On Aggregation of Fuzzy Binary Relations and Generalized Distances

[eng] Indistinguishability fuzzy relations were introduced with the aim of providing a fuzzy notion of equivalence relation. Many works have explored their relation to metrics, since they can be interpreted as a kind of measure of similarity and this is, in fact, a dual notion to dissimilarity. The problem of how to construct new indistinguishability fuzzy relations by means of aggregation has been explored in the literature. Hence, we provide new characterizations of those functions that allow us to merge a collection of indistinguishability fuzzy relations with respect to a collection of t-norms into a new one in terms of triangular triplets, and, in addition, we explore the relationship between such functions and those that aggregate extended pseudo-metrics. These results extend some already known characterizations which involve only bounded pseudo-metrics. In addition, we provide a completely new description of those indistinguishability fuzzy relations that separates points, and we show that both differ a lot. In order to extend the classical approach explored for pseudo-metrics and indistinguishability fuzzy relations, we also study the relationship between those functions that aggregate relaxed indistinguishability fuzzy relations with respect to a collection of t-norms and those functions that merge relaxed pseudo-metrics. Special attention is paid to the distinguished class of SSI-relaxed indistinguishability fuzzy relations showing that functions merging this special type of relaxed indistinguishability fuzzy relations can be expressed through functions aggregating SSD-relaxed pseudo-metrics. Outstanding differences between those functions aggregating indistinguishability fuzzy relations and those that aggregate their counterpart separating points are shown. Fuzzy preorders are a class of fuzzy transitive relations that extend the notion of indistinguishability operators to the asymmetric context. Then, we show that there is an equivalence between functions that aggregate fuzzy preorders with respect to a collection of t-norms and those functions that merge extended quasi-pseudo-metrics. It reveals that such functions must be monotone and subadditive. Special attention is paid to the case of fuzzy partial orders 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 triangular inequality. The potential utility of the developed theory in Location Analysis is outlined. Finally, we focus our effort on the study of new techniques for inducing fuzzy preorders (indistinguishability operators) through quasi-pseudo-metrics (pseudo-metrics). Hence, we extend the unique technique known in the literature based on the use of additive generators of continuous t-norms and their associated pseudo-inverses. Concretely, we characterize those functions that allow us to induce a fuzzy preorder from a quasi-pseudo-metric even when the considered t-norm is not continuous. On the one hand, we prove that they must be decreasing and fulfil a special property of dominance with respect to the ordinary addition and the t-norm T under consideration. On the other hand, we have shown that they must transform asymmetric triangular triplets into asymmetric T-triangular triplets. Moreover, we also study the case in which fuzzy preorders are exactly indistinguishability operators, showing that the monotony of the function and the previously mentioned dominance are sufficient but not necessary conditions. In addition, we prove that such functions must transform triangular triplets into T-triangular triplets, illustrating it by means of appropriate examples. Furthermore, we prove that the aforesaid celebrated technique based on the use of the pseudo-inverses of the additive generators of continuous and Archimedean t-norms is recovered as a particular case of the new one presented here.