Juan-José Miñana

Short communication: Some remarks on fuzzy contractive mappings

Recently a new concept of fuzzy contraction in fuzzy metric spaces in the sense of George and Veeramani (Wardowski, 2013 [5]) has been introduced. Here we provide some comments to this paper.

On completable fuzzy metric spaces

In this paper we construct a non-completable fuzzy metric space in the sense of George and Veeramani which allows to answer an open question related to continuity on the real parameter t. In addition, the constructed space is not strong (non-Archimedean).

Fuzzifying topology induced by a strong fuzzy metric

A construction of a fuzzifying topology induced by a strong fuzzy metric is presented. Properties of this fuzzifying topology, in particular, its convergence structure are studied. Our special interest is in the study of the relations between products of fuzzy metrics and the products of the induced fuzzifying topologies.

On probabilistic φ-contractions in Menger spaces

Abstract In this note we show that the fixed point theorems given for Menger spaces by J. Jachymski (2010) [5, Theorem 1] and J.X. Fang (2015) [2, Theorem 3.1] are equivalent .

What Is the Aggregation of a Partial Metric and a Quasi-metric?

Generalized metrics have been shown to be useful in many fields of Computer Science. In particular, partial metrics and quasi-metrics are used to develop quantitative mathematical models in denotational semantics and in asymptotic complexity analysis of algorithms, respectively. The aforesaid models are implemented independently and they are not related. However, it seems natural to consider a unique framework which remains valid for the applications to the both aforesaid fields. A first natural attempt to achieve that target suggests that the quantitative information should be obtained by means of the aggregation of a partial metric and a quasi-metric. Inspired by the preceding fact, we explore the way of merging, by means of a function, the aforementioned generalized metrics into a new one. We show that the induced generalized metric matches up with a partial quasi-metric. Thus, we characterize those functions that allow to generate partial quasi-metrics from the combination of a partial metric and a quasi-metric. Moreover, the relationship between the problem under consideration and the problems of merging partial metrics and quasi-metrics is discussed. Examples that illustrate the obtained results are also given.

A duality relationship between fuzzy metrics and metrics

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.

A technique for fuzzifying metric spaces via metric preserving mappings

Abstract In this paper we develop a new technique for constructing fuzzy metric spaces, in the sense of George and Veeramani, from metric spaces and by means of the Lukasievicz t -norm. In particular such a technique is based on the use of metric preserving functions in the sense of J. Dobos. Besides, the new generated fuzzy metric spaces are strong and completable, and if we add an extra condition, they are principal. Appropriate examples of such fuzzy metric spaces are given in order to illustrate the exposed technique.

On Indistinguishability Operators, Fuzzy Metrics and Modular Metrics

The notion of indistinguishability operator was introduced by Trillas, E. in 1982, with the aim of fuzzifying the crisp notion of equivalence relation. Such operators allow for measuring the similarity between objects when there is a limitation on the accuracy of the performed measurement or a certain degree of similarity can be only determined between the objects being compared. Since Trillas introduced such kind of operators, many authors have studied their properties and applications. In particular, an intensive research line is focused on the metric behavior of indistinguishability operators. Specifically, the existence of a duality between metrics and indistinguishability operators has been explored. In this direction, a technique to generate metrics from indistinguishability operators, and vice versa, has been developed by several authors in the literature. Nowadays, such a measurement of similarity is provided by the so-called fuzzy metrics when the degree of similarity between objects is measured relative to a parameter. The main purpose of this paper is to extend the notion of indistinguishability operator in such a way that the measurements of similarity are relative to a parameter and, thus, classical indistinguishability operators and fuzzy metrics can be retrieved as a particular case. Moreover, we discuss the relationship between the new operators and metrics. Concretely, we prove the existence of a duality between them and the so-called modular metrics, which provide a dissimilarity measurement between objects relative to a parameter. The new duality relationship allows us, on the one hand, to introduce a technique for generating the new indistinguishability operators from modular metrics and vice versa and, on the other hand, to derive, as a consequence, a technique for generating fuzzy metrics from modular metrics and vice versa. Furthermore, we yield examples that illustrate the new results.