On equivalence picture fuzzy multirelations

Abstract

A picture fuzzy multirelation over picture fuzzy multiset, defined via $(r, s, t)$-cut set provides a means of studying the relationships at various thresholds of positive, neutral, and negative membership degrees. In this paper, we have introduced the notion of picture fuzzy equivalence multirelations over picture fuzzy multisets via $(r, s, t)$-cut set of picture fuzzy multirelations and investigate some of the properties related to them. We have shown that a picture fuzzy multirelation over a picture fuzzy multiset is a picture fuzzy equivalence multirelation if and only if the $(r, s, t)$-cut set of the picture fuzzy multirelation is an equivalence relation. The picture fuzzy equivalence class with respect to the picture fuzzy equivalence multirelation on a picture fuzzy multiset was defined and it was proved that the two picture fuzzy equivalence classes of a picture fuzzy equivalence multirelation are equal with respect to the $(r, s, t)$-cut set of picture fuzzy equivalence multirelation. Furthermore, we have established that the intersection of two picture fuzzy equivalence multirelations on a picture fuzzy multiset is again a picture fuzzy equivalence multirelation on the picture fuzzy multiset but their union need not be a picture fuzzy equivalence multirelation. This development of picture fuzzy equivalence multirelation can be employed to form new methodologies for dealing with complex, multi-dimensional relationships in uncertain environments, and also its applications can be explored in decision-making problems, clustering and data analysis.