T -Equivalences: The Metric Behavior Revisited

Pilar Fuster-Parra, Javier Martín, Jordi Recasens and Óscar Valero
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Pilar Fuster-Parra: Departament de Ciències, Matemàtiques i Informàtica, Universitat de les Illes Balears, 07122 Palma de Mallorca (Illes Balears), Spain
Javier Martín: Departament de Ciències, Matemàtiques i Informàtica, Universitat de les Illes Balears, 07122 Palma de Mallorca (Illes Balears), Spain
Jordi Recasens: Department of Architecture Technology, Universitat Politècnica de Catalunya, Sant Cugat del Vallès, 08190 Barcelona, Spain
Óscar Valero: Departament de Ciències, Matemàtiques i Informàtica, Universitat de les Illes Balears, 07122 Palma de Mallorca (Illes Balears), Spain

Mathematics, 2020, vol. 8, issue 4, 1-18

Abstract: Since the notion of T -equivalence, where T is a t-norm, was introduced as a fuzzy generalization of the notion of crisp equivalence relation, many researchers have worked in the study of the metric behavior of such fuzzy relations. Concretely, a few techniques to induce metrics from T -equivalences, and vice versa, have been developed. In several fields of computer science and artificial intelligence, a generalization of pseudo-metric, known as partial pseudo-metrics, have shown to be useful. Recently, Bukatin, Kopperman and Matthews have stated that the notion of partial pseudo-metric and a type of generalized T -equivalence are linked. Inspired by the preceding fact, in this paper, we state a concrete relationship between partial pseudo-metrics and the aforesaid generalized T -equivalences. Specifically, a method for constructing partial pseudo-metrics from the new type of T -equivalences and, reciprocally, for constructing the generalized T -equivalences from partial pseudo-metrics are provided. However, important differences between the new approach and the classical one are established. Special interest is paid to the case in which the minimum, drastic, and ?ukasiewicz t-norms are under consideration.

Keywords: continuous t-norm; Archimedean t-norm; additive generator; T-equivalence; T-equality; partial pseudo-metric (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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