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

摘要

In this paper we deal with the problem of extending Zadeh's operators onfuzzy 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 thendescribe a family of lattice embeddings from FSs to SVFSs. Alternatively, ifthe former singleton viewpoint is required, we reformulate the intersection onhesitant 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. Inaddition, it allows handling together membership degrees of different natureas, for instance, closed intervals and finite sets. Finally, all theseconstructions are viewed as T2FSs forming a chain of lattices.