Functional Partial Fuzzy Relations

We study fuzzy relations that satisfy the functionality property and that their membership functions can be partial functions. Such fuzzy relations are called partial fuzzy relations, and the variable-domain fuzzy set theory is a framework that provides powerful tools for handling these objects. There, the special operations based on connectives and quantifiers of a partial fuzzy logic are in use. The undefined degrees of membership are carried via those special operations. Furthermore, we show that a suitable combination of these operations leads to a meaningful definition of the functionality property, and we investigate its basic characteristics.