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. Moreover, the problem of how to construct new indistinguishability fuzzy relations by means of aggregation has been explored in the literature. In this paper, we provide new characterizations of those functions that allow us to merge a collection of indistinguishability fuzzy relations 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, which are the natural distances associated to indistinguishability fuzzy relations. Our new 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 separate points, and we show that both differ a lot.
The main purpose of this paper is to 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, extending the classical approach explored for pseudo-metrics and indistinguishability fuzzy relations. 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.
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