The stability of local properties of fuzzy relations under ordinal equivalence

Abstract We consider a notion of equivalence relation between fuzzy relations and its properties. The relation is called ordinal equivalence since it partitions the family of fuzzy relations into equivalence classes by means of the order defined on elements of each relation. We compare fuzzy relations by their membership values and as a consequence we propose two types of local properties of fuzzy relations. New properties of fuzzy relations are more compatible with the ordinal equivalence relation, namely they are stable in the equivalence classes. Moreover, we study dependencies between these new properties of fuzzy relations. Finally, notes on applications of the presented local properties and the ordinal equivalence relation are provided.