Relational valued fuzzy sets

Abstract

A generalization of the notion of a fuzzy set is given. An R-fuzzy set A is a mapping from a set A into the relational system (S, ρ{variant}) where ρ{variant} is a binary relation on S. Necessary and sufficient conditions for a unique decomposition and synthesis of an R-fuzzy set into the family of ordinary subsets (p-cuts) are given. As a consequence, it is possible to obtain a fuzzy set as a synthesis of any collection of characteristics functions on a nonvoid set, which was impossible in classical fuzzy set theory. As a contribution to coding theory, a possibility to express any binary block code syntheticly, by a single fuzzy set, is obtained. Note that up to now only some classes of binary codes have been characterized by fuzzy sets. Suitable examples (BCD and a linear code) are given at the end of the paper, together with necessary and sufficient conditions under which an R-valued fuzzy set corresponds to a linear code.