Special Issue "New Advances in Fuzzy Metric Spaces, Soft Metric Spaces, and Other Related Structures"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Fuzzy Set Theory".

Deadline for manuscript submissions: closed (31 December 2020).

Special Issue Editors

Prof. Emeritus Ad-Honoren Dr. Salvador Romaguera
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Guest Editor
Instituto Universitario de Matemática Pura y Aplicada-IUMPA, Universitat Politècnica de Valencia, 46022 Valencia, Spain
Interests: spaces from general topology with richer structures (metric spaces, quasi-metric spaces, and uniformities); fuzzy metric spaces; fixed point theory
Prof. Dr. Manuel Sanchis
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Guest Editor
Institut de Matemàtiques i Applications de Castelló (IMAC), Universitat Jaume I de Castelló, Av. Vicent Sos Baynat, s/n 12071 Castelló de la Plana, Spain
Interests: spaces from general topology with richer structures (metric spaces, quasi-metric spaces, and uniformities); fuzy metric spaces; fixed point theory; discrete dynamical systems; topological algebra

Special Issue Information

Dear Colleagues,

General topology constitutes a fundamental piece in the development of many branches both of Fuzzy mathematics and other related disciplines that have received a strong research impulse in recent years, such as, for instance, soft set theory. In particular, fuzzy metric spaces, in their different versions and variants, as well as several related structures (Menger spaces, fuzzy normed spaces, Hutton quasi-uniformities, etc.) are key tools in such developments.

The main purpose of this Issue is the publication of high-quality papers that gather new advances in the study of (non elementary) topological properties, as well as completeness, completion, convergence, construction of hyperspaces, etc., in the realm of fuzzy metric spaces, soft metric spaces, and related structures. New and significative contributions to fixed-point theory for these spaces are also welcome. Furthermore, non-elementary applications to topological algebra, functional analysis, ordinary and partial differential equations, computer science, etc., will be also considered.

Prof. Dr. Salvador Romaguera
Prof. Dr. Manuel Sanchis
Guest Editors

Manuscript Submission Information

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Keywords

  • Fuzzy (quasi-)metric spaces
  • Fuzzy (quasi-)uniformities
  • Fuzzy normed spaces
  • Soft metric spaces
  • Fixed-point theory
  • Fuzzy set theory
  • Soft set theory
  • Applications

Published Papers (21 papers)

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Open AccessArticle
Characterization of Soft S-Open Sets in Bi-Soft Topological Structure Concerning Crisp Points
Mathematics 2020, 8(12), 2100; https://doi.org/10.3390/math8122100 - 24 Nov 2020
Abstract
In this article, a soft s-open set in soft bitopological structures is introduced. With the help of this newly defined soft s-open set, soft separation axioms are regenerated in soft bitopological structures with respect to crisp points. Soft continuity at some certain points, [...] Read more.
In this article, a soft s-open set in soft bitopological structures is introduced. With the help of this newly defined soft s-open set, soft separation axioms are regenerated in soft bitopological structures with respect to crisp points. Soft continuity at some certain points, soft bases, soft subbase, soft homeomorphism, soft first-countable and soft second-countable, soft connected, soft disconnected and soft locally connected spaces are defined with respect to crisp points under s-open sets in soft bitopological spaces. The product of two soft  axioms with respect crisp points with almost all possibilities in soft bitopological spaces relative to semiopen sets are introduced. In addition to this, soft (countability, base, subbase, finite intersection property, continuity) are addressed with respect to semiopen sets in soft bitopological spaces. Product of soft first and second coordinate spaces are addressed with respect to semiopen sets in soft bitopological spaces. The characterization of soft separation axioms with soft connectedness is addressed with respect to semiopen sets in soft bitopological spaces. In addition to this, the product of two soft topological spaces is (  space if each coordinate space is soft  space, product of two sot topological spaces is (S regular and C regular) space if each coordinate space is (S regular and C regular), the product of two soft topological spaces is connected if each coordinate space is soft connected and the product of two soft topological spaces is (first-countable, second-countable) if each coordinate space is (first countable, second-countable). Full article
Open AccessArticle
Aggregation of L-Probabilistic Quasi-Uniformities
Mathematics 2020, 8(11), 1980; https://doi.org/10.3390/math8111980 - 06 Nov 2020
Abstract
The problem of aggregating fuzzy structures, mainly fuzzy binary relations, has deserved a lot of attention in the last years due to its application in several fields. Here, we face the problem of studying which properties must satisfy a function in order to [...] Read more.
The problem of aggregating fuzzy structures, mainly fuzzy binary relations, has deserved a lot of attention in the last years due to its application in several fields. Here, we face the problem of studying which properties must satisfy a function in order to merge an arbitrary family of (bases of) L-probabilistic quasi-uniformities into a single one. These fuzzy structures are special filters of fuzzy binary relations. Hence we first make a complete study of functions between partially-ordered sets that preserve some special sets, such as filters. Afterwards, a complete characterization of those functions aggregating bases of L-probabilistic quasi-uniformities is obtained. In particular, attention is paid to the case L={0,1}, which allows one to obtain results for functions which aggregate crisp quasi-uniformities. Moreover, we provide some examples of our results including one showing that Lowen’s functor ι which transforms a probabilistic quasi-uniformity into a crisp quasi-uniformity can be constructed using this aggregation procedure. Full article
Open AccessArticle
A Notion of Convergence in Fuzzy Partially Ordered Sets
Mathematics 2020, 8(11), 1958; https://doi.org/10.3390/math8111958 - 05 Nov 2020
Abstract
The notion of sequential convergence in fuzzy partially ordered sets, under the name oF-convergence, is well known. Our aim in this paper is to introduce and study a notion of net convergence, with respect to the fuzzy order relation, named o [...] Read more.
The notion of sequential convergence in fuzzy partially ordered sets, under the name oF-convergence, is well known. Our aim in this paper is to introduce and study a notion of net convergence, with respect to the fuzzy order relation, named o-convergence, which generalizes the former notion and is also closer to our sense of the classic concept of "convergence". The main result of this article is that the two notions of convergence are identical in the area of complete F-lattices. Full article
Open AccessArticle
w-Distances on Fuzzy Metric Spaces and Fixed Points
Mathematics 2020, 8(11), 1909; https://doi.org/10.3390/math8111909 - 31 Oct 2020
Abstract
We propose a notion of w-distance for fuzzy metric spaces, in the sense of Kramosil and Michalek, which allows us to obtain a characterization of complete fuzzy metric spaces via a suitable fixed point theorem that is proved here. Our main result [...] Read more.
We propose a notion of w-distance for fuzzy metric spaces, in the sense of Kramosil and Michalek, which allows us to obtain a characterization of complete fuzzy metric spaces via a suitable fixed point theorem that is proved here. Our main result provides a fuzzy counterpart of a renowned characterization of complete metric spaces due to Suzuki and Takahashi. Full article
Open AccessArticle
Monad Metrizable Space
Mathematics 2020, 8(11), 1891; https://doi.org/10.3390/math8111891 - 31 Oct 2020
Cited by 1
Abstract
Do the topologies of each dimension have to be same and metrizable for metricization of any space? I show that this is not necessary with monad metrizable spaces. For example, a monad metrizable space may have got any indiscrete topologies, discrete topologies, different [...] Read more.
Do the topologies of each dimension have to be same and metrizable for metricization of any space? I show that this is not necessary with monad metrizable spaces. For example, a monad metrizable space may have got any indiscrete topologies, discrete topologies, different metric spaces, or any topological spaces in each different dimension. I compute the distance in real space between such topologies. First, the passing points between different topologies is defined and then a monad metric is defined. Then I provide definitions and some properties about monad metrizable spaces and PAS metric spaces. I show that any PAS metric space is also a monad metrizable space. Moreover, some properties and some examples about them are presented. Full article
Open AccessArticle
Transitivity in Fuzzy Hyperspaces
Mathematics 2020, 8(11), 1862; https://doi.org/10.3390/math8111862 - 24 Oct 2020
Abstract
Given a metric space (X,d), we deal with a classical problem in the theory of hyperspaces: how some important dynamical properties (namely, weakly mixing, transitivity and point-transitivity) between a discrete dynamical system f:(X,d [...] Read more.
Given a metric space (X,d), we deal with a classical problem in the theory of hyperspaces: how some important dynamical properties (namely, weakly mixing, transitivity and point-transitivity) between a discrete dynamical system f:(X,d)(X,d) and its natural extension to the hyperspace are related. In this context, we consider the Zadeh’s extension f^ of f to F(X), the family of all normal fuzzy sets on X, i.e., the hyperspace F(X) of all upper semicontinuous fuzzy sets on X with compact supports and non-empty levels and we endow F(X) with different metrics: the supremum metric d, the Skorokhod metric d0, the sendograph metric dS and the endograph metric dE. Among other things, the following results are presented: (1) If (X,d) is a metric space, then the following conditions are equivalent: (a) (X,f) is weakly mixing, (b) ((F(X),d),f^) is transitive, (c) ((F(X),d0),f^) is transitive and (d) ((F(X),dS)),f^) is transitive, (2) if f:(X,d)(X,d) is a continuous function, then the following hold: (a) if ((F(X),dS),f^) is transitive, then ((F(X),dE),f^) is transitive, (b) if ((F(X),dS),f^) is transitive, then (X,f) is transitive; and (3) if (X,d) be a complete metric space, then the following conditions are equivalent: (a) (X×X,f×f) is point-transitive and (b) ((F(X),d0) is point-transitive. Full article
Open AccessArticle
Classification of Complex Fuzzy Numbers and Fuzzy Inner Products
Mathematics 2020, 8(9), 1626; https://doi.org/10.3390/math8091626 - 20 Sep 2020
Abstract
The paper is concerned with complex fuzzy numbers and complex fuzzy inner product spaces. In the classical complex number set, a complex number can be expressed using the Cartesian form or polar form. Both expressions are needed because one expression is better than [...] Read more.
The paper is concerned with complex fuzzy numbers and complex fuzzy inner product spaces. In the classical complex number set, a complex number can be expressed using the Cartesian form or polar form. Both expressions are needed because one expression is better than the other depending on the situation. Likewise, the Cartesian form and the polar form can be defined in a complex fuzzy number set. First, the complex fuzzy numbers (CFNs) are categorized into two types, the polar form and the Cartesian form, as type I and type II. The properties of the complex fuzzy number set of those two expressions are discussed, and how the expressions can be used practically is shown through an example. Second, we study the complex fuzzy inner product structure in each category and find the non-existence of an inner product on CFNs of type I. Several properties of the fuzzy inner product space for type II are proposed from the modulus that is newly defined. Specfically, the Cauchy-Schwartz inequality for type II is proven in a compact way, not only the one for fuzzy real numbers. In fact, it was already discussed by Hasanhani et al; however, they proved every case in a very complicated way. In this paper, we prove the Cauchy-Schwartz inequality in a much simpler way from a general point of view. Finally, we introduce a complex fuzzy scalar product for the generalization of a complex fuzzy inner product and propose to study the condition for its existence on CFNs of type I. Full article
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Open AccessArticle
A Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-Metrics
Mathematics 2020, 8(9), 1575; https://doi.org/10.3390/math8091575 - 12 Sep 2020
Abstract
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 [...] Read more.
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. Full article
Open AccessArticle
Soft Topological Transformation Groups
Mathematics 2020, 8(9), 1545; https://doi.org/10.3390/math8091545 - 10 Sep 2020
Abstract
The aim of the present study is to introduce the concept of soft topological transformation groups by examining the topological transformation groups, which are the core subject of algebraic topology under the soft approach. Actions of soft topological groups on soft topological spaces [...] Read more.
The aim of the present study is to introduce the concept of soft topological transformation groups by examining the topological transformation groups, which are the core subject of algebraic topology under the soft approach. Actions of soft topological groups on soft topological spaces are studied, and the category of soft topological transformation groups is constructed. Also, a translation and conjugation of the soft topological groups are described. Finally, the definitions of soft orbit spaces and soft homogeneous spaces are given, and some of the properties of these concepts are examined in detail. Full article
Open AccessArticle
A Discussion on p-Geraghty Contraction on mw-Quasi-Metric Spaces
Mathematics 2020, 8(9), 1437; https://doi.org/10.3390/math8091437 - 27 Aug 2020
Cited by 1
Abstract
In this paper we consider a kind of Geraghty contractions by using mw-distances in the setting of complete quasi-metric spaces. We provide fixed point theorems for this type of mappings and illustrate with some examples the results obtained. Full article
Open AccessArticle
Completeness in Quasi-Pseudometric Spaces—A Survey
Mathematics 2020, 8(8), 1279; https://doi.org/10.3390/math8081279 - 03 Aug 2020
Abstract
The aim of this paper is to discuss the relations between various notions of sequential completeness and the corresponding notions of completeness by nets or by filters in the setting of quasi-metric spaces. We propose a new definition of right K-Cauchy net [...] Read more.
The aim of this paper is to discuss the relations between various notions of sequential completeness and the corresponding notions of completeness by nets or by filters in the setting of quasi-metric spaces. We propose a new definition of right K-Cauchy net in a quasi-metric space for which the corresponding completeness is equivalent to the sequential completeness. In this way we complete some results of R. A. Stoltenberg, Proc. London Math. Soc. 17 (1967), 226–240, and V. Gregori and J. Ferrer, Proc. Lond. Math. Soc., III Ser., 49 (1984), 36. A discussion on nets defined over ordered or pre-ordered directed sets is also included. Full article
Open AccessArticle
On Metric-Type Spaces Based on Extended T-Conorms
Mathematics 2020, 8(7), 1097; https://doi.org/10.3390/math8071097 - 05 Jul 2020
Cited by 1
Abstract
Kirk and Shahzad introduced the class of strong b-metric spaces lying between the class of b-metric spaces and the class of metric spaces. As compared with b-metric spaces, strong b-metric spaces have the advantage that open balls are open in the induced topology [...] Read more.
Kirk and Shahzad introduced the class of strong b-metric spaces lying between the class of b-metric spaces and the class of metric spaces. As compared with b-metric spaces, strong b-metric spaces have the advantage that open balls are open in the induced topology and, hence, they have many properties that are similar to the properties of classic metric spaces. Having noticed the advantages of strong b-metric spaces Kirk and Shahzad complained about the absence of non-trivial examples of such spaces. It is the main aim of this paper to construct a series of strong b-metric spaces that fail to be metric. Realizing this programme, we found it reasonable to consider these metric-type spaces in the context when the ordinary sum operation is replaced by operation ⊕, where ⊕ is an extended t-conorm satisfying certain conditions. Full article
Open AccessArticle
Fuzzy Stability Results of Finite Variable Additive Functional Equation: Direct and Fixed Point Methods
Mathematics 2020, 8(7), 1050; https://doi.org/10.3390/math8071050 - 30 Jun 2020
Cited by 3
Abstract
In this current work, we introduce the finite variable additive functional equation and we derive its solution. In fact, we investigate the Hyers–Ulam stability results for the finite variable additive functional equation in fuzzy normed space by two different approaches of direct and [...] Read more.
In this current work, we introduce the finite variable additive functional equation and we derive its solution. In fact, we investigate the Hyers–Ulam stability results for the finite variable additive functional equation in fuzzy normed space by two different approaches of direct and fixed point methods. Full article
Open AccessArticle
Some New Fuzzy Fixed Point Results with Applications
Mathematics 2020, 8(6), 995; https://doi.org/10.3390/math8060995 - 17 Jun 2020
Abstract
The aim of this article is to establish some fixed point results for fuzzy mappings and derive some corresponding multivalued mappings results of literature. For this purpose, we define some new and generalized contractions in the setting of b-metric spaces. As applications, [...] Read more.
The aim of this article is to establish some fixed point results for fuzzy mappings and derive some corresponding multivalued mappings results of literature. For this purpose, we define some new and generalized contractions in the setting of b-metric spaces. As applications, we find solutions of integral inclusions by our obtained results. Full article
Open AccessArticle
A Characterization of Strong Completeness in Fuzzy Metric Spaces
Mathematics 2020, 8(6), 861; https://doi.org/10.3390/math8060861 - 26 May 2020
Abstract
Here, we deal with the concept of fuzzy metric space (X,M,), due to George and Veeramani. Based on the fuzzy diameter for a subset of X, we introduce the notion of strong fuzzy diameter zero [...] Read more.
Here, we deal with the concept of fuzzy metric space ( X , M , ) , due to George and Veeramani. Based on the fuzzy diameter for a subset of X , we introduce the notion of strong fuzzy diameter zero for a family of subsets. Then, we characterize nested sequences of subsets having strong fuzzy diameter zero using their fuzzy diameter. Examples of sequences of subsets which do or do not have strong fuzzy diameter zero are provided. Our main result is the following characterization: a fuzzy metric space is strongly complete if and only if every nested sequence of close subsets which has strong fuzzy diameter zero has a singleton intersection. Moreover, the standard fuzzy metric is studied as a particular case. Finally, this work points out a route of research in fuzzy fixed point theory. Full article
Open AccessArticle
On Hybrid Contractions in the Context of Quasi-Metric Spaces
Mathematics 2020, 8(5), 675; https://doi.org/10.3390/math8050675 - 29 Apr 2020
Cited by 2
Abstract
In this manuscript, we will investigate the existence of fixed points for mappings that satisfy some hybrid type contraction conditions in the setting of quasi-metric spaces. We provide examples to assure the validity of the given results. The results of this paper generalize [...] Read more.
In this manuscript, we will investigate the existence of fixed points for mappings that satisfy some hybrid type contraction conditions in the setting of quasi-metric spaces. We provide examples to assure the validity of the given results. The results of this paper generalize several known theorems in the recent literature. Full article
Open AccessArticle
Soft Open Bases and a Novel Construction of Soft Topologies from Bases for Topologies
Mathematics 2020, 8(5), 672; https://doi.org/10.3390/math8050672 - 29 Apr 2020
Cited by 4
Abstract
Soft topology studies a structure on the collection of all soft sets on a given set of alternatives (the relevant attributes being fixed). It is directly inspired by the axioms of a topological space. This paper contributes to the theoretical bases of soft [...] Read more.
Soft topology studies a structure on the collection of all soft sets on a given set of alternatives (the relevant attributes being fixed). It is directly inspired by the axioms of a topological space. This paper contributes to the theoretical bases of soft topology in various ways. We extend a general construction of soft topologies from topologies on the set of alternatives in two different directions. An extensive discussion with criteria about what a soft counterpart of “topological separability” should satisfy is also given. The interactions of the properties that arise with separability, and of second-countability and its soft counterpart, are studied under the general mechanisms that generate soft topological spaces. The first non-trivial examples of soft second-countable soft topological spaces are produced as a consequence. Full article
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Open AccessArticle
T-Equivalences: The Metric Behavior Revisited
Mathematics 2020, 8(4), 495; https://doi.org/10.3390/math8040495 - 02 Apr 2020
Cited by 1
Abstract
Since the notion of T-equivalence, where T is a t-norm, was introduced as a fuzzy generalization of the notion of crisp equivalence relation, many researchers have worked in the study of the metric behavior of such fuzzy relations. Concretely, a few techniques [...] Read more.
Since the notion of T-equivalence, where T is a t-norm, was introduced as a fuzzy generalization of the notion of crisp equivalence relation, many researchers have worked in the study of the metric behavior of such fuzzy relations. Concretely, a few techniques to induce metrics from T-equivalences, and vice versa, have been developed. In several fields of computer science and artificial intelligence, a generalization of pseudo-metric, known as partial pseudo-metrics, have shown to be useful. Recently, Bukatin, Kopperman and Matthews have stated that the notion of partial pseudo-metric and a type of generalized T-equivalence are linked. Inspired by the preceding fact, in this paper, we state a concrete relationship between partial pseudo-metrics and the aforesaid generalized T-equivalences. Specifically, a method for constructing partial pseudo-metrics from the new type of T-equivalences and, reciprocally, for constructing the generalized T-equivalences from partial pseudo-metrics are provided. However, important differences between the new approach and the classical one are established. Special interest is paid to the case in which the minimum, drastic, and Łukasiewicz t-norms are under consideration. Full article
Open AccessFeature PaperArticle
Characterizing Complete Fuzzy Metric Spaces Via Fixed Point Results
Mathematics 2020, 8(2), 273; https://doi.org/10.3390/math8020273 - 18 Feb 2020
Cited by 3
Abstract
With the help of C-contractions having a fixed point, we obtain a characterization of complete fuzzy metric spaces, in the sense of Kramosil and Michalek, that extends the classical theorem of H. Hu (see “Am. Math. Month. 1967, 74, 436–437”) that a [...] Read more.
With the help of C-contractions having a fixed point, we obtain a characterization of complete fuzzy metric spaces, in the sense of Kramosil and Michalek, that extends the classical theorem of H. Hu (see “Am. Math. Month. 1967, 74, 436–437”) that a metric space is complete if and only if any Banach contraction on any of its closed subsets has a fixed point. We apply our main result to deduce that a well-known fixed point theorem due to D. Mihet (see “Fixed Point Theory 2005, 6, 71–78”) also allows us to characterize the fuzzy metric completeness. Full article
Open AccessArticle
A Note on Extended Z-Contraction
Mathematics 2020, 8(2), 195; https://doi.org/10.3390/math8020195 - 05 Feb 2020
Cited by 12
Abstract
In this article, we aim to evaluate and merge the as-scattered-as-possible results in fixed point theory from a general framework. In particular, we considered a common fixed point theorem via extended Z-contraction with respect to ψ-simulation function over an auxiliary function [...] Read more.
In this article, we aim to evaluate and merge the as-scattered-as-possible results in fixed point theory from a general framework. In particular, we considered a common fixed point theorem via extended Z-contraction with respect to ψ -simulation function over an auxiliary function ξ in the setting of b-metric space. We investigated both the existence and uniqueness of common fixed points of such mappings. We used an example to illustrate the main result observed. Our main results cover several existing results in the corresponding literature. Full article
Open AccessArticle
A Study of Approximation Properties in Felbin-Fuzzy Normed Spaces
Mathematics 2020, 8(2), 161; https://doi.org/10.3390/math8020161 - 23 Jan 2020
Cited by 1
Abstract
In this paper, approximation properties in Felbin-fuzzy normed spaces are studied. These approximation properties have been recently introduced in Felbin-fuzzy normed spaces. We make topological tools to analyze such approximation properties. We especially develop the representation of dual spaces related to our contexts. [...] Read more.
In this paper, approximation properties in Felbin-fuzzy normed spaces are studied. These approximation properties have been recently introduced in Felbin-fuzzy normed spaces. We make topological tools to analyze such approximation properties. We especially develop the representation of dual spaces related to our contexts. By using this representation, we establish characterizations of approximation properties in terms of infinite sequences. Finally, we provide dual problems for approximation properties and their results in our contexts. Full article
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