On fuzzy uniformities induced by a fuzzy metric space

Abstract

Different types of fuzzy uniformities have been introduced in the literature standing out the notions due to Hutton, Höhle and Lowen. The main purpose of this paper is to study several methods to endow a fuzzy metric space (X, M, *), in the sense of George and Veeramani, with a probabilistic uniformity and with a Hutton [0, 1](-quasi)-uniformity. We will show the functorial behaviour of these constructions as well as its relation with respect to Lowen's functors and Katsaras's functors, which establish a relationship between the categories of probabilistic uniformities and Hutton [0, 1](-quasi)-uniformities with the category of classical uniformities respectively. Furthermore, we also study the fuzzy topologies associated with these fuzzy uniformities.

Fuzzy partial metric spaces

“…A (GV -)fuzzy metric M on a set X induces a topology T M on X, and this topology is metrizable (see George and Veeramani (1995); Gregori and Romaguera (2000)). Since both concepts of fuzzy metrics were defined, several authors have contributed to their study from the mathematical point of view (see, for instance, the following current references Gutiérrez-Gracía et al (2018); Miñana and Valero (2018); Shukla et al (2016), and references therein). Moreover, fuzzy metrics have been used successfully in engineering applications such as colour image filtering (see Camarena et al (2010), and references therein) and perceptual colour difference (see Grečova and Morillas (2016); Gregori et al (2012)).…”

Aggregation of L-Probabilistic Quasi-Uniformities

“…Since they are also isotone and maps L I to S then they * -aggregate bases of L- * -probabilistic quasi-uniformities on products and on sets. Some authors have studied how to construct adjoint functors between the category of probabilistic quasi-uniformities and the category of classical quasi-uniformities [29,39]. Although their results have been obtained for L-probabilistic quasi-uniformities when L = [0, 1], some of them, as the following one, are also valid when L is an arbitrary complete lattices.…”

“…Moreover, he established adjoint functors between the categories of Lowen uniformities and classical uniformities. These functors can also be established in the more general categories of probabilistic quasi-uniformities and crisp quasi-uniformities [39]. In particular, if (X, U, * ) is a probabilistic quasi-uniform space then:…”

Fuzzy Metrics in Terms of Fuzzy Relations

“…Currently, there is growing interest in exploring the topological properties of fuzzy metrics, as this line of study holds promise not only for theoretical constructions but also for fixed-point theorems and various practical applications. Regarding the investigation of the topological properties of classical fuzzy metrics, extensive references can be found in [9][10][11][12][13][14][15][16][17][18][19][20][21]. Although fuzzy metrics have demonstrated successful applications in image processing problems [22][23][24], their full potential remains untapped.…”