Annals of Fuzzy Mathematics and Informatics @ FMI Volume x, No. x, (Month 201y), pp. 1{xx ISSN: 2093{9310 (print version) c Kyung Moon Sa Co. ISSN: 2287{6235 (electronic version) http://www.kyungmoon.com http://www.afmi.or.kr Zadeh extension principle: A note Pawan Kumar Tiwari, ; Arun K. Srivastava ; Received 29 January 2014 Revised 12 June 2014 Accepted 02 July 2014 Abstract. In this note, we examine a connection of the Zadeh extension principle with the notion of matrix theories (as used in category theory) in the sense of E.G. Manes. In particular, we employ matrix theories over a complete idempotent semiring, for this purpose. 03E72, 16D90. Keywords: Zadeh extension principle, Complete semiring, Adjoint functors, Algebraic theory, Complete distributive lattice. Corresponding Author: Author 1 ([email protected]) 2010 AMS Classication: 1. Introduction The well- known Zadeh extension principle (ZEP) of fuzzy set theory is regarded as an important tool in fuzzy set theory and its applications (e.g. in fuzzy arithmetic). ZEP is given, e.g., in Zadeh [13], as follows: For every function f : X ! Y between sets, there exist functions f ! : I X ! I Y and f : I Y ! I X , (where I = [0; 1]; elements of I X are called fuzzy sets in X ) dened as: f ! ()(y) = _ (x) : x 2 f 1 (y) ; 8 2 I X and f ( ) = f; 8 2 I Y such that ! (f f )(b) b; 8b 2 I Y and (f f ! )(a) a; 8a 2 I X : (f ! and f are frequently referred to as the forward and the backward lifting operators): An introductory account of ZEP has been given by Kerre [5], while a somewhat detailed study of the extension principle for fuzzy sets has been made, e.g., by Gerla and Scarpati [2], Nguyen [9] and Yager [12]. It may be pointed out that the ZEP was initially introduced in the context of fuzzy sets but has since been extended in related areas also (e.g., to dene the image of an intuitionistic fuzzy sets under a functions, ZEP is used; see, e.g., Kang et al [4] and Saleh [11]). The ZEP has been also looked into from a category theoretic point of view by a few authors, e.g., Rodabaugh [10], Kotze [6] and Barone [1]. In this note, we point out a Author 1 et al./Ann. Fuzzy Math. Inform. x (201y), No. x, xx{xx connection of the ZEP with the notion of a matrix theory over a complete semiring S , as introduced by Manes [8]. In fact, we show that given such a matrix theory, each function f : X ! Y between sets, naturally gives rise to a function S X ! S Y which resembles the forward lifting operators f ! in the ZEP, in case the complete semiring S is additionally assumed to be idempotent. 2. Preliminaries We shall use the following result from category theory; see, e.g., Herrlich et al [3] and Maclane [7]. L and M be posets g : L ! M preserve arbitrary joins. Then there exists an order-preserving map f : M ! L such that f is right adjoint of g , i.e., gf (a) a and fg (b) b hold 8a 2 M and 8b 2 L. Moreover, f is given Theorem 2.1. (Freyd's adjoint functor theorem for posets) Let with L closed under arbitrary _ . Let a function by f (a) = _ fb 2 L : a g(b)g : Denition 2.2. A semiring (S; ; ) is a non empty set S on which two commu- tative binary operations and are dened such that the following conditions are satised: (1) (S; ) is a monoid (with identity element 0), (2) (S; ) is a monoid, (3) Multiplication distributes over addition from either side, (4) 0 a = 0 = a 0; 8a 2 S , Further, a semiring (S; ; ) is called (i) idempotent if is idempotent and (ii) complete if for every family fai 2 S : i 2 g; there exists an element i2 ai of S such that the following conditions hold: (i)i2 ai = 0; if = ; (ii)i2 ai = a1 a2 : : : an ; if = f1; 2; : : : ; ng; (iii)b i2 ai = i2 (b ai ) and (i2 ai ) b = i2 (ai b); 8b 2 S S (iv)i2 ai = j2(k2 ak ) for every partition = j of . j 2 j Remark 2.3. We can easily verify that If (S; ; ) is a complete idempotent semiring and X is any set, then (S X ; +; :) is also a complete idempotent semiring with : and + dened as follows: for all f; g 2 S X ; ffi : i 2 g S and x 2 X; (f:g)(x) = f (x):g(x); (f + g)(x) = f (x) + g(x); 8x 2 X; and (+i2 fi )(a) = i2 fi (a): Denition 2.4. (Manes [8]) An algebraic theory in a category K (in extension form) is a triple (T; ; ( )# ) where : (1) T is a function assigning to each K-object X , a K-object T (X ), (2) is an assignment, assigning to each K -object X , a K-morphism, X T X, X ! 2 Author 1 et al./Ann. Fuzzy Math. Inform. x (201y), No. x, xx{xx (3) ( )# assigns to each K-morphism X ! T (Y ), a K-morphism # T (X ) ! T (Y ) such that for all K-objects X and K-morphisms X ! T (Y ), Y ! T (Z ); (i) # :X = ; (ii) X# = IdT (X ) ; (iii) ( # :)# = # :# : Given a complete semiring (S; ; ); an algebraic theory (T; ; ( )# ) in the category SET is called a matrix theory of S if, for every set X (1) T (X ) = S X , (2) X : X ! T (X ) is dened by X (x)(x0 ) = xx0 ; # (3) for sets X; Y and any X ! T (Y ); T (X ) ! T (Y ) is dened by: (# (p))(y) = x2X ((x)(y)) p(x); 8p 2 T (X ), 8y 2 Y . X ! T (Y ), may be thought as a matrix with entries in S with X indexing columns and Y indexing rows. As is known, T gives rise to a functor T : SET ! SET, given by T (X ) = S X and for any f : X ! Y , T (f ) : S X ! S Y is dened by (T (f )(p))(y) = x2f1(y)p(x); 8p 2 S X ; 8y 2 Y . 3. Main Observation Proposition 3.1. Let (T; ; ( )# ) be a matrix theory of a complete semiring (S; ; ). Then for any function f : X ! Y , between sets T (f )(+ pj ) = + T (f )(pj ); 8pj 2 S X ; j 2 J: j 2J j 2J Let y 2 Y . Then for every y 2 Y; (T (f )(+j2J pj ))(y) = (+j2J pj )(x); x2f 1(y) = x2f1 (y)(j2J pj (x)); = j2J ( pj (x)); x2f 1(y) = j2J (T (f )(pj )(y)); = (+j2J T (f )(pj ))(y); 8y 2 Y , Hence, (T (f )(+j2J pj ) = +j2J T (f )(pj ): Proof. We now conne to a complete semiring (S; ; ) which is idempotent also (i.e., is idempotent). Then we can dene a relation on S as follows: a b i a b = b. It is easy to see that is a partial order on S and the operation coincides with join operation induced by . Theorem 3.2. Let (S; ; ) be a complete idempotent semiring, f : X ! Y a map ! . Then there exists a function between sets X and Y , and T (f ) be denoted by f Y X f : S ! S such that (f ! f )(q) q; 8q 2 S Y ; and (f f ! )(p) p; 8p 2 S X : 3 Author 1 et al./Ann. Fuzzy Math. Inform. x (201y), No. x, xx{xx Moreover, f is given by, f (q) = q f; 8q 2 S Y : ! : S X ! S Y ; as a functor from the poset S X to the Proof. For a proof, we view f Y poset S , considered as categories. Then from Proposition 3.1 and Theorem 2.1, 9 an order-preserving map f : S Y ! S X such that f , considered as functor, is right adjoint of (f ! ); i.e., (f ! f )(q) q; 8q 2 S Y ; and (f f ! )(p) p; 8p 2 S X : We now show that f (q) = qf . By Theorem 2.1, 8q 2 S Y ; f (q) = + p 2 S X : q (f ! )(p) = +p2 p; where q = p 2 S X : q (f ! )(p) . q First, we show that f (q) q f: Now, 8y 2 Y; (f ! (q f ))(y) = x2f1 (y)(q f )(x) = x2f1 (y)q(f (x)) = q(y) = q(y). Hence q f 2 q , whereby f (q) = +p2 p q f (1) q Now we have to show that q f f (q), i.e., (q f )(x) f (q)(x); 8x 2 X We note that, p 2 q ) q(y) f ! (p)(y); 8y 2 Y ) q(y) 1 p(z ); 8y 2 Y: z2f (y) In particular, for any given x 2 X and p 2 q ; q(f (x)) 1 p(z ); 8p 2 q : z2f (f (x)) Taking summation over p 2 q ; we get q(f (x)) ( p2 p(z)): q z2f 1 (f (x)) But as 1 p(z ) p(x); z2f (f (x)) p2 q ( 1 p(z)) p2 q p(x); whereby q(f (x)) p2 q p(x); z2f (f (x)) i.e., (q f )(x) (+p2 p)(x) = (f (q))(x); 8x 2 X: q Hence (q f ) f (q) (2); From (1) and(2), we nd that f (q) = q f . Remark 3.3. In view of Theorem 3.2 and in particular due to the properties (f ! f )(q) q; 8q 2 S Y ; and (f f ! )(p) p; 8p 2 S X ; it appears appropriate to refer to the functions f ! and f as the forward and backward operators, as in the case of the ZEP. Acknowledgements. The authors are grateful to the reviewers for making suggestions for improving the paper. 4 Author 1 et al./Ann. Fuzzy Math. Inform. x (201y), No. x, xx{xx References [1] J. M. Barone, Fuzzy points and fuzzy prototypes, AFSS-2002, Springer Lecture Notes in Articial Intelligence, 2275 (2002) 466{470. [2] G. Gerla and L. Scarpati, Extension principles for fuzzy set theory, Inform. Sci. 106 (1998) 49{69. [3] H. Herrlich and G. E. Strecker, Category Theory, Sigma Series in Pure Math., 1, HeldermannVerlag, 1979. [4] H. W. Kang, J. G. Lee and K. Hur, Intuitionistic fuzzy mappings and intuitionistic fuzzy equivalence relations, Ann. fuzzy Math. Inform. 3 (2012) 61-87. [5] E. E. Kerre, A tribute to Zadeh's extension principle, Scientia Iranica 18 (2011) 593-595. [6] W. Kotze, Justication of the Zadeh extension principle (preprint). [7] S. MacLane, Categories for the Working Mathematician, Springer-Verlag, 1971. [8] E. G. Manes, A class of fuzzy theories, J. Math. Anal. Appl. 85 (1982) 409{451. [9] H. T. Nguyen, A note on the extension principle for fuzzy sets, J. Math. Anal. Appl. 64 (1978) 369{380. [10] S. E. Rodabaugh, Powerset operator based foundation for point-set lattice- theoretic (poslat) fuzzy set theories and topologies, Quaest. Math. 3 (1997) 463{530. [11] S. Saleh, On category of interval valued fuzzy topological spaces, Ann. fuzzy Math. Inform. 4 (2012) 385-392. [12] R. R. Yager, A characterization of the extension principle, Fuzzy Sets and Systems 18 (1986) 205{217. [13] L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338{353. Author 1 (pawan47 [email protected]) School of Computer & Systems Sciences, Jawaharlal Nehru University, New Delhi 110067, India Author 2 ([email protected]) Department of Mathematics, Faculty of Science, Banaras Hindu University, Varanasi 221005, India & Centre for Interdisciplinary Mathematical Sciences, Banaras Hindu University, Varanasi 221005, India 5