Fuzzy Relations and Fuzzy Functions in Partial Fuzzy Set Theory

In this paper, we will study partially-defined fuzzy relations i.e., fuzzy relations with membership functions that are not necessarily defined everywhere. We will handle them in a suitable framework that is a partial fuzzy set theory. It provides tools for dealing with undefined values of the membership by means of special operations based on the connectives and quantifiers of a background fuzzy logic. We analyze a suitability of operations of a partial fuzzy set theory for a meaningful definition of the functionality property. This property determines fuzzy functions and we will be concerned with their properties.