The aggregation of transitive fuzzy relations revisited

Abstract

[eng] We consider the aggregation problem of a collection of fuzzy binary rela- tions when each one belongs to the same class and the aggregation process provides an output fuzzy binary relation in such a class. Concretely we focus our efforts on the characterization of those functions that aggregates a collection of fuzzy binary relations which are transitive with respect to a collection of t-norms preserving the transitivity. We characterize them in terms of triangular triplets. The relationship with monotony and an appro- priate dominance notion is explicitly stated. Special attention is paid to a few classes of transitive fuzzy binary relations that are relevant in the literature. In particular, we describe, in terms of triangular triplets the functions that aggregate a collection of fuzzy preorders, fuzzy partial orders, relaxed indistinguishability operators, indistinguishability operators and equalities. A surprising link between functions that aggregate transitive fuzzy relations into a TM -transitive fuzzy relation and those that aggregate relaxed indistinguishability operators is obtained. A few consequences of the new results are provided for those cases in which all the t-norms belonging to the collection are exactly the same. Some celebrated results are retrieved as a particular case from the exposed theory.