Lattice embeddings between types of fuzzy sets. Closed-valued fuzzy sets

TL;DR

This paper explores lattice embeddings for extending Zadeh's fuzzy set operators to more complex fuzzy set types, introducing closed-valued fuzzy sets to unify different membership degrees and maintain order-preserving properties.

Contribution

It introduces a family of lattice embeddings from fuzzy sets to set-valued and type-2 fuzzy sets, and proposes closed-valued fuzzy sets to unify membership degrees of various natures.

Findings

Established order-preserving lattice embeddings for fuzzy set extensions

Reformulated intersection for hesitant fuzzy sets

Introduced closed-valued fuzzy sets for unified membership handling

Abstract

In this paper we deal with the problem of extending Zadeh's operators on fuzzy sets (FSs) to interval-valued (IVFSs), set-valued (SVFSs) and type-2 (T2FSs) fuzzy sets. Namely, it is known that seeing FSs as SVFSs, or T2FSs, whose membership degrees are singletons is not order-preserving. We then describe a family of lattice embeddings from FSs to SVFSs. Alternatively, if the former singleton viewpoint is required, we reformulate the intersection on hesitant fuzzy sets and introduce what we have called closed-valued fuzzy sets. This new type of fuzzy sets extends standard union and intersection on FSs. In addition, it allows handling together membership degrees of different nature as, for instance, closed intervals and finite sets. Finally, all these constructions are viewed as T2FSs forming a chain of lattices.