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Fuzzy Functions and Their Applications Journal of Mathematical Analysis and Applications 252, 495᎐517Ž. 2000 doi:10.1006rjmaa.2000.7185, available online at http:rrwww.idealibrary.com on Fuzzy Functions and Their Applications 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 generaliza- tion of the fuzzy equality given inwx 2, 3 by integrals, commutative cl-mono- ids, 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 investi- gated. 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 respec- tively 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 rela- tion 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 generaliza- tion 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.
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