Aggregation of fuzzy quasi-metrics

Abstract

[eng] In the last years fuzzy (quasi-)metrics and indistinguishability operators have been used as a mathematical tool in order to develop appropriate models useful in applied sciences as, for instance, image processing, clustering analysis and multi-criteria decision making. The both aforesaid similarities allow us to fuzzify the crisp notion of equivalence relation when a certain degree of similarity can be only provided between the compared objects. However, the applicability of fuzzy (quasi-)metrics is reduced because the difficulty of generating examples. One technique to generate new fuzzy binary relations is based on merging a collection of them into a new one by means of the use of a function. Inspired, in part, by the preceding fact, this paper is devoted to study which functions allow us to merge a collection of fuzzy (quasi-)metrics into a single one. We present a characterization of such functions in terms of ∗-triangular triplets and also in terms of isotonicity and ∗-supmultiplicativity, where ∗ is a t-norm. We also show that this characterization does not depend on the symmetry of the fuzzy quasi-metrics. The same problem for stationary fuzzy (quasi-)metrics is studied and, as a consequence, characterizations of those functions aggregating fuzzy preorders and indistinguishability operators are obtained.