A duality relationship between fuzzy metrics and metrics. Issue 6 (18th August 2018)

• Record Type: Journal Article
• Authors: Miñana, Juan-José; Valero, Oscar
• Is Part Of: International journal of general systems. Volume 47:Issue 6(2018)
• Page Start: 593
• Page End: 612
• Publication Date: 2018-08-18
• Subjects: Fuzzy (pseudo-)metric -- indistinguishability operator -- continuous Archimedean t-norm -- additive generator -- pseudo-inverse -- (pseudo-)metric; System theory -- Periodicals; 003
• DOI: 10.1080/03081079.2018.1487417
• Languages: English
• ISSNs: 0308-1079

Abstract

ABSTRACT: Based on the duality relationship between indistinguishability operators and (pseudo-)metrics, we address the problem of establishing whether there is a relationship between the last ones and fuzzy (pseudo-)metrics. We give a positive answer to the posed question. Concretely, we yield a method for generating fuzzy (pseudo-)metrics from (pseudo)-metrics and vice versa. The aforementioned methods involve the use of the pseudo-inverse of the additive generator of a continuous Archimedean t -norm. As a consequence, we get a method to generate non-strong fuzzy (pseudo-)metrics from (pseudo-)metrics. Examples that illustrate the exposed methods are also given. Finally, we show that the classical duality relationship between indistinguishability operators and (pseudo)-metrics can be retrieved as a particular case of our results when continuous Archimedean t -norms are under consideration.