1 =t2 is a term of type a ! b. If t1 and t2 are terms of type a ! b then t1 t2 is a formula. If u1 and u2 are formulas then u1 ^ u2 is a formula. If u is a formula then :u is a formula. If u is a formula and r : a ! b is a relation variable then 8r : a ! bu is a formula. If u is a formula and a is an object variable then 8au is a formula.
1. 2. 3. 4.
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237
Given a Dedekind category R, an environment r is a function mapping each object variable a to an object A of R and each relation variable r : a ! b to a relation R : r a ! r b. The update rA=a, respectively rR=r : a ! b, of r at the object variable a, respectively at the relation variable r : a ! b, with the object A, respectively with the relation R : r a ! r b, is de®ned by r b iff a 6 b; rA=a b : A iff a b; 8 iff a c; < Ar d rA=a r : c ! d : iff a d; r cA : r r : c ! d iff a 6 c ^ a 6 d; rR=r : a ! b c : r c;
rR=r : a ! b s : c ! d :
r s : c ! d R
iff r : a ! b 6 s : c ! d; iff r : a ! b s : c ! d:
As usual we denote a sequence of updates rA=aB=bR=r on an environment r by rA=a; B=b; R=r. Furthermore, given a class F of mappings FA : A ! A for every object A of R and an environment r we denote with F r the environment de®ned by 1. F r a r a for all object variables a, ^ 2. F r r : a ! b : Fr a ; r r : a ! b; Fr b r : a ! b.
for
all
relation
variables
^ Notice that F rR=r : a ! b F rFr a ; R; Fr b =r : a ! b. Furthermore, if F is a class of isomorphisms, i.e., every FA is an isomorphism, the function r 7! F r is a bijective function on the class of environments.
De®nition 6. The value VR t r of a term t of type a ! b and the validity R r u of a formula u in a Dedekind category R under an environment r are de®ned inductively as follows: 1. VR r : a ! b r : r r : a ! b, 2. VR Ia r : Ir a , 4. VR
F
:
ab r
r ar b ,
ab r
3. VR
:
r ar b ,
5. VR fr : a ! b j ug r :
F
fR j R rR=r:a!b ug,
6. VR wfr : a ! b j ug r : wfR j R rR=r:a!b ug, ^
7. VR t^ r : VR t r , 8. VR t1 ; t2 r : VR t1 r; VR t2 r,
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9. VR t1 =t2 r : VR t1 r=VR t2 r, 10. R r t1 t2 i VR t1 r VR t2 r, 11. R r u1 ^ u2 i R r u1 and R r u2 , 12. R r :u i R 2r u, 13. R r 8r : a ! bu i R rR=r:a!b u for all R : r a ! r b, 14. R r 8au i R rA=a u for all objects A. Later on, the following lemma plays an important role in the proof of Theorem 1. Lemma 7. Let t be a term of type a ! b, let u be a formula, let R be a Dedekind category, let r be an environment, and let F be a class of isomorphisms. Then the following hold: ^ 1. VR t r Fr a ; VR t F r; Fr b . 2. R r u iff R F r u.
Proof. The assertions are shown simultaneously by structural induction. · t r : a ! b ^ ^ VR t r r r : a ! b Fr a ; Fr a ; r r : a ! b; Fr b ; Fr b ^ ^ Fr a ; F r r : a ! b; Fr b Fr a ; VR t F r; Fr b :
^ ^ ; Ir a ; Fr a Fr a ; VR t F r; Fr a . · t Ia VR t r Ir a Fr a · t Fab or t ab g Analogously to t Ia . · t fr : a ! b j ug G G VR t r fR j R rR=r:a!b ug fR j R F rR=r:a!b ug G ^ =r:a!b ug fR j R F rFr a ;R;Fr b G ^ Fr a ; fS j R F rS=r:a!b ug; Fr b ^ Fr a ; VR t F r; Fr b :
· t wfr : a ! b j ug Analogously to t of isomorphisms. · t t1^
F fr : a ! b j ug since F is a class
^ VR t r VR t1 r^ Fr b ; VR t1 F r; Fr a ^ ^
^ ^ Fr a ; VR t1 F r ; Fr b Fr a ; VR t F r; Fr b :
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239
· t t1 ; t2 VR t r VR t1 r; VR t2 r ^ ^ ; VR t1 F r; Fr c ; Fr c ; VR t2 F r; Fr b Fr a ^ Fr a ; VR t1 F r; VR t2 F r; Fr b ^ Fr a ; VR t F r; Fr b :
· t t1 =t2 . Using Lemma 4 (2)±(4) we get VR t r VR t1 r=VR t2 r ^ ^ ; VR t1 F r; Fr c = Fr b ; VR t2 F r; Fr c Fr a ^ Fr a ; VR t1 F r=VR t2 F r; Fr b ^ Fr a ; VR t F r; Fr b :
· u t1 t2 R r u () VR t1 r VR t2 r ^ ^ () Fr a ; VR t1 F r; Fr b Fr a ; VR t2 F r; Fr b () VR t1 F r VR t2 F r () R F r u: · u u1 ^ u2 R r u () R r u1 and R r u2 () R F r u1 and R F r u2 () R F r u: u1 () R F r u. · u :u1 R r u () R 2 u1 () R 2 r F r · u 8r : a ! bu1 R r u () R rR=r:a!b u1 for all R : r a ! r b () R F rR=r:a!b u1 for all R : r a ! r b ^ =r:a!b u () R F rFr a ;R;Fr b 1 for all R : r a ! r b () R F rS=r:a!b u1 for all S : r a ! r b () R F r u: · u 8au1 R r u () R rA=a u1 for all objects A () R F rA=a u1 for all objects A () R F rA=a u1 for all objects A () R F r u:
3. Matrix-algebras, fuzzy- and L-relations Given a Dedekind category R, an algebra of matrices with coecients from R may be de®ned.
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De®nition 8. Let R be a Dedekind category. The algebra R of matrices with coecients from R is de®ned by 1. The class of objects of R is the collection of all functions from an arbitrary set I into the class of objects ObjR of R. 2. For every pair f : I ! ObjR ; g : J ! ObjR of objects from R , a morphism R : f ! g is a function from I J into the class of all morphisms MorR of R such that R i; j : f i ! g j holds. 3. For R : f ! g and S : g ! h, composition is de®ned by G R; S i; k : R i; j; S j; k: j2J
4. For R : f ! g, conversion is de®ned by R^ j; i : R i; j^ . 5. For R; S : f ! g, join and meet are de®ned by R t S i; j : R i; j t S i; j;
R u S i; j : R i; j u S i; j:
6. The identity, zero and universal elements are de®ned by : i1 6 i2 ; f i1 f i2 fg i; j : f ig j ; If i1 ; i2 : If i1 : i1 i2 ; fg i; j : f ig j : Obviously, a morphism in R may be seen as a (in general non®nite) matrix indexed by objects from R. If R has only one object the objects f : I ! ObjR of R may be identi®ed with the set I. Lemma 9. R is a Dedekind category. Proof (sketch). The critical point is the de®nition of the left residual in R . As shown in Lemma 2 the in®nite variants of meet-subdistributivity and joindistributivity hold in R. A straightforward computation shows that these properties are also valid in R . Now, it is possible to de®ne the residual of S and R as the join of all elements Q satisfying Q; R v S. Let L L; _; ^; 0; 1 be a complete distributive lattice with least element 0 and greatest element 1. The induced ordering is denoted by 6 . Notice that we have pseudo-complements in L, i.e., for all elements k and l there is an element k ! l such that k ^ m 6 l () m 6 k ! l. L may also be considered as an oneobject Dedekind category with identity 1 and composition ^ (the residual is given by the pseudo-complement). Consequently, L is a Dedekind category, called the full category of L-relations. This category (or every subcategory closed under all operations) seems to be a good model for generalized fuzzy relations, also called L-fuzzy relations in [3]. An L-relation R is called 0±1 crisp i R i; j 0 or R i; j 1 holds for all i and j.
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Let 0; 1 R be the unit interval, i.e., the set of all real numbers r such that 0 6 r 6 1 holds. Then 0; 1; inf; sup; 0; 1 is a complete distributive lattice. Hence, we get the full category of fuzzy relations 0; 1 as a special case.
4. Crispness in Dedekind categories In some sense a relation of a Dedekind category may be seen as an L-relation. The lattice L may be equivalently characterized by the ideal relations, i.e., a relation J : A ! B satisfying AA ; J ; BB J , or by the scalar relations.
De®nition 10. A relation a : A ! A is called a scalar on A i a v IA and 0 AA ; a a; AA . Furthermore, a is called linear i for every scalar a : A ! A, 0 0 a u a AA implies a AA . Notice that the notion of ideal was introduced by J onsson and Tarski [4] and the notion of scalar was introduced by Kawahara and Furusawa [5]. Given a complete distributive lattice L, the ideal elements J and the scalars a are exactly these relations such that k iff x y; J x; y k for all x and y; a x; y 0 iff x 6 y
for an element k 2 L. Notice that the collection of ideal elements on A is isomorphic to the collection of scalars on A via the mappings / J : J u IA and / 1 a : a; AA . Furthermore, for all S : A ! B, the relation AA ; S; BA is an ideal element and IA u AA ; S; BA is a scalar. If S 6 AB then both relations are nonzero since otherwise we have BB
IA u
AA ; S;
BA
;
AB
AA ;
AB
AB :
AA ; S;
Sv
Lemma 11. Let a; b be scalars on A, and let R : A ! B. Then 1. 2. 3. 4.
a; R R u a; a; b a u b, a; a a, IA u AA ; a;
AB ,
AA
a.
All proofs not given in this section (including the proof of the lemma above) may be found in [2,5]. Using the scalars, there are two notions of crispness in an arbitrary Dedekind category.
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De®nition 12. A relation R : A ! B is called (l-crisp) s-crisp i a; Q v R implies Q v R for all (linear) nonzero scalars a and all relations Q : A ! B.
Obviously, AB is s-crisp. Furthermore, a little computation shows that the class of s-crisp relations is closed under meet. But, unfortunately, AB may be not s-crisp. Dedekind categories in which AB is s-crisp are characterized by the following lemma: Lemma 13. In a Dedekind category the following statements are equivalent:
1. All nonzero scalars are linear. 2. AB is s-crisp. 3. For every R : A ! B, its pseudo-complement R ! 4. Every complemented R : A ! B is s-crisp.
AB
is s-crisp.
Considering L-relations the following hold. Lemma 14. In L all s-crisp relations are 0±1 crisp. The converse is, in general, not true. A property of the underlying lattice L equivalent for L-relations to Lemma 13 (1) is required. Lemma 15. Let L be a complete distributive lattice. Then the following properties are equivalent: 1. In L all 0±1 crisp relations are s-crisp. 2. k ^ l 0 implies k 0 or l 0 for all elements k; l 2 L. The following lemma shows that s-crispness grasp 0±1 crispness i the properties listed in Lemma 13 are valid. Lemma 16. Let A be an object of a Dedekind category. If there is a nonzero and nonlinear scalar a on A then there is no s-crisp relation R : A ! B except AB for all objects B.
Proof. Suppose R : A ! B is s-crisp and a; a0 are nonzero scalars such that a u a0 AA holds. Then a; a0 ; AB a u a0 ; AB AB v R. We conclude a0 ; AB v R and further AB v R since R is s-crisp. We have seen that the notion of s-crispness does not grasp 0±1 crispness of L-relations. Also, the notion of l-crispness does not work. If L is a Boolean lattice the identities are the only linear scalars in L , and hence, all relations are l-crisp. But the following hold:
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243
Lemma 17. In L all 0±1 crisp relations are l-crisp. Unfortunately, we are able to show that the theory of Dedekind categories is too weak to express the notion of 0±1 crispness of L-relations. Theorem 18. There is no formula u in the language of Dedekind categories such that for all lattices L and L-relations R : A ! B, L rA=a;B=b;R=r u for all environments r () R is 0±1 crisp: Proof. Consider the Boolean algebra B4 : P fa; bg, i.e., the powerset of the set fa; bg. Let X fxg and Y fx; yg be the sets. Then the relation FY :
fag fbg
fbg fag
is an isomorphism in B 4 Y ; Y . Now, take F as the class of isomorphisms given by FY on Y and IZ on all other sets Z. Suppose there is such a formula u. Then B 4 rX =a;Y =b;R=r u for all environments r where R is the 0±1 crisp relation R : fa; bg ; . Lemma 7 shows that B 4 F rX =a;Y =b;R=r u. Since F rX =a; Y =b; R=r r FX ; R; FY^ R; FY^ fag fbg fa; bg ; ; fag fbg ; fbg fag we conclude B 4 F rX =a;Y =b;S=r u where S is the relation above. Since the function r7!F r is bijective, B 4 rX =a;Y =b;S=r u for all environments r. From the de®nition of u we conclude that the relation S is 0±1 crisp, a contradiction.
Nevertheless, we are able to map every relation R : A ! B to the least s-crisp relation Rs : wfQ : A ! B j R v Q and Q is s-crispg containing R. There is another possible way to characterize Rs . Consider the function G U R : a n R: a6
AA a scalar
This function is monotone, and R IA n R v a n R, since a is a subidentity. F This gives us R v a6 a n R U R. Altogether we conclude that for every R AA there is a least ®xpoint lR of U greater than R.
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Theorem 19. Let R be a Dedekind category. Then Rs lR for all relations R.
Proof. v: It is sucient to show that lR is s-crisp. Suppose there is a nonzero scalar a andF a relation Q such that a; Q v lR . Then we conclude Q v a n lR v a6 a n lR U lR lR . AA w: It is sucient to show that Rs is a ®xpoint of U. For every nonzero scalar s s a we have a; a n Rs v Rs and Rs is s-crisp. This implies F a n R v sR since s s U R a6 a n R v R . The other inclusion Rs v U Rs is trivial. AA
5. Goguen categories In the last section we have shown that we need an additional concept to de®ne a suitable algebraic theory of L-fuzzy relations. Our approach introduces two operations one which maps each relation to the greatest 0±1 crisp relation it contains and one which maps each relation to the least 0±1 crisp relation which includes it. We now give the abstract de®nition. De®nition 20. A Goguen category is a Dedekind category together with two operations " and # satisfying the following:
1. R" ; R# : A ! B for all R : A ! B. 2. " ; # is a Galois correspondence, i.e., R" v S () R v S # for all R; S : A ! B. ^ " 3. R^ ; S # R" ; S # for all R : B ! A and S : B ! C. 4. If a 6 AA is a nonzero scalar then a" IA . 5. For any set fXa : A ! B j a : A ! A scalarg such that (a) Xa Xa" for all a : A ! A. (b) w Xa XF M for all sets M of scalars in GA; A, a2M
and all R : A ! B the following equivalence holds: G a; Xa () a n R# v Xa for all a : A ! A: Rv a:A!A a scalar
Furthermore, a Goguen category G is called Boolean based i for all R : A ! B the relation R" is complemented, i.e., R" t R" ! AB AB . G is called linear i all nonzero scalars are linear and nonlinear i G is not linear. The obvious de®nitions of
"
and
#
for L-relations give the standard model.
Theorem 21. Let L L; _; ^; 0; 1 be a complete distributive lattice. Then L together with the operations
M. Winter / Information Sciences 139 (2001) 233±252
"
R x; y :
1 0
iff R x; y 6 0; iff R x; y 0;
#
R x; y :
1 0
245
iff R x; y 1; iff R x; y 6 1
is a Boolean based Goguen category. Furthermore, for a relation R in L we have R" R R# iff R 0±1 crisp. Proof. The assertion R" R R# i R 0±1 crisp follows immediately from the de®nitions of " and # . Since every scalar a : A ! A is of the form a x; y
k 0
iff x y; iff x 6 y
for an element k 2 L, we will identify throughout this proof the scalar a with the element k and denote it by a. 1. Axiom 1 is true by de®nition. 2. Suppose R" v S and R" x; y 6 0. Then by the de®nition of " we have 1 R" x; y 6 S x; y and hence R x; y 6 1 S # x; y. The other implication is shown analogously. 3. De®ne two operations " ; # : L ! L on the lattice L by a" 0 if a 0 and a" 1 otherwise, and a# 1 if aW 1 and a#W 0 otherwise. The operations satisfy a ^ b# " a" ^ b# and a2M a" a2M a" for all subsets M of L. Then we conclude _
# "
R^ ; S x; z
_
!" R^ x; y ^ S # y; z
y "
#
R y; x ^ S y; z
y
_
_
R y; x ^ S # y; z
"
y "^
^
R x; y ^ S # y; z R" ; S # :
y
4. By the de®nition of " we conclude for every nonzero scalar a : A ! A 1 iff x y a" x; y IA x; y: 0 iff x 6 y 5. Suppose fXa : A ! B j a : A ! A scalarg is a set of 0±1 crisp relations with the required properties. ()) Suppose a n R# x; y 1. Then the 0±1 crisp relation Q de®ned by 1 iff x0 x and y 0 y; 0 0 Q x ; y : 0 else is included in a n R# . We conclude Q v a n R# () Q v a n R () a; Q v R () a a; Q x; y 6 R x; y:
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M. Winter / Information Sciences 139 (2001) 233±252
F By the assumption we get a 6 b b; Xb x; y. Let M be the set of all elements b such that Xb x; y 1. Then G _ _ a 6 b; Xb x; y b ^ Xb x; y b ^ Xb x; y b
b
_
b
_
b2M
M:
b2M
W 0 W Now, let M 0 : fa ^ b j b 2 Mg. We have M a ^ M a and 1 Xb x; y 6 Xaub x; y for all b 2 M since Xaub u Xb X aubtb Xb . Finally, we conclude x; y XW 0 x; y Xa x; y: 1 w Xb x; y XF b2M 0
b2M 0
b
M
(() Suppose R x; y a. Using the relation Q de®ned above, we conclude # # assumption we get Q v a n R and hence 1 6 a n R x; y. From the F 1 Xa x; y. This gives us R x; y a a; Xa x; y 6 b b; Xb x; y. According to our intuition we de®ne crispness in an arbitrary Goguen category as follows: De®nition 22. A relation R : A ! B of a Goguen category is called crisp i R" R. The crisp fragment G" of G is de®ned as the collection of all crisp relations of G. Notice that the fact that " ; # is a Galois correspondence implies 1. 2. 3.
R v R"# and R w R#" , " # #"# RF R"#" and F R " R , " # i2I Ri i2I Ri and wi2I Ri wi2I R#i .
In the following lemma we have collected some basic properties of Goguen categories. Lemma 23. Let G be a Goguen category. Then the following hold:
1. I"A IA . 2. R#" R# . 3. R"# R" . 4. " is a closure and # a co-closure operation, i.e., R v R" , R"" R" and R# v R, R## R# . 5. R R" () R# R. " 6. AB AB and "AB AB . ^ " 7. R^ ; S " R" ; S " . ^ # ^" "^ 8. R R and R^ R# .
M. Winter / Information Sciences 139 (2001) 233±252 "
247
"
9. R; S " R" ; S " and R" ; S R" ; S " . 10. For all nonzero ideal relations J, J "
AB .
Proof. 1. If IA 6 AA then this is a special case of Axiom 4. Otherwise, R IA ; R AA ; R AA follows for all relations R : A ! B and we conclude I"A AA IA . "^ # " # # # 2. Using 1 and Axiom 3 we compute R#" I^ A ; R IA ; R IA ; R R . " "#" "# 3. R R R by 2. 4. First, we conclude R v R"# R" and R" R"#" R"" using 2 and 3. Analogously, we get R w R#" R# and R# R#"# R## . 5. Follows from R v R# () R" v R and 4. # # 6. By 4 AB v AB and AB v "AB , and hence AB AB and "AB AB . Using 5 we get the assertion. ^ ^ " " 7. We immediately conclude R^ ; S " R^ ; S "# R" ; S "# R" ; S " . ^ ^ ^ " " " 8. Using 7 we get R^ R^ ; IA R^ ; I"A R" ; I"A R" ; IA R" . Furthermore, the computation ^
#
"
X v R^ () X " v R^ () X " v R () X ^ v R () X ^ v R# () X v R#
^
gives us the second assertion. 9. We immediately conclude ^
"
^^
"^
"
R; S " R^ ; S " R^ ; S " R"
; S " R" ; S "
and ^^
"^
R" ; S" R" ; S" ^
"
R" ; S^
^
^
"
^
S ^ ; R" " S ^ ; R^ "
"^
^
S " ; R^ R^ ; S " R" ; S " :
J " a;
AB
" a;
" " AB
a" ;
BA
" AB
such that J a;
IA ;
AB
AB
:
AB
and a 6
10. Suppose J 6 AB , and let a : IA u J ; holds. We conclude
AA
Notice that the last lemma shows that R is crisp i R# R. Lemma 24. Let G be a Goguen category, and let fXa j a : A ! A scalarg be a set of crisp relations satisfying wa2M Xa XF M for all sets M of scalars in GA; A. Then for all R : A ! B the following hold:
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M. Winter / Information Sciences 139 (2001) 233±252
G
R
#
a; Xa () a n R Xa for all a : A ! A:
a:A!A a scalar
Proof. ()) It is sucient to show that Xa v a n R# for all a. This property follows from G a; Xa R ) Xa v a n R () Xa" v a n R () Xa v a n R# : a; Xa v a
(() It is sucient to show that
F a
a; Xa v R. This property follows from
#
a; Xa a; a n R v a; a n R v R:
#
As indicated in the proof of Theorem 21 in L the relation a n R is # characterized by a n R x; y 1 () a 6 R x; y. In fuzzy theory this relation is called the a-cut of R. We are now able to prove the so-called a-cut Theorem of fuzzy theory in an arbitrary Goguen category.
Theorem 25 (a-cut Theorem). Let G be a Goguen category. Then for all R : A ! B the following hold: F # 1. R a:A!A a; a n R , a scalar F # 2. R" a6 a n R . AA a scalar
Proof. 1. By Lemma 24 the assertion follows from !# !# G # G # w a n R w a n R anR M nR : a2M
a2M
a2M
2. We immediately compute !" " G G G # # # " R a; a n R a; a n R a" ; a n R a
a #
a n R :
a
G a6
AA
The last theorem shows another interesting relation between Rs and R" since R is the least ®xpoint above R of U de®ned in Section 4. s
Lemma 26. Let G be a Goguen category, let Q; R : A ! B be relations, and let a 6 AA be a scalar. Then the following hold:
M. Winter / Information Sciences 139 (2001) 233±252 #
a n R" R" . " Q" u R" Q u R" . " IA u AA ; R; BA IA u AA ; R" ; BA . If AB 6 AB then AB ; BC AC . If R 6 AB then CA ; R" ; BD CD for all objects C and D. R" v R s . " If a is linear then R" a; R .
1. 2. 3. 4. 5. 6. 7.
249
Proof. 1. The assertion follows from #
G
#
#
a n R" R"" R" :
R" R"# IA n R" v a n R" v
a6
AA
a6
AA
"
#
a6
G
a n Q u a n R
AA
G
#
"
a n Q u R" Q u R" :
a6
2. Using 1 and the a-cut Theorem we get G G # # # Q " u R" a n Q u R" a n Q u a n R"
a6
AA
AA
3. It suces to consider the following computation: IA u
AA ; R;
BA
"
I"A u IA u
AA ; R;
BA
" AA ; R;
IA u
AA ; R;
IA u
AA ; R
"
;
"
AA ; R;
BA
" IA u
" AA ; R;
BA
"
BA
" " BA BA
"
I"A u IA u
AA ; R
"
;
" BA
:
a6
AA
4. As AB 6 AB the composite AB ; BA is a nonzero ideal on A and so AB ; BA " AA by Lemma 23 (10). On the other hand " " AB ; BA "AB ; BA "AB ; "BA AB ; BA . 5. Since R 6 AB the composite CA ; R; BD is a nonzero ideal. It follows " " ; R; ; R ; . CD CA BD CA BD 6. From the monotonicity of U we conclude G G R" a n R# v a n R U R v lR Rs a6
AA
since R v lR implies U R v U lR lR . " 7. It is sucient to show R" v a; R . First, a u b; b n R a; b; b n R # v a; R, which proves b n R v a u b n a; R and hence b n R v a u bn # a; R . Since a u b 6 AA i b 6 AA and the set of all scalars fa u b j b : A ! Ag is a subset of all scalars in GA; A we conclude
M. Winter / Information Sciences 139 (2001) 233±252
G
#
b n R v
R"
b6
a u b n a; R
#
AA
#
"
b n a; R a; R :
v
b6
AA
G
G
250
b6
AA
Notice that property 4 of the last lemma implies that for all A, the sets Sc A of scalar on A are isomorphic via the mapping a 7! BA ; a; AB uIB . For this reason we identify these sets. We call this set the underlying lattice of G and denote it by LG . From properties 6 and 7 of the last lemma we are able to conclude a theorem corresponding to Lemmas 14 and 17 for L-relations in an arbitrary Goguen category. Theorem 27. Let G be a Goguen category. Then the following hold: 1. All s-crisp relations are crisp. 2. All crisp relations are l-crisp. Proof. 1. Suppose R is s-crisp. Then we conclude R" v Rs R v R" . 2. Suppose k; Q v R for a linear scalar k and a crisp relation R. Then we con" clude Q v k; Q v R" R. Furthermore, we have a result similar to Lemma 15 concerning the characterization of these Goguen categories such that the notions of s-crispness and crispness coincide. Theorem 28. Let G be a Goguen category. Then the following statements are equivalent: 1. G is linear. 2. All crisp relations are s-crisp, i.e., R" Rs .
Proof. 1 ) 2: By Theorem 27 since the class of l-crisp and the class of scrisp relations are equal. 2 ) 1: Let a be a nonzero scalar and a0 a scalar such that a u a0 AA holds. Then a; a0 a u a0 AA . We conclude a0 AA since AA is crisp, and hence s-crisp. We have collected closure properties of the class of crisp relations in the following lemma:
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Lemma 29. Let G be a Goguen category, and for i 2 I, let Qi ; Q : A ! B, R : A ! C and S : B ! C be crisp relations. Then the following hold: F 1. i2I Qi and wi2I Qi are crisp. 2. Q^ is crisp. 3. Q; S is crisp. 4. R=S and Q n R are crisp. 5. If Q u T and Q t T are crisp then T is crisp. 6. If Q is complemented then Q is crisp. Proof. F F F " # 1. i2I Qi i2I Q"i i2I Qi and wi2I Qi wi2I Q#i wi2I Qi . ^ " 2. Q^ Q" Q^ . " " 3. Q; S Q; S " Q" ; S " Q; S. " 4. It is sucient to show R=S v R=S. This follows from "
"
"
"
R=S ; R R=S ; R" R=S; R" R=S; R v S " S: "
5. First, Q u T Q" u T Q" u T " Q u T " . Furthermore, Q t T Q t T " Q" t T " Q t T " . Since GA; B is a distributive lattice, we conclude T " T . 6. By 4. since AB Q u Q and AB Q t Q are crisp. The last lemma gives us the following corollary: Corollary 30. If G is a (Boolean based) Goguen category then G" is a simple (Schr oder) Dedekind category.
Acknowledgements I am grateful to Hitoshi Furusawa, Yasuo Kawahara and Gunther Schmidt for reading, commenting and contributing to this paper. Last but not least, I wish to thank the anonymous referees for their helpful comments.
References [1] P. Freyd, A. Scedrov, Categories, Allegories, North-Holland, Amsterdam, 1990. [2] H. Furusawa, Algebraic formalisations of fuzzy relations and their representation theorems, Ph.D. Thesis, Kyushu University, 1998. [3] J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967) 145±157. [4] B. J onsson, A. Tarski, Boolean algebras with operators, part I, Amer. J. Math. 73 (1951) 891± 939;
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M. Winter / Information Sciences 139 (2001) 233±252 B. J onsson, A. Tarski, Boolean algebras with operators, part II, Amer. J. Math. 74 (1952) 127±162. Y. Kawahara, H. Furusawa, Crispness and representation theorems in Dedekind categories, DOI-TR 143, Kyushu University, 1997. J.P. Olivier, D. Serrato, Categories de Dedekind. Morphismes dans les Categories de Schr oder, C. R. Acad. Sci. Paris 290 (1980) 939±941. J.P. Olivier, D. Serrato, Squares and rectangles in relational categories ± three cases: semilattice, distributive lattice and boolean non-unitary, Fuzzy sets and systems 72 (1995) 167± 178. G. Schmidt, T. Str ohlein, Relationen und Graphen, Springer, 1989 (English version: Relations and Graphs. Discrete Mathematics for Computer Scientists, EATCS Monographs on Theoret. Comput. Sci., Springer, 1993). G. Schmidt, C. Hattensperger, M. Winter, Heterogeneous relation algebras, in: C. Brink, W. Kahl, G. Schmidt (Eds.), Relational Methods in Computer Science, Advances in Computer Science, Springer, Vienna, 1997. L.H. Chin, A. Tarski, Distributive and Modular Laws in the Arithmetic of Relation Algebras, University of California Press, Berkeley, CA, 1951. M. Winter, Strukturtheorie heterogener Relationenalgebren mit Anwendung auf Nichtdetermismus in Programmiersprachen, Dissertationsverlag NG Kopierladen GmbH, M unchen, 1998. L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338±353.