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Open AccessArticle

T-Equivalences: The Metric Behavior Revisited

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Departament de Ciències, Matemàtiques i Informàtica, Universitat de les Illes Balears, 07122 Palma de Mallorca (Illes Balears), Spain
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Institut d’ Investigació Sanitària Illes Balears (IdISBa), Hospital Universitari Son Espases, 07120 Palma de Mallorca (Illes Balears), Spain
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Department of Architecture Technology, Universitat Politècnica de Catalunya, Sant Cugat del Vallès, 08190 Barcelona, Spain
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Author to whom correspondence should be addressed.
Mathematics 2020, 8(4), 495; https://doi.org/10.3390/math8040495
Received: 17 February 2020 / Revised: 20 March 2020 / Accepted: 21 March 2020 / Published: 2 April 2020
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 Łukasiewicz t-norms are under consideration. View Full-Text
Keywords: continuous t-norm; Archimedean t-norm; additive generator; T-equivalence; T-equality; partial pseudo-metric continuous t-norm; Archimedean t-norm; additive generator; T-equivalence; T-equality; partial pseudo-metric
MDPI and ACS Style

Fuster-Parra, P.; Martín, J.; Recasens, J.; Valero, Ó. T-Equivalences: The Metric Behavior Revisited. Mathematics 2020, 8, 495.

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