A General Framework for Ordering Fuzzy Sets

Abstract

Abstract. Orderings and rankings of fuzzy sets have turned out to play a fundamental role in various disciplines. Throughout the previous 25 years, a lot a different approaches to this issue have been introduced, ranging from rather simple ones to quite exotic ones. The aim of this paper is to present a new framework for comparing fuzzy sets with respect to a general class of fuzzy orderings. This approach includes several known techniques based on generalizing the crisp linear ordering of real numbers by means of the extension principle, however, in its general form, it is applicable to any fuzzy subsets of any kind of universe for which a fuzzy ordering is known -no matter whether linear or partial.

A Formal Model of Interpretability of Linguistic Variables

“…The following basic properties hold for all normalized fuzzy sets Since the relations ~ and.$ are reflexive and transitive [4,5]' it is sufficient to prove the relations indicated by arrows in the two Hasse diagrams (see Figs. 1 and 2).…”

“…It is easy to see that ATL always yields the smallest superset with non-decreasing membership function, while ATM yields the smallest superset with non-increasing membership function. For more details about the particular properties of the ordering relation :s and the two operators ATL and ATM, see [4,5].…”

“…We start with the usual inclusion of fuzzy sets according to Zadeh [37]. For defining a meaningful counterpart of the ordering relation ::S, we adopt a simple variant of the general framework for ordering fuzzy sets proposed in [4,5]' which includes well-known orderings of fuzzy numbers based on the extension principle [23,25]. where the operators ATL and ATM are defined as follows:…”