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T -Equivalences: The Metric Behavior Revisited

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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
    Institut d’ Investigació Sanitària Illes Balears (IdISBa), Hospital Universitari Son Espases, 07120 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
    Institut d’ Investigació Sanitària Illes Balears (IdISBa), Hospital Universitari Son Espases, 07120 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
    Institut d’ Investigació Sanitària Illes Balears (IdISBa), Hospital Universitari Son Espases, 07120 Palma de Mallorca (Illes Balears), Spain)

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 Łukasiewicz t-norms are under consideration.

Suggested Citation

  • Pilar Fuster-Parra & Javier Martín & Jordi Recasens & Óscar Valero, 2020. "T -Equivalences: The Metric Behavior Revisited," Mathematics, MDPI, vol. 8(4), pages 1-18, April.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:4:p:495-:d:340422
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    References listed on IDEAS

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    1. Shahzad, Naseer & Valero, Oscar & Alghamdi, Mohammed A. & Alghamdi, Maryam A., 2015. "A fixed point theorem in partial quasi-metric spaces and an application to Software Engineering," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 1292-1301.
    2. N. Shahzad & O. Valero, 2013. "On 0-Complete Partial Metric Spaces and Quantitative Fixed Point Techniques in Denotational Semantics," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-11, December.
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