Fuzzy Functions and Their Applications

Home , Fuzzy set

Journal of Mathematical Analysis and Applications 252, 495᎐517Ž. 2000 doi:10.1006rjmaa.2000.7185, available online at http:rrwww.idealibrary.com on

Mustafa Demirci

Akdeniz Uni¨ersity, Department of Mathematics, 07058 Antalya, Turkey E-mail: [email protected] View metadata, citation and similar papers at core.ac.uk brought to you by CORE Submitted by Ulrich Hohle¨ provided by Elsevier - Publisher Connector

Received February 7, 2000

1. INTRODUCTION

The concept of fuzzy equalitywx 7 , which is also called the equality relationwx 10, 14 , the fuzzy equivalence relation wx 12 , the similarity relation wx6, 17, 21 , the indistinguishability operator wx 18, 19 , and the fuzzy functions based on fuzzy equalities, has a significant concern in various fieldsw 1, 2, 7, 8, 14x . The fuzzy functions and fuzzy equalities had been successfully applied in the category theoretical framewx 1, 6, 7, 9 and fuzzy control w 10, 13x . Vagueness on equality of points is mainly associated with Poincare’s´ paradox on physical continuumwx 7, 9, 11 , and it is shown as an example of Poincare’s paradox the ignorance of two very near numbers representing an attribute of a thing, for example, the temperature of weather, the cost of a car, the mass of a particle, etc., as the same number leads us some physical paradoxical situationswx 11 . The concept of fuzzy equality proves a useful mathematical tool to avoid Poincare-like´ paradoxeswx 7 . In this paper, adopting a slightly modified definition of the equality relation instead of that presented inwx 14 , which is nothing but a generalization of the fuzzy equality given inwx 2, 3 by integrals, commutative cl-monoids, and reformulating the definition of fuzzy function inwx 1, 14 under the name of the perfect fuzzy function, the connections between the perfect fuzzy function and the generalized notation of strong fuzzy function introduced inwx 3 are pointed out, and two theorems providing powerful representations for perfect fuzzy functions and strong fuzzy functions are established, and the various properties of these fuzzy functions are investigated. Improving the results presented to apply the fuzzy functions in fuzzy control and approximate reasoning inwx 14 , it will be shown in this direction that perfect fuzzy functions provide better results than those established inwx 14 .

495 0022-247Xr00 $35.00 Copyright ᮊ 2000 by Academic Press All rights of reproduction in any form reserved. 496 MUSTAFA DEMIRCI

2. PRELIMINARIES

Now we first introduce the lattice-theoretic base of our work, quoted fromwx 6, 8 . Let Ž.L, F be a lattice with the bottom element 0 and the top element 1, where conjunction and disjunction operations in L are respectively denoted by n and k. The triple Ž.L, F , ) is called a residuated, commutative l-monoid iff Ž.L, ) is a commutative monoid and there exists a binary operation ª on L such that Ž.AD x) y F z m x F y ª z, ᭙ x, y, z g L. The conditionŽ. AD determines ª , uniquely. The binary operation l on L is defined by

x l y s Ž.Ž.x ª y n y ª x , ᭙ x, y g L.

A residuated, commutative l-monoid Ž.L, F , ) is called a commutative cl-monoid iff Ž.L, F is a complete lattice. A residuated, commutative l-monoid Ž.L, F , ) is called an integral iff the top element 1 of L acts as unit element w.r.t. ); i.e., x)1 s x for all x g L. An integral, residuated, commutative l-monoid Ž.L, F , ) is called divisible iff for each x, y g L with x F y there exists z g L such that x s y) z. An integral, divisible, commutative cl-monoid is said to be a GL-monoid. The following are examples of integral commutative cl-monoidswx 6 : Ž.i the triple Žwx 0, 1 , F , )., where F is the usual order onwx 0, 1 and ) is a left continuous t-norm, Ž.ii a complete Heyting algebra ŽL, F ., where ) is simply the conjunction n in this case, Ž.iii the ideal lattice of a commutative and associative ring R with unit where ) is given the ideal multiplication. Furthermore, for a continuous t-norm ),0,1,ŽwxF , ). is a GL-monoid wx12 , and the operation ª is called the quasi inverse of ), and denoted by )ˆˆ, and the notation ) Ž.x ¬ y is used instead of x ª y wx19 . Now let us collect some useful properties of integral commutative cl-monoids, which will be needed in the sequel.

PROPOSITION 2.1. In any integral commutati¨ecl-monoidŽ. L, F , ) , the following properties are ¨alid: Ž.i x F y « x) z F y) z, ᭙ x, y, z g L, Ž.ii x s 1 ª x, ᭙ x g L, Ž.iii 1 s x ª y m x F y, ᭙ x, y g L, FUZZY FUNCTIONS 497

Ž.iv 1 s x l y m x s y, ᭙ x, y g L, Ž.v x ª y s EÄ4z g L : x) z F y , ᭙ x, y g L, ) ¨ ¨ ¬ g Ž.vi is distributi eo er arbitrary joins; i.e., for each subsetÄ4 yi i I g )E s E ) of L and for each x L, x Ž.ig Iiy ig Ii Ž.x y . Proof. The propertiesŽ. i ᎐ Žiii . are respectively given byw 8, Proposition 2.1 and Lemma 2.2x .Ž. iv is easily gotten from the definition of l and Ž.Ž.iii . v and Ž. vi are presented inwx 8, Lemma 5.1 . For an integral commutative cl-monoid M s Ž.L, F , ) and for a nonempty usual set X, a map E: X = X ª L having the properties Ž.E.1 Ex Ž, x .s 1, Ž.E.2 Ex Ž, y .s Ey Ž, x ., Ž.Ž.Ž.E.3 Ex, y ) Ey, z F Ex Ž., z , for all x, y, z g X is called an M-valued similarity relation on X wx6. If Ž.L, F , ) is particularly a GL-monoid, then an M-valued similarity relation on X is called an equality relationŽ. an L-fuzzy equivalence relation w.r.t. ) on X wx12, 14 . If Ž.L, F , ) is the special GL-monoidŽwx 0, 1 , F , )., where ) is a continuous t-norm, then an equality relation w.r.t. the t-norm ) on X is called an )-indistinguishability operator on X wx19 . For an integral commutative cl-monoid M s Ž.L, F , ) ,an M-valued similarity relation satisfying the following condition is called a global M-valued equality on X wx6: X ŽE.1 .Ž.Ex, y s 1 « x s y, ᭙ x, ᭙ y g X. In this paper, the notations X, Y, Z always stand for nonempty usual sets; the triple Ž.L, F , ) is assumed to be an integral commutative cl-monoid unless otherwise stated. An M-valued similarity relation on X and a global M-valued equality on X will be respectively renamed an equality relation on X w.r.t. ) and a )-fuzzy equality on X for simplicity. The order F and the binary operation ) in the tripleŽwx 0, 1 , F , ). always respectively denote the usual order on real numbers and a continuous t-norm. The main difference between an equality relation on X w.r.t. ) and )-fuzzy equality on X results from the fact that the former is a generalization of an equivalence relation on X in the classical sense, although the latter is a generalization of classical equality relation on X. In other words, the classical sense equality on a nonempty set X can be conceivable as a N = ª map EX : X X Ä40, 1 , defined by 1 : x s y ExN Ž., y s , X ½50 : x / y 498 MUSTAFA DEMIRCI where 0 and 1 respectively stand for the bottom and the top elements of the lattice L, and if an equality relation E w.r.t. ) on X Ž)-fuzzy equality E on X .Ž.is restricted to be a crisp relation on X, i.e., EX= X : Ä40, 1 , then it will exactly correspond to an equivalence relation on X in the classical senseŽ the equality relation on X in the classical sense, i.e., s N ) E EX .. Therefore, for a given equality relation E w.r.t. on X and for a given )-fuzzy equality F on X, it is convenient to say that ExŽ., y shows the similarity or the equivalency degree of x and y, and FxŽ., y represents the grade for which x equals to y. An L-fuzzy set of X is a mapping ␮: X ª L, and the set of all L-fuzzy sets of X is denoted by LX. For L s wx0, 1 , we will call an L-fuzzy set of X a fuzzy set of X. The notation ␮c will represent the complement of a fuzzy set ␮ of X in Zadeh’s sensewx 15, 20 . For a given function f: X ª Y and for a given L-fuzzy set ␪ g LX, the image of ␪ under f will be given by

f Ž.Ž.␪ y s E ␪ Ž.x xgfyy1Ž. as an analogy of the corresponding definition inwx 15 . Now, let us recall some concepts introduced inwx 12, 14 , where Ž.L, F , ) is assumed as a GL-monoid. For a given equality relation E w.r.t. ) on X, an L-fuzzy set ␮ g LX is said to be extensional w.r.t. E on X if the inequality ␮Ž.x ) Ex Ž, y .F ␮ Ž.y is fulfilled for all x, y g X. The L-fuzzy set ␮ˆ s HÄ␯ ¬ ␮ F ␯ and ␯ is extensional w.r.t. E4 is called the extensional hull of ␮ w.r.t. E. The extensional hull of an L-fuzzy set ␮ g LX, the extensional hull of a crisp subsetŽ. an ordinary subset M of X, and the extensional hull of an element u of X w.r.t. E are respectively formulated by

␮ˆŽ.x s EEÄ4␮ Ž.z ) Ez Ž, x ., Mxˆ Ž.s Ex Ž, m ., and zgXmgM Ž.1 uxˆŽ.s Eu Ž, x . for all x g X. For a given equality relation E w.r.t. on X,an L-fuzzy set ␮ g LX is extensional w.r.t. E on X iff the inequality ExŽ., y F Ž.␮ Ž.Ž.x l ␮ y Ž.2 is satisfied for all x, y g X. It is obvious that this equivalency is also valid if a )-fuzzy equality on X is taken instead of an equality relation E w.r.t. ) on X. The concepts of an extensional L-fuzzy set, an extensional hull of an L-fuzzy set, and the resultsŽ. 1 and Ž. 2 are obviously also valid for an FUZZY FUNCTIONS 499 integral commutative cl-monoid Ž.L, F , ) instead of a GL-monoid ŽL, F , ).Ž., since the divisibility condition in the GL-monoid L, F , ) has no effect on them. For this reason, we will also use all of these concepts in an integral commutative cl-monoid Ž.L, F , ) .

3. )-FUZZY EQUALITIES AND FUZZY FUNCTIONS

In this section, it is first shown that all results established inwx 3, Sect. 3 can be restated in terms of )-fuzzy equalities by adding slight changes on the hypotheses of these results.

THEOREM 3.1. LetŽ. L, F , ) be a GL-monoid, and let F : LX be a set of L-fuzzy sets of X such that, for all x, y g X, x / y, there exists ␮ g F satisfying the inequality ŽŽ.␮ x l ␮ Ž..y / 1, or equi¨alently the inequality ␮ / ␮ = ª ¨ Ž.x Ž.y . Then the map EF : X X L, gi en by

s ␮ l ␮ ExF Ž, y .H Ž. Ž.x Ž.y , ␮gF is the greatest )-fuzzy equality on X such that all L-fuzzy sets in F are extensional w.r.t. EF . X Proof. We only need to prove that EF satisfiesŽ E.1. since the other assertions follow fromwx 14, Theorem 1 and the equivalencyŽ. 2 . Indeed, for g / s x, y X, x y,if ExF Ž., y 1, then by considering the definition of EF , we obviously have ␮Ž.x l ␮ Ž.y s 1 for all ␮ g F. This contradicts with X the assumption on F, and soŽ E.1. follows. Furthermore, by Proposition 2.1Ž. iv , the inequality Ž␮ Ž.x l ␮ Ž..y / 1 is equivalent to the inequality ␮Ž.x / ␮ Ž.y . In a similar fashion to Theorem 3.1, adding the assumption ‘‘for each g / g ␮ / ␮ x, y X, x y, there exists k I such that kkŽ.x Ž.y ’’ to the hypotheses ofwx 14, Theorem 2, Corollary 1, Theorem 3, and Corollary 2 , it is easily seen that all of these results are also valid for )-fuzzy equalities on X instead of equality relations w.r.t. ) on X.

DEFINITION 3.2. Let E and F be two equality relations w.r.t. ) on X and Y, respectively. Ž.i A fuzzy relation ␳ g LX=Y is called extensional w.r.t. E iff the inequality X X Ž.Ž.ŽEX.1 ␳ x, y ) Ex, x .ŽF ␳ x , y. X is satisfied for all x, x g X and for all y g Y. 500 MUSTAFA DEMIRCI

Ž.ii A fuzzy relation ␳ g LX=Y is called extensional w.r.t. F iff the inequality X X Ž.Ž.ŽEX.2 ␳ x, y ) Fy, y .ŽF ␳ x, y . X is satisfied for all x g X and for all y, y g Y. Ž.iii A fuzzy relation ␳ g LX=Y is called extensional w.r.t. E and F iff ␳ satisfies bothŽ.Ž. EX.1 and EX.2 . Remark 3.3. For the given equality relations E on X and F on Y w.r.t. ) = = = ª , the map GX=Y : Ž.Ž.X Y X Y L, given by s ) GxX=Y Ž.Ž.Ž.Ž.Ž.11, y , x 22, y Ex 12, x Fy 12, y , is obviously an equality relation on X = Y w.r.t. ), and a fuzzy relation ␳ g LX=Y is extensional w.r.t. E and F iff ␳ is an extensional L-fuzzy set = of X Y w.r.t. GX=Y .

DEFINITION 3.4. Let E and F be two equality relations w.r.t. ) on X and Y, respectively, and let a fuzzy relation ␳ g LX=Y be extensional w.r.t. E and F. Then Ž.i ␳ is called a partial fuzzy function if X X Ž.PF ␳ Žx, y .) ␳ Žx, y .ŽF Fy, y . X is satisfied for all x g X and for all y, y g Y. Ž.ii A fuzzy relation ␳ g LX=Y is said to be fully defined if ␳ fulfills E ␳ s g the condition y g Y Ž.x, y 1 for all x X. Ž.iii A fully defined partial fuzzy function is called a fuzzy function. The conditions in Definitions 3.2 and 3.4 were first established inwx 1 via a different terminology. Later on, Klawonnwx 14 introduced these definitions by taking Ž.L, F , ) as a GL-monoid. The only difference between our definitions and the corresponding definitions inwx 14 is the selection of an integral commutative cl-monoid Ž.L, F , ) instead of a GL-monoid. It should be noted that if the equality relations E on X, F on Y and the fuzzy function ␳ in Definition 3.4 are respectively restricted to the crisp case, i.e., EXŽ.= X : Ä40, 1 , FYŽ.= Y : Ä40, 1 , and ␳Ž.X = Y : Ä40, 1 , then since E and F correspond to exactly two equivalence relations but / N not to classical equalities on X and Y, respectively, i.e., E EX and / N F FY in general, the fuzzy function is not really a generalization of function in classical sense. In the following definitions, two sorts of fuzzy functions satisfying the requirement that they are generalizations of classical functions by means of integral commutative cl-monoids are introduced, and it is shown that these functions have powerful representations by classical functions. FUZZY FUNCTIONS 501

DEFINITION 3.5. Let E and F be two )-fuzzy equalities on X and Y, respectively. A partial fuzzy function ␳ g LX=Y is called a perfect fuzzy function w.r.t. E and F if ␳ satisfies the condition Ž.F.1 For each x g X, ᭚ y g Y such that ␳ Žx, y .s 1. It is obvious that a perfect fuzzy function is also a fuzzy function in the sense of Definition 3.4Ž. iii .

DEFINITION 3.6. Let E and F be two )-fuzzy equalities on X and Y, respectively. A fuzzy relation ␳ g LX=Y is called a strong fuzzy function w.r.t. E and F,if ␳ satisfies the conditionŽ. F.1 and the condition XX X X Ž.F.2 ␳ Žx, y .) ␳ Žx , y .Ž) Ex, x .ŽF Fy, y . X X for all x, x g X and for all y, y g Y. A strong fuzzy function w.r.t.wŽ 0, 1x , F , n . is exactly the strong fuzzy function in the sense ofwx 3 .

DEFINITION 3.7. Let E and F be two equality relations w.r.t. ) on X and Y, respectively, and let f: X ª Y be a function in the ordinary sense. f is called extensional w.r.t. E and F if X X ExŽ., x F FfxŽ. Ž.Ž., fx X holds for all x, x g X. If Ž.L, F , ) is particularly taken as a GL-monoid, then Definition 3.7 will be nothing butwx 14, Definition 9 .

THEOREM 3.8. Let E and F be two )-fuzzy equalities on X and Y, respectiëly. For a giën ordinary function f: X ª Y extensional w.r.t. E and F, a fuzzy relation ␳ g LX=Y, which satisfies the conditions ␳ Ž.x, fxŽ.s 1 and ␳ Žx, y .F FfxŽ. Ž., y , ᭙ x g X, ᭙ y g Y, Ž.3 is a strong fuzzy function w.r.t. E and F. Conërsely, for a giën strong fuzzy function ␳ g LX=Y w.r.t. E and F, there exists a unique ordinary function f: X ª Y extensional w.r.t. E and F satisfying the conditions Ž.3. Proof. For a given ordinary function f: X ª Y extensional w.r.t. E and F, let a fuzzy relation ␳ g LX=Y satisfy the conditionsŽ. 3 . It is obvious that ␳ has the propertyŽ. F.1 . By using the extensionality of f w.r.t. E and F and taking into considerationŽ. 3 , we may write X X X ␳ Ž.Žx, y ) ␳ x , y .Ž) Ex, x . X X X F FfxŽ.Ž.Ž.Ž., y ) Ffx Ž., y ) Ffx Ž.Ž., fx X X F FfxŽ.Ž.Ž., y ) Ffx Ž., y F Fy Ž, y . 502 MUSTAFA DEMIRCI

XX for all x, x g X and for all y, y g Y; i.e., ␳ satisfiesŽ. F.2 , and so ␳ is a strong fuzzy function w.r.t. E and F. Conversely, for a given strong fuzzy function ␳ g LX=Y w.r.t. E and F, let us define the subset F s ÄŽ.Ž.x, y ¬ ␳ x, y s 14 of X = Y. ByŽ. F.1 , for X each x g X, ᭚ y g Y such that Ž.x, y g F. For Ž.Žx, y , x, y . g F, exploiting the definition of F and applying the conditionŽ. F.2 we obtain X X X ␳ Ž.Ž.x, y ) ␳ x, y s 1 F Fy Ž, y ., i.e., y s y . Thus F is a graph of a classical function f: X ª Y where, for each x g X, the unique y g Y satisfying Ž.x, y g F is denoted by fx Ž.. On the other hand, byŽ. F.2 , and by the definition of f, we observe that

X X X X X ExŽ., x s ␳ Ž.Ž.x, fx Ž.) ␳ x , fx Ž.Ž.) Ex, x F Ffx Ž Ž.Ž., fx ., X ᭙ x, x g X ; i.e., f is an extensional function w.r.t. E and F. To confirm the uniqueness of the extensional function f: X ª Y w.r.t. E and F satisfyingŽ. 3 , let us consider a function g: X ª Y possessing the propertyŽ. 3 . Then, for each x g X, we have ␳ Ž.x, gxŽ.s 1, i.e., Ž.x, gxŽ.g F, i.e., gxŽ.s fx Ž., i.e., f s g.

PROPOSITION 3.9. Let E and F be two )-fuzzy equalities on X and Y, respecti¨ely. Then Ž.i A fuzzy relation ␳ g LX=Y extensional w.r.t. E is a strong fuzzy function, if ␳ holds the conditions Ž.F.1 and Ž.PF . Ž.ii A perfect fuzzy function is a strong fuzzy function. Proof. Noticing thatŽ. EX.1 together with Ž. PF implies the condition Ž.Ž.F.2 , i is easily acquired. Ž. ii follows from Ž. i and Definition 3.5 at once.

THEOREM 3.10. Let E and F be two )-fuzzy equalities on X and Y, respectiëly. For a giën ordinary function f: X ª Y extensional w.r.t. E and F, a fuzzy relation ␳ g LX=Y, defined by the formula ␳ Ž.x, y s FfxŽ. Ž., y , ᭙ x g X, ᭙ y g Y,4Ž. is a perfect fuzzy function w.r.t. E and F. Conërsely, for a giën perfect fuzzy function ␳ g LX=Y w.r.t. E and F, there exists a unique ordinary function f: X ª Y fulfilling the equality Ž.4. Proof. To verify the first part of theorem, for a given ordinary function f: X ª Y extensional w.r.t. E and F, let us consider the fuzzy relation ␳ g LX=Y expressed by the equalityŽ. 4 . From Theorem 3.8, ␳ is obviously a strong fuzzy function w.r.t. E and F; i.e., the propertiesŽ. PF and Ž. F.1 FUZZY FUNCTIONS 503 are satisfied. Thus, it is adequate to show that ␳ is extensional w.r.t. E and F. Using the fact that f: X ª Y is an extensional function w.r.t. E and F, and considering the equalityŽ. 4 , we see that X ␳ Žx, y .) Ex Ž, x .F FfxŽ.Ž. Ž., y ) Ffx Ž., fx Ž. X X X F FfxŽ.Ž., y s ␳ Žx , y ., ᭙ x, ᭙ x g X, ᭙ y g Y and X X ␳ Ž.Ž.x, y ) Fy, y s FfxŽ. Ž.Ž., y ) Fy, y X X X F FfxŽ.Ž., y s ␳ Žx, y ., ᭙ x g X, ᭙ y, ᭙ y g Y. Now let us ratify the converse part of the theorem. For a given perfect fuzzy function ␳ w.r.t. E and F, by Proposition 3.9Ž. ii and Theorem 3.8, there exists a unique ordinary function f: X ª Y extensional w.r.t. E and F satisfyingŽ. 3 . Thus, for all x g X and for all y g Y, using Ž EX.2 . and Ž. 3 we may write FfxŽ.Ž.Ž.Ž., y s ␳ x, fxŽ.) Ffx Ž., y F ␳ Žx, y ., and so ␳Ž.x, y s Ffx ŽŽ.., y follows from Ž. 3 . In the first part of Theorem 3.10, by taking equality relations E and F w.r.t. ) instead of the )-fuzzy equalities E and F and using the notation ␳ ␳ F ) f instead of the fuzzy relation formulated byŽ. 4 , where ŽL, , .is wx ␳ assumed as a GL-monoid, it is proven in 14 that f is a fuzzy function and ␳ s ) f Ž.x, y E Ex Ž., z FfzŽ. Ž., y . zgX By consideringwx 14, Proof of Lemma 2 , it is obvious that this equality is also valid in any integral commutative cl-monoid Ž.L, F , ) .

COROLLARY 3.11. For gi¨en )-fuzzy equalities E on X and F on Y, if a strong fuzzy function ␳ g LX=Y w.r.t. E and F is an extensional fuzzy relation w.r.t. F, then ␳ is a perfect fuzzy function w.r.t. E and F. Proof. For given )-fuzzy equalities E on X and F on Y, suppose that a strong fuzzy function ␳ g LX=Y w.r.t. E and F is an extensional fuzzy relation w.r.t. F. Then, by Theorem 3.8, there exists a unique ordinary function f: X ª Y extensional w.r.t. E and F fulfillingŽ. 3 . In a similar fashion to that in converse part of Theorem 3.10,Ž.Ž. EX.2 and 3 imply the equality ␳Ž.x, y s Ffx ŽŽ.., y for all x g X and for all y g Y. Hence, Theorem 3.10 directly justifies that ␳ is a perfect fuzzy function w.r.t. E and F. 504 MUSTAFA DEMIRCI

DEFINITION 3.12. Let E and F be two )-fuzzy equalities on X and Y, respectively. Ž.i For a given function f: X ª Y extensional w.r.t. E and F, the perfect fuzzy function ␳ g LX=Y w.r.t. E and F, given by the formulaŽ. 4 , is called an E y F vague description of f, and it is denoted by vagŽ.f . Ž.ii For a given perfectŽ resp., a strong. fuzzy function ␳ g LX=Y w.r.t. E and F, the ordinary function f: X ª Y extensional w.r.t. E and F satisfying the equalityŽ.Ž 4 resp., the property Ž.. 3 is called an ordinary description of ␳, and it is denoted by ordŽ.␳ . Considering Theorems 3.8 and 3.10 one can easily perceive that, for a given function f: X ª Y extensional w.r.t. E and F and for a perfect fuzzy function ␳ g LX=Y w.r.t. E and F,

f s ordŽ. vagŽ.f and ␳ s vagŽ. ordŽ.␳ , although, for a strong fuzzy function ␺ g LX=Y w.r.t. E and F, only the inequality ␺ F vagŽŽ.. ord ␺ is true. Furthermore

␳ Ž.x, y s E Ex Ž., z ) FŽ.ord Ž.Ž.␳ z , y . zgX

4. PROPERTIES OF FUZZY FUNCTIONS

In this section, various rudimentary properties of perfect fuzzy functions and strong fuzzy functions will be introduced. We condense only the perfect fuzzy functions and strong fuzzy functions and do not pay attention to fuzzy functions due to the fact that these sorts of fuzzy functions have more desirable properties than fuzzy functions. = = DEFINITION 4.1wx 4 . Let ␳ g LX YYand ␺ g L Z. Then Ž.i The k-) composition ␺ ( ␳ of ␺ and ␳ is a fuzzy relation ␳ (␺ g LX=Z, defined by

Ž␺ ( ␳ .Ž.x, z s E ␳ Ž.Ž.x, y )␺ y, z , ᭙ x g X, ᭙z g Z, ygY

Ž.ii The inverse ␳y1 g LY=X of ␳ is given by

␳y1 Ž.y, x s ␳ Ž.x, y , ᭙ x g X, ᭙ y g Y.

DEFINITION 4.2. Let E and F be two )-fuzzy equalities on X and Y, respectively. A perfect fuzzy function ␳ g LX=Y w.r.t. E and F is said to FUZZY FUNCTIONS 505 be injective iff ␳ holds: X X X Ž.Ž.ŽIPF ␳ x, y ) ␳ x , y.ŽF Ex, x ., ᭙ x, ᭙ x g X, ᭙ y g Y. A strong fuzzy function ␳ g LX=Y w.r.t. E and F is said to be injective iff ␳ holds: XX X X XX Ž.ISF ␳ Žx, y .Ž) ␳ x , y .Ž) Fy, y .ŽF Ex, x ., ᭙ x, ᭙ x g X, ᭙ y, ᭙ y g Y. A perfectŽ. or strong fuzzy function ␳ g LX=Y w.r.t. E and F is surjective if ␳ holds: Ž.SF ᭙ y g Y, ᭚ x g X such that ␳ Žx, y .s 1. A perfectŽ. or strong fuzzy function is bijective if it is both surjective and injective.

PROPOSITION 4.3. Let E and F be two )-fuzzy equalities on X and Y, respecti¨ely. Then ␳y1 g LY=X is a perfectŽ. resp., strong fuzzy function w.r.t. F and E iff ␳ is bijecti¨e perfectŽ. resp., strong fuzzy function w.r.t. E and F.

PROPOSITION 4.4. For )-fuzzy equalities E on X and F on Y, if ␳ is a bijecti¨e perfect fuzzy function w.r.t. E and F, then

␳y1 ( ␳ s E and ␳ ( ␳y1 s F.

Proof. By the injectivity of ␳ and by Definition 4.1Ž. i we may write

X X X X ␳ Ž.x, y ) ␳y1 Žy, x .s ␳ Ž.Ž.x, y ) ␳ x , y F Ex Ž, x ., ᭙ x, ᭙ x g X, X U ᭙ y g Y, i.e., Ž.␳y1 ( ␳ Ž.x, x s E ␳ Ž.x, y ) ␳y1 Žy, x . ygY X X F ExŽ., x , ᭙ x, ᭙ x g X.5Ž. On the other hand, applyingŽ. EX.1 for ␳ and using Definition 3.12 we observe that

X X X X X ExŽ., x s Ex Ž., x ) ␳ Ž.Ž.x , ord Ž.Ž.␳ x F ␳ x, ord Ž.Ž.␳ x X X X s ␳ Ž.Ž.x, ordŽ.Ž.␳ x ) ␳ x , ord Ž.Ž.␳ x X X X s ␳ Ž.Ž.x, ordŽ.Ž.␳ x ) ␳y1 ord Ž.Ž.␳ x , x X X F E ␳ Ž.x, y ) ␳y1 Žy, x .s Ž.␳y1 ( ␳ Žx, x ..6 Ž. ygY

FromŽ. 5 and Ž. 6 the equality Ž␳y1 ( ␳. s E follows. By Proposition 4.3, since ␳y1 g LY=X is a perfect fuzzy function w.r.t. F and E, and since we have Ž ␳y1 .y1 s ␳, the equality ␳ ( ␳y1 s F is directly obtained from the 506 MUSTAFA DEMIRCI

equality Ž ␳y1 ( ␳. s E by simply substituting ␳ and E with ␳y1 and F, respectively.

PROPOSITION 4.5. LetŽ. L, F , ) be the particular GL-monoid Žwx0, 1 , F , )., and let E, F, and G be )-fuzzy equalities on X, Y, and Z, ¨ ␳ g wxX=Y ␳ g wxY=Z respecti ely. If the fuzzy relations 120, 1 , 0, 1 are both perfectŽ. resp., strong fuzzy functions w.r.t. E, F, and G, then the k-) ␳ ( ␳ ␳ ␳ composition 21of 1and 2is also a perfectŽ. resp., strong fuzzy function w.r.t. E and G. ␳ g wxX=Y ␳ g wxY=Z Proof. If the fuzzy relations 120, 1 , 0, 1 are strong fuzzy functions w.r.t. E, F, and G, then, in a similar fashion tow 3, x ␳ ( ␳ Proposition 2.1 , it can be easily seen that 21is a strong fuzzy function ␳ ( ␳ w.r.t. E and G. We now only show that 21is a perfect fuzzy function ␳ g wxX=Y ␳ g wxY=Z w.r.t. E and G under the assumption that 120, 1 and 0, 1 ␳ g wxX=Y are perfect fuzzy functions w.r.t. E, F, and G. Suppose that 1 0, 1 , ␳ g wxY=Z ␳ ( ␳ 2 0, 1 are perfect fuzzy functions w.r.t. E, F, and G; i.e., 21is a strong fuzzy function w.r.t. E and G. By Corollary 3.11, for the verifica- ␳ ( ␳ tion of the required result, it is adequate to see that 21is an ␳ extensional fuzzy relation w.r.t. G. Now using the conditionŽ. EX.2 for 2 we may write

X X Ž␳ ( ␳ .Ž.Žx, z )Gz, z .s E ␳ Ž.x, y ) ␳ Ž.y, z )Gz Ž, z . 21 ž/1 2 ygY s ␳ ) ␳ ) X E 12Ž.x, y Ž.Žy, z Gz, z . ygY F ␳ ) ␳ X E 12Ž.x, y Žy, z . ygY s ␳ ( ␳ X ᭙ g ᭙ ᭙ X g Ž.Ž.21x, z , x X, z, z Z; ␳ ( ␳ i.e., 21is an extensional fuzzy relation w.r.t. G.

DEFINITION 4.6wx 4 . The image of an L-fuzzy set ␮ of X under a fuzzy relation ␳ g LX=Y is an L-fuzzy set of Y, defined by

␳␮wxŽ.y s E Ä4␮ Ž.x ) ␳ Žx, y ., ᭙ y g Y. xgX

PROPOSITION 4.7. For )-fuzzy equalities E and X and F on Y, let ␳ g LX=Y be a perfectŽ. or strong fuzzy function w.r.t. E and F. Then Ž.i For all ␮ g LX, ␮ F ␳␳␮y1ww xx. Ž.ii If ␳ is surjecti¨e, then, for all ␯ g LY, ␯ F ␳␳w y1wxx ␯ . FUZZY FUNCTIONS 507

Proof. Ž.i For ␮ g LX, by Definition 4.6, we may write

␳␳␮y1 wxŽ.x s EEÄ4␮Ž.u ) ␳ Žu, y .) ␳ Žx, y . g ½5ž/g y YuX s EEÄ4␮Ž.u ) ␳ Žu, y .) ␳ Žx, y ., ᭙ x g X.7Ž. ygYugX g ᭚ g ␳ s By the conditionŽ. F.1 , for each x X, y00Y such that Ž.x, y 1. ThereforeŽ. 7 directly yields

␮ s ␮ ) ␳ ) ␳ F ␳␳␮y1 wx Ž.x Ž.x Žx, y00 . Žx, y . Ž.x . Ž.ii This is easy. = PROPOSITION 4.8. LetŽ.Ž L, F , ) be wx0, 1 , F , )., and let ␳ g wx0, 1 X Y be a perfectŽ. or strong fuzzy function w.r.t. a )-fuzzy equality E on X and a )-fuzzy equality F on Y. Then Ž.i If ␳ is injecti¨e and ␮ is an extensional fuzzy set of X w.r.t. E, then ␳␳␮y1ww xxF ␮. Ž.ii If ␯ is an extensional fuzzy set of Y w.r.t. F, then ␳␳w y1wxx ␯ F ␯. Proof. Ž.i Let ␳ g wx0, 1X=Y be an injective perfectŽ. or strong fuzzy function w.r.t. E and F, and let ␮ be an extensional fuzzy set of X w.r.t. E. For an arbitrarily fixed x g X, letting ␣ s ␳␳␮y1ww xxŽ.x , our aim is to confirm that ␣ F ␮Ž.x . We may assume ␣ ) 0 since the other case is trivial. The equalityŽ. 7 gives that, for an arbitrarily chosen ␧ ) 0, ᭚u Ž␧ . g X and ᭚ yŽ.␧ g Y such that ␮Ž.ŽuŽ.␧ ) ␳ u Ž.␧ , y Ž.␧ .Ž) ␳ x, y Ž.␧ .) ␣ y ␧ . Therefore, since ␳ is injective and ␮ is an extensional fuzzy set of X w.r.t. E, we may write ␣ y ␧ - ␮Ž.ŽuŽ.␧ ) ␳ u Ž.␧ , y Ž.␧ .Ž) ␳ x, y Ž.␧ . F ␮Ž.ŽuŽ.␧ ) Eu Ž.␧ , x . F ␮Ž.x . Since the selection of ␧ is arbitrary, the inequality ␣ y ␧ - ␮Ž.x gives the required inequality. The proof ofŽ. ii is similar to Ž. i .

PROPOSITION 4.9. LetŽ.Ž L, F , ) be wx0, 1 , F , n .. For n-fuzzy equalities E on X and F on Y, let ␳ g wx0, 1 X=Y be a perfectŽ. or strong fuzzy function w.r.t. E and F. 508 MUSTAFA DEMIRCI

Ž.i If ␳ is injecti¨e, then, for all fuzzy sets ␮ of X possessing the property that ␮ and ␮c are both extensional w.r.t. E, ␳␮w c x F Ž ␳␮wx.c. Ž.ii For all fuzzy sets ¨ of Y ha¨ing the property that ¨ and ¨ c are both extensional w.r.t. F, ␳y1w¨ c x F Ž ␳␯y1wx.c. ␮ ¬ g ␮ Ž.iii For a family Ä4iii I of fuzzy sets of X, let each be F ␳␮ F ␳ F ␮ extensional w.r.t. E. Then ig IiŽ. Ž ig Ii .. ␯ ¬ g ¨ Ž.iv For a family Ä4iii I of fuzzy sets of Y, let each be extensional F ␳y1 ¨ F ␳y1 F ¨ w.r.t. F. Then ig IiŽ. Ž. ig Ii. Proof. Ž.i Let a fuzzy set ␮ of X have the property that ␮ and ␮c are both extensional w.r.t. E, and let ␳ be an injective perfectŽ. or strong fuzzy function w.r.t. E and F. For an arbitrarily fixed y g Y, using the notations ␣ s ␳␮w c xŽ.y s E Ä4␮c Ž.x n ␳ Žx, y . and xgX ␤ s ␳␮wxŽ.y s E Ä4␮ Ž.x n ␳ Žx, y ., xgX one can easily notice that the inequality ␳␮w c xŽ.y F Ž␳␮wx.cŽ.y is equivalent to ␣ q ␤ F 1.Ž. 8 We may suppose that ␣ ) 0 and ␤ ) 0 sinceŽ. 8 is obvious otherwise. By the definition of ␣ and ␤, for an arbitrarily selected ␧ such that 0 - ␧ F ␣ n ␤ ᭚ ␧ g ᭚ ␧ g , x12Ž.X and x Ž. X such that ␮c ␧ n ␳ ␧ ) ␣ y ␧ Ž.Žx11Ž.x Ž., y . and 9 ␮ ␧ n ␳ ␧ ) ␤ y ␧ Ž. Ž.Žx22Ž.x Ž., y . . Now let us condense the cases ␣ y ␧ G ␤ y ␧ and ␣ y ␧ - ␤ y ␧, separately. Let us first suppose that ␣ y ␧ G ␤ y ␧. By the first inequality inŽ. 9 , we have ␳ ␧ ) ␣ y ␧ G ␤ y ␧ Ž.x1Ž., y .1Ž.0 Therefore the second inequality inŽ. 9 and the inequalityŽ. 10 together yield ␮ ␧ n ␳ ␧ n ␳ ␧ ) ␤ y ␧ Ž.Žx22Ž.x Ž., y .Žx 1 Ž., y . . Now applying the injectivity of ␳ and using the extensionality of ␮ w.r.t. E in the last inequality we secure ␤ y ␧ - ␮ ␧ n ␳ ␧ n ␳ ␧ Ž.Žx22Ž.x Ž., y .Žx 1 Ž., y . F ␮ ␧ n ␧ ␧ Ž.Žx221Ž.Ex Ž., x Ž. . F ␮ ␧ Ž.x1Ž..1Ž.1 FUZZY FUNCTIONS 509

By considering the first inequality inŽ. 9 and using Ž 11 . we find ␣ y ␧ q ␤ y ␧ - ␮c ␧ n ␳ ␧ q ␮ ␧ Ž . Ž .Ž.Ž.Žx11 Ž.x Ž., y .Ž.x 1 Ž. F ␮c ␧ q ␮ ␧ s Ž.Ž.x11Ž.x Ž. 1. Now let us assume that ␣ y ␧ - ␤ y ␧. From the second inequality in Ž.9 it is clear that ␳ ␧ ) ␤ y ␧ ) ␣ y ␧ Ž.x2 Ž., y , and so the inequality ␮c ␧ n ␳ ␧ n ␳ ␧ ) ␣ y ␧ Ž.Žx11Ž.x Ž., y .Žx 2 Ž., y . follows from the second inequality inŽ. 9 . Similarly, using the injectivity of ␳ and the extensionality of ␮ w.r.t. E it is easily seen that ␣ y ␧ - ␮c ␧ n ␳ ␧ n ␳ ␧ Ž.Žx11Ž.x Ž., y .Žx 2 Ž., y . F ␮c ␧ n ␧ ␧ Ž.Žx121Ž.Ex Ž., x Ž. . F ␮c ␧ Ž.x2 Ž..1Ž.2 Therefore the second inequality inŽ. 9 and Ž 12 . together yields ␣ y ␧ q ␤ y ␧ - ␮ ␧ n ␳ ␧ q ␮ x ␧ Ž . Ž .Ž.Ž.Žx22 Ž.x Ž., y . Ž.x 2Ž. F ␮ ␧ q ␮c ␧ s Ž.Ž.x22Ž.x Ž. 1. Hence the inequality ␣ q ␤ F 1 is obtained from the inequality Ž.␣ y ␧ q Ž.␤ y ␧ - 1 and the choice of ␧. Ž.ii can be easily verified in a similar way to that in Ž. i . g ␮ g wxX Ž.iii For each i I, let i 0, 1 be extensional w.r.t. E on X. Let us assume that ␳ is an injective perfectŽ. or strong fuzzy function w.r.t. g F ␳␮ ␥ E and F. For y Y, denoting ig IiŽ.Ž.y by our aim is to confirm ␳ F ␮ G ␥ F ␳␮ that Žig Ii .Ž.y . By the definition of ig IiŽ .Ž.y , for each g ␧ ) ᭚ ␧ g i I, and for each arbitrarily chosen 0, xiŽ. X such that ␮ ␧ n ␳ ␧ ) ␥ y ␧ iiŽ.Žx Ž.xi Ž., y . .1Ž.3 FromŽ. 13 we may write ␮ ␧ n ␳ ␧ n ␳ ␧ ) ␥ y ␧ ᭙ ᭙ g iiŽ.Žx Ž.xi Ž., y .Ž.x j Ž., y , i, j I. ␳ ␮ Now utilizing the injectivity of and the extensionality i w.r.t. E we observe that ␥ y ␧ - ␮ ␧ n ␳ ␧ n ␳ ␧ iiŽ.Žx Ž.xi Ž., y .Ž.x j Ž., y F ␮ ␧ n ␧ ␧ iiŽ.x Ž.ExŽ.i Ž., x j Ž. F ␮ ␧ ᭙ ᭙ g ijŽ.x Ž., i, j I.14Ž. 510 MUSTAFA DEMIRCI

␳ ␧ ) ␥ y ␧ g Furthermore the inequalityŽ. 13 gives ŽŽ..x j , y for all j I. Thus consideringŽ. 14 we acquire ␮ ␧ n ␳ ␧ ) ␥ y ␧ ᭙ ᭙ g ijŽ.Žx Ž.x j Ž., y . , i, j I.15Ž. g ␳ F ␮ For an arbitrarily fixed k I, by the definition of Ž.ig Iiand by using Ž.15 , we see that

␳␮ s ␮ n ␳ FEFiiŽ.y Ž.x Žx, y . ž/g g ½5 ž/g i IxXiI

s EH␮ x n ␳ x, y ½5i Ž. Ž . xgXigI G ␮ ␧ n ␳ ␧ H Ä4iiŽ.Žx Ž.xk Ž., y . igI

G ␥ y ␧ ; i.e., ␳␮F Ž.y G ␥ . ž/i igI In a similar fashion toŽ.Ž. iii , iv can be easily deduced. The converse inclusions in Proposition 4.9Ž.Ž ii ᎐ iv . are also valid for a perfectŽ. or strong fuzzy function ␳ without the need for any assertions on ␯ g wxY ␮ ¬ g : wxX ␯ ¬ g : wxY w 0, 1 , Ä4iii I 0, 1 and Ä4i I 0, 1Ž see 3, Proposi- tions 2.6Ž. c and 2.7 Ž c, d .x.Ž . Furthermore, for a surjective perfect or strong. fuzzy function ␳ and for all ␮ g wx0, 1X , Ž ␳␮wx.c F ␳␮w c x is trueŽ seew 3, Proposition 2.6Ž. bx. .

5. APPLICATIONS OF FUZZY FUNCTIONS IN FUZZY CONTROL

In approximate reasoning and in fuzzy controlwx 4, 16, 22 , a collection of if-then rules of the form ␰ ␩ g if is Aii, then is BiŽ.I Ž.16 is extensively considered where ␰ and ␩ stand for variables in X and Y, ␮ g X and the linguistic terms Aiiand B are modelled by L-fuzzy sets iL , ␯ g Y i L , respectively. The collection of if-then rulesŽ. 16 is described by the system of fuzzy relational equations ␳␮wxs ␯ g iiŽ.i I ,17Ž. ␮ g X ␯ g Y where the L-fuzzy sets iiL , L expressing the linguistic terms Ai and Bi are known and solution of the system in the form of a fuzzy FUZZY FUNCTIONS 511

relation ␳ g LX=Y has to be determined. After the construction of a solution ␳ to the systemŽ. 17 of fuzzy relational equations, for a given ␮ g X inputŽ. antecedent L-fuzzy set input L corresponding to an input ␯ g Y linguistic term Ainput, the output fuzzy set output L representing output ␳␮wxwx linguistic term Boutput is given by input 4, 14, 16 . It is well known that if there exists a solution ofŽ. 17 , then the fuzzy ␳ g X=Y relation U L , defined by

␳ s ␮ ª ␯ ᭙ g ᭙ g UiŽ.x, y H Ž. Ž.Ž.x iy , x X, y Y, igI

is the greatest solution ofŽ. 17wx 4, 5 . In this section, the solutions ofŽ. 17 from the point of view of fuzzy functions will be investigated, and more desirable results than those established inwx 14 will also be stated.

THEOREM 5.1. Let E and F be equality relations w.r.t. ) on X and Y, ¨ ␪ ¬ g ª respecti ely, let Ä4i i I be a family of L-fuzzy sets of X, and let f: X Y ␮ g X ␯ g Y be an extensional function w.r.t. E and F. Let iiL , L be exten- ␪ ␪ ¨ g sional hulls of iiand fŽ.w.r.t. E and F, respecti elyŽ. i I . Then the fuzzy ␳ ␳ function fUis a solution of Ž.17 , i.e., is the greatest solution of Ž.17 . Proof. For all y g Y and for all i g I, we may write

␳␮wx s ␪ ) ) ␳ fiŽ.y EEiŽ.z Ez Ž, x .f Žx, y . g ½5ž/g x XzX

s ␪ ) ) EEi Ž.z Ez Ž, x .FfxŽ. Ž., y g ½5ž/g x XzX s ␪ ) ) EEÄ4i Ž.z Ez Ž, x .FfxŽ. Ž., y .18 Ž . xgXzgX

The equalityŽ. 18 directly implies that

␳␮wx G ␪ ) ) fiŽ.y i Ž.z Ex Ž, z .FfxŽ. Ž., y , ᭙ x, ᭙z g X, ᭙i g I, ᭙ y g Y, ␳␮wx G ␪ ) ) i.e., fiŽ.y i Ž.z Ez Ž, z .FfzŽ. Ž., y s ␪ ) ᭙ g ᭙ g ᭙ g i Ž.z FfzŽ. Ž., y , z X, i I, y Y, ␳␮wx G ␪ ) ᭙ g ᭙ g i.e., fiŽ.y E i Ž.z FfzŽ. Ž., y , i I, y Y.19Ž. zgX 512 MUSTAFA DEMIRCI

␯ ␪ Since iiis the extensional hull of fŽ.we may write $ ␪ s ␯ s ␪ ) f Ž.Ž.iiy Ž.y E Ä4f Ž.Ž. iu Fu Ž, y . ugY

s ␪ ) EEi Ž.t Fu Ž, y . g ž/y1 u Y ½5g Ž. t fu

s ␪ ) EEŽ.i Ž.t FftŽ. Ž., y ½5y ugY tgfu1Ž. s ␪ ) E Ž.i Ž.t FftŽ. Ž., y gD y1 t ug Y fuŽ. s ␪ ) ᭙ g ᭙ g E i Ž.z FfzŽ. Ž., y , i I, y Y.20Ž. zgX

CombiningŽ. 19 and Ž. 20 we get ␳␮wxG ␯ ᭙ g fi i, i I.21Ž.

On the other hand, exploiting the extensionality of f w.r.t. E and F inŽ. 18 and consideringŽ. 20 , we observe that ␳␮wx s ␪ ) ) fiŽ.y EEÄ4iŽ.z Ez Ž, x .FfxŽ. Ž., y xgXzgX F ␪ ) ) EEÄ4i Ž.z FfxŽ.Ž. Ž., fz Ž.Ffx Ž., y xgXzgX F ␪ ) EE i Ž.z FfzŽ. Ž., y xgXzgX s ␪ ) s ␯ ᭙ g ᭙ g E iiŽ.z FfzŽ. Ž., y Ž.y , i I, y Y. zgX Ž.22

␳␮wxs ␯ Thus the equality fi iis immediately acquired fromŽ. 21 and Ž. 22 ; ␳ ␳ i.e., the fuzzy function fUis a solution ofŽ. 17 and is the greatest solution ofŽ. 17 .

THEOREM 5.2. Let E and F be equality relations w.r.t. ) on X and Y, ¨ s ¬ g ª respecti ely, let C Ä4xi i I be an ordinary subset of X, and let f: X Y ␮ g X ␯ g Y be an extensional function w.r.t. E and F. Let iiL , L be exten- ¨ g sional hulls of xii and fŽ. x w.r.t. E and F, respecti elyŽ. i I . Then the fuzzy ␳ g X=Y ¨ ␳ s E ␮ )␯ relation LLL , gi en by Ž.x, y ig Ii ŽŽ.Ž..x iy , satisfies the ␳ s ␳ equality L< = f < . C Y C FUZZY FUNCTIONS 513

Proof. It is obvious that f < : C ª Y is extensional w.r.t. the )-fuzzy C equality E< on C and F on Y; i.e., C= C ␳ g C=Y ␳ s E ) ff<

s E Ex< , x ) Ffx , y Ž.C= CŽ.iiŽ. Ž. igI s ␳ ᭙ g ᭙ g ␳ s ␳ fL< Ž.x, y , x C, y Y, i.e., < = f < . C C Y C

THEOREM 5.3wx 14 . LetŽ. L, F , ) stand for a GL-monoid. Let E and F ) ¨ ␮ ¬ g be equality relations w.r.t. on X and Y, respecti ely. Let Ä4i i I and ␯ ¬ g Ä4i i I be two family of extensional L-fuzzy sets of X w.r.t. E and ¨ ␳ extensional L-fuzzy sets of Y w.r.t. F, respecti ely. Then U is an extensional fuzzy relation w.r.t. E and F.

THEOREM 5.4. LetŽ. L, F , ) denote a GL-monoid, and let E and F be ) ¨ ␮ ¬ g equality relations w.r.t. on X and Y, respecti ely. Let Ä4i i Ibea family of extensional L-fuzzy sets of X w.r.t. Eha¨ing the property that, for g ᭚ g ␮ s ¬ g each x X, k I such that kiŽ.x 1. For any ordinary subsetÄ4 y i I ␯ g Y g of Y, let iiL be the extensional hull of yŽ. i I . Then the fuzzy relation ␳ U is a partial fuzzy function w.r.t. E and F. ␳ Proof. It is adequate to see that U holdsŽ. PF . Then the required result follows from Theorem 5.3 at once. Considering the assumption that, for each x g X, ᭚k g I such that ␮ s kŽ.x 1, and using Proposition 2.1 Ž. ii , it can be easily seen that ␳ ) ␳ X UUŽ.x, y Žx, y .

X s HH␮ Ž.x ª ␯ Ž.y ) ␮ Ž.x ª ␯ Žy . ž/ž/Ž.Ž.ii ii igIigI F ␮ ª ␯ ) ␮ ª ␯ X Ž.kkŽ.x Ž.y Ž. kk Ž.x Žy . s ª ␯ ) ª ␯ X Ž.Ž.1 kkŽ.y 1 Žy . s ␯ )␯ X s ) X F X kkŽ.y Žy .Fy Ž, y k .Fy Ž k, y .Fy Ž, y ., X ᭙ x g X, ᭙ y, ᭙ y g Y. 514 MUSTAFA DEMIRCI

THEOREM 5.5. LetŽ. L, F , ) represent a GL-monoid, and let E and F be equality relations w.r.t. ) on X and Y, respecti¨ely. For an ordinary subset ¬ g ª Ä4yi i I of Y and for an extensional function f: X Y w.r.t. E and F, let ¬ g : y1 s ÄCiiii I, C fŽÄ4 y.4 be a family of ordinary subsets of X and S D ␮ g X ␯ g Y ig IiC . If iL , iL are extensional hulls of C i and y iw.r.t. E and S=Y F, respecti¨elyŽ. i g I , then the fuzzy relation ␳ < g L is a partial fuzzy U S= Y < ␳ function w.r.t. ES=S on S and F on Y and U is the greatest solution of Ž.17 . ␮ g X ␯ g Y Proof. Let iiL and L be extensional hulls of Ciiand y w.r.t. E and F, respectively Ž.i g I . Then it is obvious that the family Ä4␮ < ¬ i g I i S ␮ g of L-fuzzy sets of S obtained from restriction of i on SiŽ.I satisfies S=Y the hypothesis of Theorem 5.4; i.e., ␳ < g L is a partial fuzzy U S= Y function w.r.t. E< on S and F on Y. Furthermore, since Ä4␮ ¬ i g I and S= S i ␯ ¬ g ␳ Ä4i i I also satisfies the hypothesis of Theorem 5.1, it is clear that U is the solution ofŽ. 17 .

COROLLARY 5.6. LetŽ. L, F , ) beaGL-monoid. Let E and F be ) ¨ ¬ g -fuzzy equalities on X and Y, respecti ely, letÄ4 yi i I be an ordinary subset of Y, and let f: X ª Y be an extensional function w.r.t. E and F. ¬ g : y1 LetÄ Ciiii I, C fŽÄ4 y.4 be a family of ordinary subsets of X and s D ␮ g X ␯ g Y S ig IiC . Let iL , iL be extensional hulls of C i and y iw.r.t. E S=Y and F, respecti¨elyŽ. i g I . Then the fuzzy relation ␳ < g L is a perfect U S= Y < ␳ fuzzy function w.r.t. ES=S on S and F on Y, and U is the greatest solution of Ž.17 . ␳ Proof. If it is proven that U holdsŽ. F.1 , then the required results ␳ s follow from Theorem 5.5, immediately. Now we show that U ŽŽ..x, fx 1 for all x g X; i.e., ␳ fulfills the conditionŽ. F.1 ; i.e., ␳ < holdsŽ. F.1 . By UUS= Y ␳ ␳ s the definition of UU, the equality ŽŽ..x, fx 1 is equivalent to the ␮ ª ␯ s g equality Žii Žx . Žfx Ž ... 1 for all i I. By Proposition 2.1Ž. iii , since ␮ ª ␯ s ␮ the equality Žii Žx . Žfx Ž ... 1 is equivalent to the inequality iŽ.x F ␯ ␮ F ␯ iiŽŽ..fx, it is adequate to show the inequality Ž.x i ŽŽ..fx for all i g I and for all x g X. Taking into consideration the hypothesis on ␮ g X ␯ g Y iiL , L and applying the extensionality of f, we easily see that ␮ s F i Ž.x EEEx Ž, z .FfxŽ. Ž., fz Ž. g g z CziiC s s s ␯ E FuŽ., fxŽ.FyŽ.ii, fx Ž.Ž.fx Ž., g u fCŽ.i ᭙i g I, ᭙ x g X. THEOREM 5.7. Let E and F be )-fuzzy equalities on X and Y, respec- ¨ s ¬ g ª ti ely, let C Ä4xi i I an ordinary subset of X, and let f: X Ybean ␮ ¬ g ␯ ¬ g extensional function w.r.t. E and F. Let Ä4Ä4iii I and i Ibean FUZZY FUNCTIONS 515

g extensional hull of xiiw.r.t. E and the extensional hull of fŽ. x w.r.t. Fi ŽI .. Then, for any perfect fuzzy function ␳ g LX=Y w.r.t. E and F satisfying Ž.17 , we ha¨e s ␳ s ␳ s ␳ s ␳ f < ord <

Proof. Let a perfect fuzzy function ␳ g LX=Y w.r.t. E and F satisfy Ž.17 . Then, recalling that a perfect fuzzy function ␳ g LX=Y w.r.t. E and F can be written as

␳ Ž.x, y s FŽ.ord Ž.Ž.␳ x , y s E Ä4Ex Ž., z ) F Ž.ord Ž.Ž.␳ z , y , zgX ᭙ x g X, ᭙ y g Y ␳␮wxs ␯ g and using the assumption iiŽ.i I , we easily see that ␳␮wx s ␮ ) ␳ iiŽ.y E Ä4 Ž.x Žx, y . xgX s ) ␳ E Ä4ExŽ., xi F Ž.ord Ž.Ž.x , y xgX s ␳ s ␯ s F Ž.Ž.ordŽ.Ž.xii, y Ž.y Ffx Ž. i, y , ᭙i g I, ᭙ y g Y.

Thus we get ␳ s ᭙ g ᭙ g F Ž.Ž.ordŽ.Ž.xii, y Ffx Ž., y , i I, y Y.23Ž. g s Now, for i I, taking y fxŽ.i in Ž. 23 , it is observed that s ␳ s FfxŽ.Ž.Ž.ii, fx Ž.F ord Ž.Ž.Ž.xii, fx 1, ␳ s i.e., ordŽ.Ž.Ž.xiifx.

Therefore, we have f <

C=Y On the other hand, since vagŽ.f < g L is a perfect fuzzy function C w.r.t. the )-fuzzy equality E< on C and F on Y it is obvious that C= C C=Y ␳ < g L is a perfect fuzzy function w.r.t. E< and F. Thus the L C= Y C=C equality ␳ <

~~f < s ord ␳ < s ord ␳ < . CCŽ.C ŽL =Y .~~

COROLLARY 5.8. Let E and F be )-fuzzy equalities on X and Y, respec- ¨ s ¬ g ª ti ely. Let us suppose that X Ä4xi i I and f: X Y is an extensional ␮ ¬ g ␯ ¬ g function w.r.t. E and F. Let Ä4Ä4iii I and i I be the extensional g hull of xiiw.r.t. E and the extensional hull of fŽ. x w.r.t. Fi ŽI .. Then vagŽ.f is the unique perfect fuzzy function w.r.t. E and F satisfying Ž.17 . Proof. The required result is a direct consequence of Theorems 5.1 and 5.7. F ) s ¬ g COROLLARY 5.9. LetŽ. L, , be a GL-monoid, and let X Ä xi i I4. Let E and F be )-fuzzy equalities on X and Y, respecti¨ely, and let f: ª ␮ ¬ g X Y be an extensional function w.r.t. E and F. Let Ä4i i I and ␯ ¬ g ¨ Ä4iii I be extensional hulls of x and fŽ. xiw.r.t. E and F, respecti ely. Then s ␳ s ␳ vagŽ.f UL. Proof. The required result is easily deduced from Corollary 5.6, Theo- rem 5.7, and Corollary 5.8.

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