A Similarity-Based Generalization of Fuzzy Orderings Preserving the Classical Axioms

Abstract

Equivalence relations and orderings are key concepts of mathematics. For both types of relations, formulations within the framework of fuzzy relations have been proposed already in the early days of fuzzy set theory. While similarity (indistinguishability) relations have turned out to be very useful tools, e.g. for the interpretation of fuzzy partitions and fuzzy controllers, the utilization of fuzzy orderings is still lagging far behind, although there are a lot of possible applications, for instance, in fuzzy preference modeling and fuzzy control. The present paper is devoted to this missing link. After a brief motivation, we will critically analyze the existing approach to fuzzy orderings. In the main part, an alternative approach to fuzzy orderings, which also takes the strong connection to gradual similarity into account, is proposed and studied in detail, including several constructions and representation results.

Left and right compatibility of strict orders with fuzzy tolerance and fuzzy equivalence relations

“…for any x, y, z ∈ X; (iii) < is called compatible with E, if it holds that τ (x < y) * E(x, z) * E(y, t) ≤ τ (z < t) , (6) for any x, y, z ∈ X.…”

“…Another notion of compatibility has been introduced by Bodenhofer and colleagues [5,6,7,8]. Bodenhofer has defined the compatibility of a fuzzy equivalence relation with a crisp order as follows: the two outer elements of an ascending threeelement chain are at most as similar as any two elements of this chain.…”

A General Framework for Ordering Fuzzy Sets

“…Since, according to the above discussions, the ordering method should be able to cope with vagueness and indistinguishability, all studies in this paper will be based on the similarity-based definition of fuzzy orderings (for extensive studies, see [3,4,11]). We only recall the very basic definitions.…”

“…It is worth to mention that there is a one-to-one correspondence between direct fuzzifications of crisp linear orderings and fuzzy orderings L which additionally fulfill strong completeness [3,4] max L(x, y), L(y, x) = 1 .…”

“…Correspondingly, for any fuzzy set A, RTL(A) is the smallest superset whose membership function is non-increasing. In direct analogy, ATL(A) can be regarded as the smallest superset of A the membership function of which is non-decreasing with respect to the fuzzy ordering L; analogously for ATM(A) (for a detailed and mathematically exact argumentation, see [3]). …”