A Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-Metrics
In 1994, Matthews introduced the notion of partial metric and established a duality relationship between partial metrics and quasi-metrics defined on a set <inline-formula><math display="inline"><semantics><mi mathvariant="script">X</mi></semantic...
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MDPI AG
2020-09-01
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| Online Access: | https://www.mdpi.com/2227-7390/8/9/1575 |
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| author | Valentín Gregori Juan-José Miñana David Miravet |
| author_facet | Valentín Gregori Juan-José Miñana David Miravet |
| author_sort | Valentín Gregori |
| collection | DOAJ |
| description | In 1994, Matthews introduced the notion of partial metric and established a duality relationship between partial metrics and quasi-metrics defined on a set <inline-formula><math display="inline"><semantics><mi mathvariant="script">X</mi></semantics></math></inline-formula>. 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 <inline-formula><math display="inline"><semantics><mi mathvariant="script">X</mi></semantics></math></inline-formula>, defined using the residuum operator of a continuous <i>t</i>-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 <i>t</i>-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. |
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| id | doaj.art-399781614e4b4f0a9f29225f2e1f9399 |
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| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T16:22:20Z |
| publishDate | 2020-09-01 |
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| spelling | doaj.art-399781614e4b4f0a9f29225f2e1f93992023-11-20T13:30:53ZengMDPI AGMathematics2227-73902020-09-0189157510.3390/math8091575A Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-MetricsValentín Gregori0Juan-José Miñana1David Miravet2Instituto de Investigación para la Gestión Integrada de Zonas Costeras, Universitat Politècnica de València, C/ Paranimf, 1, 46730 Grao de Gandia, SpainDepartament de Ciències Matemàtiques i Informàtica, Universitat de les Illes Balears, Carretera de Valldemossa km. 7.5, 07122 Palma, SpainInstituto de Investigación para la Gestión Integrada de Zonas Costeras, Universitat Politècnica de València, C/ Paranimf, 1, 46730 Grao de Gandia, SpainIn 1994, Matthews introduced the notion of partial metric and established a duality relationship between partial metrics and quasi-metrics defined on a set <inline-formula><math display="inline"><semantics><mi mathvariant="script">X</mi></semantics></math></inline-formula>. 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 <inline-formula><math display="inline"><semantics><mi mathvariant="script">X</mi></semantics></math></inline-formula>, defined using the residuum operator of a continuous <i>t</i>-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 <i>t</i>-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.https://www.mdpi.com/2227-7390/8/9/1575fuzzy quasi-metricfuzzy partial metricadditive generatorresiduum operatorArchimedean <i>t</i>-norm |
| spellingShingle | Valentín Gregori Juan-José Miñana David Miravet A Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-Metrics Mathematics fuzzy quasi-metric fuzzy partial metric additive generator residuum operator Archimedean <i>t</i>-norm |
| title | A Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-Metrics |
| title_full | A Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-Metrics |
| title_fullStr | A Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-Metrics |
| title_full_unstemmed | A Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-Metrics |
| title_short | A Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-Metrics |
| title_sort | duality relationship between fuzzy partial metrics and fuzzy quasi metrics |
| topic | fuzzy quasi-metric fuzzy partial metric additive generator residuum operator Archimedean <i>t</i>-norm |
| url | https://www.mdpi.com/2227-7390/8/9/1575 |
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