Fuzzy points of equivalent fuzzy subsets

A natural equivalence relation on the set of all fuzzy subsets, generalizing the equality of crisp sets, was recently introduced and studied in various contexts such as finite fuzzy subsets, fuzzy vector space, finite fuzzy subfields and in the classification of finite Abelian groups. In this paper, we restrict this equivalence to the set of fuzzy points and study its effect on the relationship between fuzzy points and fuzzy subsets. All fuzzy subsets take a finite number of membership values in the real unit interval. An important tool for studying this equivalence relation is that of a keychain. This notion gives rise to the idea of a pinned-flag, their equivalences and index of fuzzy subsets. 2003 Elsevier Inc. All rights reserved.