Fuzzy functions: a fuzzy extension of the category SET and some related categories

Abstract

In research Works where fuzzy sets are used, mostly certain usual functions are taken as morphisms. On the other hand, the aim of this paper is to fuzzify the concept of a function itself. Namely, a certain class of L-relations F : X x Y -> L is distinguished which could be considered as fuzzy functions from an L-valued set (X,Ex) to an L-valued set (Y,Ey). We study basic properties of these functions, consider some properties of the corresponding category of L-valued sets and fuzzy functions as well as briefly describe some categories related to algebra and topology with fuzzy functions in the role of morphisms.