Partially ordered and relational valued fuzzy relations I

Abstract

The aim of the paper is to define and investigate some special properties of partially ordered and relational valued fuzzy relations. We use the concept of a fuzzy set as the mapping from an unempty set into a partially ordered set or into a suitable relational system (see [2, 3] ). Fuzzy equivalence and fuzzy order are defined by means of ordinary equivalence and ordering relations as the corresponding level relations, since the direct definitions (see [1], for example) are useless because of the absence of lattice operations. Necessary and sufficient conditions under which a collection of equivalence or ordering relations can be synthesized into the above-mentioned partially ordered fuzzy relation are given. For the relational valued fuzzy relations, it turns out that any collection of equivalences or orderings gives a relational valued fuzzy equivalence or ordering.