<i>T</i>-Equivalences: The Metric Behavior Revisited
Since the notion of <i>T</i>-equivalence, where <i>T</i> 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 t...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/4/495 |
| _version_ | 1797571646773526528 |
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| author | Pilar Fuster-Parra Javier Martín Jordi Recasens Óscar Valero |
| author_facet | Pilar Fuster-Parra Javier Martín Jordi Recasens Óscar Valero |
| author_sort | Pilar Fuster-Parra |
| collection | DOAJ |
| description | Since the notion of <i>T</i>-equivalence, where <i>T</i> 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 <i>T</i>-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 <i>T</i>-equivalence are linked. Inspired by the preceding fact, in this paper, we state a concrete relationship between partial pseudo-metrics and the aforesaid generalized <i>T</i>-equivalences. Specifically, a method for constructing partial pseudo-metrics from the new type of <i>T</i>-equivalences and, reciprocally, for constructing the generalized <i>T</i>-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. |
| first_indexed | 2024-03-10T20:43:36Z |
| format | Article |
| id | doaj.art-2038e3a063ae46cca878036a38981114 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T20:43:36Z |
| publishDate | 2020-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-2038e3a063ae46cca878036a389811142023-11-19T20:29:04ZengMDPI AGMathematics2227-73902020-04-018449510.3390/math8040495<i>T</i>-Equivalences: The Metric Behavior RevisitedPilar Fuster-Parra0Javier Martín1Jordi Recasens2Óscar Valero3Departament de Ciències, Matemàtiques i Informàtica, Universitat de les Illes Balears, 07122 Palma de Mallorca (Illes Balears), SpainDepartament de Ciències, Matemàtiques i Informàtica, Universitat de les Illes Balears, 07122 Palma de Mallorca (Illes Balears), SpainDepartment of Architecture Technology, Universitat Politècnica de Catalunya, Sant Cugat del Vallès, 08190 Barcelona, SpainDepartament de Ciències, Matemàtiques i Informàtica, Universitat de les Illes Balears, 07122 Palma de Mallorca (Illes Balears), SpainSince the notion of <i>T</i>-equivalence, where <i>T</i> 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 <i>T</i>-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 <i>T</i>-equivalence are linked. Inspired by the preceding fact, in this paper, we state a concrete relationship between partial pseudo-metrics and the aforesaid generalized <i>T</i>-equivalences. Specifically, a method for constructing partial pseudo-metrics from the new type of <i>T</i>-equivalences and, reciprocally, for constructing the generalized <i>T</i>-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.https://www.mdpi.com/2227-7390/8/4/495continuous t-normArchimedean t-normadditive generatorT-equivalenceT-equalitypartial pseudo-metric |
| spellingShingle | Pilar Fuster-Parra Javier Martín Jordi Recasens Óscar Valero <i>T</i>-Equivalences: The Metric Behavior Revisited Mathematics continuous t-norm Archimedean t-norm additive generator T-equivalence T-equality partial pseudo-metric |
| title | <i>T</i>-Equivalences: The Metric Behavior Revisited |
| title_full | <i>T</i>-Equivalences: The Metric Behavior Revisited |
| title_fullStr | <i>T</i>-Equivalences: The Metric Behavior Revisited |
| title_full_unstemmed | <i>T</i>-Equivalences: The Metric Behavior Revisited |
| title_short | <i>T</i>-Equivalences: The Metric Behavior Revisited |
| title_sort | i t i equivalences the metric behavior revisited |
| topic | continuous t-norm Archimedean t-norm additive generator T-equivalence T-equality partial pseudo-metric |
| url | https://www.mdpi.com/2227-7390/8/4/495 |
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