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Open AccessArticle

A Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-Metrics

1
Instituto de Investigación para la Gestión Integrada de Zonas Costeras, Universitat Politècnica de València, C/ Paranimf, 1, 46730 Grao de Gandia, Spain
2
Departament de Ciències Matemàtiques i Informàtica, Universitat de les Illes Balears, Carretera de Valldemossa km. 7.5, 07122 Palma, Spain
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(9), 1575; https://doi.org/10.3390/math8091575
Received: 17 August 2020 / Revised: 7 September 2020 / Accepted: 9 September 2020 / Published: 12 September 2020
In 1994, Matthews introduced the notion of partial metric and established a duality relationship between partial metrics and quasi-metrics defined on a set X. In this paper, we adapt such a relationship to the fuzzy context, in the sense of George and Veeramani, by establishing a duality relationship between fuzzy quasi-metrics and fuzzy partial metrics on a set X, defined using the residuum operator of a continuous t-norm ∗. Concretely, we provide a method to construct a fuzzy quasi-metric from a fuzzy partial one. Subsequently, we introduce the notion of fuzzy weighted quasi-metric and obtain a way to construct a fuzzy partial metric from a fuzzy weighted quasi-metric. Such constructions are restricted to the case in which the continuous t-norm ∗ is Archimedean and we show that such a restriction cannot be deleted. Moreover, in both cases, the topology is preserved, i.e., the topology of the fuzzy quasi-metric obtained coincides with the topology of the fuzzy partial metric from which it is constructed and vice versa. Besides, different examples to illustrate the exposed theory are provided, which, in addition, show the consistence of our constructions comparing it with the classical duality relationship. View Full-Text
Keywords: fuzzy quasi-metric; fuzzy partial metric; additive generator; residuum operator; Archimedean t-norm fuzzy quasi-metric; fuzzy partial metric; additive generator; residuum operator; Archimedean t-norm
MDPI and ACS Style

Gregori, V.; Miñana, J.-J.; Miravet, D. A Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-Metrics. Mathematics 2020, 8, 1575. https://doi.org/10.3390/math8091575

AMA Style

Gregori V, Miñana J-J, Miravet D. A Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-Metrics. Mathematics. 2020; 8(9):1575. https://doi.org/10.3390/math8091575

Chicago/Turabian Style

Gregori, Valentín; Miñana, Juan-José; Miravet, David. 2020. "A Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-Metrics" Mathematics 8, no. 9: 1575. https://doi.org/10.3390/math8091575

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