Variable-domain fuzzy sets - Part II: Apparatus

Abstract This is the second part of a two-part paper on operations on fuzzy sets that may not share the same domain of definition. Part I described several ways of representing variable-domain fuzzy sets and introduced a few basic variable-domain fuzzy set operations. The present (largely self-contained) Part II further develops the theory of variable-domain fuzzy sets, based primarily on the representation of variable domains by a dummy membership degree that indicates the out-of-domain assignment error. We first describe several families of algebraic operations on the extended set of membership degrees; these are later employed in the investigation of basic notions of variable-domain fuzzy set theory. The fuzzy set operations studied in this paper include variable-domain variants of unions and intersections, kernels and supports, heights and plinths, equalities and inclusions, Cartesian products, and fuzzy relational compositions. Besides the initial examination of these variable-domain notions, our aim is to demonstrate the viability of variable-domain fuzzy set theory and highlight the similarity as well as differences between the fixed- and variable-domain treatments of fuzzy sets.