Fuzzy preorders and generalized distances: The aggregation problem revisited

Transitive fuzzy relations play a central role in fuzzy research activity. A special type of this class of fuzzy relations is known as indistinguishability operators. Many works have focused their efforts on the study of the properties of such operators. One celebrated problem is to find out the class of functions that allows to aggregate a collection of indistinguishability operators into a new single one. Characterizations of this class of functions have been obtained and the relationship with those functions that aggregate a collection of extended pseudo-metrics into a single one has been revealed in the literature. Moreover, fuzzy preorders are a class of fuzzy transitive relations that extend the notion of indistinguishability operators to the asymmetric context. In this paper we show that there is an equivalence between functions that aggregate fuzzy preorders and those functions that merge extended quasi-pseudo-metrics. The new provided characterizations reveal that, in essence, such functions must be monotone and subadditive. Special attention is paid to the case of fuzzy partial orders (a special case of fuzzy preorder) showing that there is a correspondence between those functions aggregating fuzzy partial orders and those that aggregate extended quasi-metrics. Since all the aforesaid fuzzy relations are transitive we also provide new information about those functions that preserve the class of transitive fuzzy relations in terms of those that preserve the so-called ordinary triangular triplets and those that preserve the triangle inequality. The potential applicability of the exposed theory to multi-criteria decision making problems has been discussed.