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Rough sets (RSs) and fuzzy sets (FSs) are designed to tackle the uncertainty in the data. By taking into account the control or reference parameters, the linear Diophantine fuzzy set (LD-FS) is a novel approach to decision making (DM), broadens the previously dominant theories of the intuitionistic fuzzy set (IFS), Pythagorean fuzzy set (PyFS), and q-rung orthopair fuzzy set (q-ROFS), and allows for a more flexible representation of uncertain data. A promising avenue for RS theory is to investigate RSs within the context of LD-FS, where LD-FSs are approximated by an intuitionistic fuzzy relation (IFR). The major goal of this article is to create a novel method of roughness for LD-FSs employing an IFR over dual universes. The notions of lower and upper approximations of an LD-FS are established by using an IFR, and some axiomatic systems are carefully investigated in detail. Moreover, a link between LD-FRSs and linear Diophantine fuzzy topology (LDF-topology) has been established. Eventually, based on lower and upper approximations of an LD-FS, several similarity relations are investigated. Meanwhile, we apply the recommended model of LD-FRSs over dual universes for solving the DM problem. Furthermore, a real-life case study is given to demonstrate the practicality and feasibility of our designed approach. Finally, we conduct a detailed comparative analysis with certain existing methods to explore the effectiveness and superiority of the established technique. [ABSTRACT FROM AUTHOR]

Manifold learning-based approaches have emerged as prominent techniques for dimensionality reduction. Among these methods, t-Distributed Stochastic Neighbor Embedding (t-SNE) and Uniform Manifold Approximation and Projection (UMAP) stand out as two of the most widely used and effective approaches. While both methods share similar underlying procedures, empirical observations indicate two distinctive properties: global data structure preservation and computational efficiency. However, the underlying mathematical principles behind these distinctions remain elusive. To address this gap, this study presents a comparative analysis of the subprocesses involved in these methods, aiming to elucidate the mathematical mechanisms underlying the observed distinctions. By meticulously examining the equation formulations, the mathematical mechanisms contributing to global data structure preservation and computational efficiency are elucidated. To validate the theoretical analysis, data are collected through a laboratory experiment, and an open-source dataset is utilized for validation across different datasets. The consistent alignment of results obtained from both balanced and unbalanced datasets robustly confirms the study's findings. The insights gained from this study provide a deeper understanding of the mathematical underpinnings of t-SNE and UMAP, enabling more informed and effective use of these dimensionality reduction techniques in various applications, such as anomaly detection, natural language processing, and bioinformatics. [ABSTRACT FROM AUTHOR]

Since the 1970s and 1980s, significant contributions have been made by Weiss, Butnariu, Heilpern, Chitra, Subrahmanyam, and others, extending fixed-point theorems to fuzzy mappings and topological spaces. This paper provides two generalizations of two important fixed-point theorems, one provided by Butnariu and the other provided by Chitra. The first generalization ensures that, under certain conditions, an acyclic fuzzy mapping has a fixed point. The second result ensures the existence of a point in the intersection of two or more fuzzy mappings considering contractible finite dimensional ANR spaces, which is a generalization of the statement provided by Chitra. [ABSTRACT FROM AUTHOR]

In this paper, an attempt is made to define a Type-2 fuzzy metric on a nonempty set X by allowing it to take fuzzy numbers as values of distance of a pair of points under a membership grade which is also a fuzzy number. Its type-1 counterpart is a fuzzy metric mostly similar to those defined by Kramosil and Michalek c i t e 1 0 and also by George and Veeramani c i t e 6. It is shown that the topology induced by this fuzzy metric is Hausdorff in nature. In this type of fuzzy metric space, Banach's fixed point theorem and Edelstein's fixed point theorems are extended. Finally decomposition theorems for this fuzzy metric are proved from which a justification of type-2 behavior of this fuzzy metric can also be interpreted. [ABSTRACT FROM AUTHOR]

In this paper, for fuzzy metric spaces we will attempt to demonstrate some novel fixed-point theorems that is common. To support our findings with evidence, we employ the concept of compatibility. The previous review found several useful mutual fixedpoint theorems for metric and fuzzy metric spaces. Our results greatly expand and strengthen these theorems. Reconsideration of fuzzy metric spaces is done by this study so that fuzzy scalars are utilized to construct fuzzy metric spaces rather than real numbers or fuzzy numbers. In contrast to the preceding work, which defined fuzzy metric spaces using fuzzy or real numbers, this definition uses fuzzy numbers. It has been demonstrated that any complete regular metric space can, under certain conditions, complete fuzzy metric space are given. In addition, we provide evidence that the fuzzy metric spaces generate fuzzy topology that is specified in this study and is consistent with previously presented fuzzy topologies. In the coming years, these findings will pave the way for additional research into imperfect optimization and pattern recognition. [ABSTRACT FROM AUTHOR]

The role of fuzzy 훿-open set is highly significant in the study of fuzzy topology initiated by Ganguly and Saha [S. Ganguly and S. Saha, A note on 훿-continuity and 훿-connected sets in fuzzy set theory, Simon Stevin62 (1988), 2, 127–141]. This article begins with the introduction of 훿-ℐ-open covers in a mixed fuzzy ideal topological space. After that, we introduce 훿-ℐ-compactness and then some properties of its are discussed therein. It is shown that the aforesaid compactness is the weaker form of fuzzy compactness. Moreover, we show that if we retopologize the fuzzy topology then in the new environment fuzzy 훿-ℐ-compactness and fuzzy compactness are equivalent. In addition, we introduce two different notions of continuity and investigate the behavior between fuzzy 훿-ℐ-compactness and fuzzy compactness. [ABSTRACT FROM AUTHOR]

The goal of this paper is to examine various neutrosophic topological spaces and to learn about the differences between them. This paper provides the list of application of Neutrosophic topology in diverse fields like computer science, operational research, artificial intelligence and many sciences field. The importance of neutrosophic topological space in real time applications are listed in this article. [ABSTRACT FROM AUTHOR]

In this paper we have introduced intuitionistic fuzzy semi γ* generalized irresolute mappings and investigated some of their properties. Also we have provided some characterization of intuitionistic fuzzy semi γ* generalized irresolute mappings. [ABSTRACT FROM AUTHOR]

Fuzzy β-covering(Fβ-C) plays a key role in processing real-valued data sets and covering plays an important role in the topological spaces. Thus they have attracted much attention. But the relationship between Fβ-C and topology has not been studied. This inspires the research of Fβ-C from the perspective of topology. In this paper, we construct Fβ-C rough continuous and homeomorphism mappings by using Fβ-C operator. We not only obtain some equivalent descriptions of the mappings but also profoundly reveal the relationship of two Fβ-C approximation spaces. We give the classification method of Fβ-C approximation spaces with the help of homeomorphism mapping, propose a new method to construct topology induced by Fβ-C operator and investigate the properties in the topological spaces further. Finally, we obtain the necessary and sufficient conditions for Fβ-C operators to be topological closure operators. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce and discuss a new set of separation properties in fuzzy soft topological spaces called F S δ -separation and F S δ -regularity axioms by using fuzzy soft δ -open sets and the quasi-coincident relation. We provide a comprehensive study of their properties with some supporting examples. Our analysis includes more characterizations, results, and theorems related to these notions, which contributes to a deeper understanding of fuzzy soft separability properties. We show that the F S δ -separation and F S δ -regularity axioms are harmonic and heredity property. Additionally, we examine the connections between F S δ ∗ -compactness and F S δ -separation axioms and explore the relationships between them. Overall, this work offers a new perspective on the theory of separation properties in fuzzy soft topological spaces, as well as provides a robust foundation for further research in the transmission of properties from fuzzy soft topologies to fuzzy and soft topologies and vice-versa by swapping between the membership function and characteristic function in the case of fuzzy topology and the set of parameters and a singleton set in the case of soft topology. In the present paper, we introduce and discuss a new set of separation properties in fuzzy soft topological spaces called -separation and -regularity axioms by using fuzzy soft -open sets and the quasi-coincident relation. We provide a comprehensive study of their properties with some supporting examples. Our analysis includes more characterizations, results, and theorems related to these notions, which contributes to a deeper understanding of fuzzy soft separability properties. We show that the -separation and -regularity axioms are harmonic and heredity property. Additionally, we examine the connections between -compactness and -separation axioms and explore the relationships between them. Overall, this work offers a new perspective on the theory of separation properties in fuzzy soft topological spaces, as well as provides a robust foundation for further research in the transmission of properties from fuzzy soft topologies to fuzzy and soft topologies and vice-versa by swapping between the membership function and characteristic function in the case of fuzzy topology and the set of parameters and a singleton set in the case of soft topology. [ABSTRACT FROM AUTHOR]

This paper's aim is to extend and generalize the classic results regarding spectral radii and the corresponding resolvent sets for some different classes of bounded operators acting on fuzzy normed spaces. In this context, the fuzzy norm definition introduced giving shape to a new topology for a fuzzy space, namely a fuzzy topology, also gives the opportunity to study the behavior of various types of operators defined between fuzzy normed spaces, along with their spectral properties. There are several definitions for resolvent sets and consequently, several corresponding definitions of spectral radii that will be considered in this work, since these are non-equivalent ways of defining such notions. Spectral radii are calculated for a fuzzy Volterra type operator acting between fuzzy normed spaces. [ABSTRACT FROM AUTHOR]

The aim of this study is to provide neighborhood structures in bipolar fuzzy supra topological space and to show the applicability of bipolar fuzzy supra topology to the medical diagnosis problem. Firstly, we give some properties related to bipolar fuzzy points and their neighborhood structure in bipolar fuzzy supra topological spaces. Then, we consider another important structure, "quasi-coincident", in the case of bipolar fuzzy points and bipolar fuzzy sets. Then, we introduce the corresponding neighborhood structure called "Q-neighborhood system" by using the quasi-coincident relations. Furthermore, we also investigate the characterization of bipolar fuzzy supra topological space in terms of quasi-neighborhoods. Finally, we present a new method to solve medical diagnosis problems by using the bipolar fuzzy score function. [ABSTRACT FROM AUTHOR]

This article investigates the fully distributed secure observation and consensus problem for networked multi-agent systems (MASs) with uncertain communication topology. The complexities of dual-channel false data injection (FDI) attacks and fuzzy communication among agents are considered, bringing direct challenges to the acquisition of system states and the design of consensus protocols. To address these difficulties, on the one hand, adaptive coupling weights that vary with observation and consensus errors are neatly designed to avoid the use of global topology information in the whole mechanism. On the other hand, an auxiliary observation system is constructed based on intermediate variables to realize the simultaneous estimation of system states and dual-channel FDI attacks. After that, a distributed attack compensation controller that can guarantee secure consensus among agents is proposed. Finally, a simulation example and an experiment compared with existing results are given to examine the feasibility and advantages of the developed strategy. • The fully distributed secure observation and consensus mechanism is proposed. • Real-time estimations of system states and more general FDI attacks are achieved. • Constraints on the membership functions of the fuzzy topology model are relaxed. [ABSTRACT FROM AUTHOR]

Fuzzy rough set models are useful tools for dealing with fuzzy and real-valued data. They have been used in many real-world applications. In this paper, we investigate the fuzzy rough set model based on triangular norms and fuzzy implications. First, we extend some results in the published literature by removing the condition that is the continuity of triangular norms, and obtain more general conclusion about fuzzy upper approximation operators. Then, for the fuzzy neighborhood and the fuzzy lower approximation operator based on fuzzy implications, we investigate their characterization with each other. Finally, we establish the relationships between fuzzy rough sets and fuzzy topology. In this work, researches on the properties of fuzzy rough sets based on triangular norms which need not be continuous provide generalization results for fuzzy rough set theory from viewpoint of mathematics. [ABSTRACT FROM AUTHOR]

P.M. Pu and Y.M. Liu extended Moore-Smith's convergence of nets to fuzzy topology and Y.M. Liu provided analogous results to J. Kelley's classical characterization theorem of net convergence by introducing the notion of fuzzy convergence classes. In a previous paper, the authors of this study provided modified versions of this characterization by using an alternative notion of convergence of fuzzy nets, introduced by B.M.U. Afsan, named fuzzy net ideal convergence. Our main scope here is to generalize and simplify the preceding results. Specifically, we insert the concept of a fuzzy function ideal convergence class, L , on a non-empty set, X, consisting of triads (f , e , I) , where f is a function from a non-empty set, D, to the set FP (X) of fuzzy points in X, which we call fuzzy function, e ∈ FP (X) , and I is a proper ideal on D, and we provide necessary and sufficient conditions to establish the existence of a unique fuzzy topology, δ , on X, such that (f , e , I) ∈ L iff f I -converges to e, relative to the fuzzy topology δ. [ABSTRACT FROM AUTHOR]

Spatial objects in a geographical information system are often vague. Fuzzy topology-based intersection methods were introduced to obtain the topological relation between such vague objects by considering the objects as fuzzy sets. The elements of the intersection matrices express the nature of the intersections between topological parts of the objects. In reality, it may only be possible to calculate some of the required intersections due to the unavailability of data. This paper introduces a method to construct a fuzzy 9-intersection matrix under incomplete observations. Further, it computes the possible fuzzy topological relation using a quantitative fuzzy 9-intersection method based on fuzzy similarity measure. Some special cases of the incomplete observations and fuzzy topological relation under such situations are discussed. Finally, few applications are presented, where the proposed method is used to obtain relation between two uncertain objects under the condition that the topological properties of the objects are partially known. [ABSTRACT FROM AUTHOR]

In this paper, we introduce three new structures of fuzzy group sets and three new structures of fuzzy group topological spaces and we study the relation among them. [ABSTRACT FROM AUTHOR]

In this paper, a new approach for fuzzy gyronorms on gyrogroups is presented. The relations between fuzzy metrics(in the sense of Morsi), fuzzy gyronorms, gyronorms on gyrogroups are studied. Also, some sufficient conditions, which can make a fuzzy normed gyrogroup to be a topological gyrogroup and a fuzzy topological gyrogroup, are found. Meanwhile, the relations between topological gyrogroups, fuzzy topological gyrogroups and stratified fuzzy topological gyrogroups are studied. Finally, the properties of fuzzifying topological gyrogroups are studied. [ABSTRACT FROM AUTHOR]

Manifold learning-based approaches have emerged as prominent techniques for dimensionality reduction. Among these methods, t-Distributed Stochastic Neighbor Embedding (t-SNE) and Uniform Manifold Approximation and Projection (UMAP) stand out as two of the most widely used and effective approaches. While both methods share similar underlying procedures, empirical observations indicate two distinctive properties: global data structure preservation and computational efficiency. However, the underlying mathematical principles behind these distinctions remain elusive. To address this gap, this study presents a comparative analysis of the subprocesses involved in these methods, aiming to elucidate the mathematical mechanisms underlying the observed distinctions. By meticulously examining the equation formulations, the mathematical mechanisms contributing to global data structure preservation and computational efficiency are elucidated. To validate the theoretical analysis, data are collected through a laboratory experiment, and an open-source dataset is utilized for validation across different datasets. The consistent alignment of results obtained from both balanced and unbalanced datasets robustly confirms the study’s findings. The insights gained from this study provide a deeper understanding of the mathematical underpinnings of t-SNE and UMAP, enabling more informed and effective use of these dimensionality reduction techniques in various applications, such as anomaly detection, natural language processing, and bioinformatics.

Since the 1970s and 1980s, significant contributions have been made by Weiss, Butnariu, Heilpern, Chitra, Subrahmanyam, and others, extending fixed-point theorems to fuzzy mappings and topological spaces. This paper provides two generalizations of two important fixed-point theorems, one provided by Butnariu and the other provided by Chitra. The first generalization ensures that, under certain conditions, an acyclic fuzzy mapping has a fixed point. The second result ensures the existence of a point in the intersection of two or more fuzzy mappings considering contractible finite dimensional ANR spaces, which is a generalization of the statement provided by Chitra.

In the category of ⊤-quasi-Cauchy spaces, completeness and completion can be studied in a non-symmetric framework encompassing ⊤-quasi-uniform (limit) spaces. Based on constructions by E.E. Reed in the category of Cauchy spaces and, recently, by L. Reid and G. Richardson in the category of ⊤-Cauchy spaces, we give a family of completions for a non-complete ⊤-quasi Cauchy space. As particular instances we study pretopological and topological completions of ⊤-quasi-Cauchy spaces. [ABSTRACT FROM AUTHOR]

It is common knowledge that fuzzy topology contributes to developing techniques to address real-life applications in various areas like information systems and optimal choices. The building blocks of fuzzy topology are fuzzy open sets, but other extended families of fuzzy open sets, like fuzzy pre-open sets, can contribute to the growth of fuzzy topology. In the present work, we create some classifications of fuzzy topologies which enable us to obtain several desirable features and relationships. At first, we introduce and analyze stronger forms of fuzzy pre-separation and regularity properties in fuzzy topology called fuzzy pre- T i , i = 0 , 1 2 , 1 , 2 , 3 , 4 , fuzzy pre-symmetric, and fuzzy pre- R i , i = 0 , 1 , 2 , 3 by utilizing the concepts of fuzzy pre-open sets and quasi-coincident relation. We investigate more novel properties of these classes and uncover their unique characteristics. By presenting a wide array of related theorems and interconnections, we structure a comprehensive framework for understanding these classes and interrelationships with other separation axioms in this setting. Moreover, the relations between these classes and those in some induced topological structures are examined. Additionally, we explore the hereditary and harmonic properties of these classes. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce the notion of fuzzy Sβ - compactness. Some of the basic properties and characterization theorems would be investigated of this newly defined compactness in fuzzy setting. We would also introduce and study fuzzy Sβ - closed spaces. [ABSTRACT FROM AUTHOR]

Rough sets, as a tool for handling uncertain data, have been successfully applied to solve practical problems. Many scholars have conducted various studies on rough sets, especially on various rough set models and upper and lower approximation operators. Recently, these studies have gradually begun to expand on lattice values. The other side of the shield, overlap and grouping functions, come as two distinct from the ordinary binary aggregation functions, due to their not necessarily associative property, have become two new mathematical models for processing information and have been successfully applied in practice. As a result, in this study, (G O , O) - fuzzy rough sets based on the overlap and grouping functions on the complete lattices are introduced and their topological properties are evaluated, further promoting the concept of rough sets. First, the G O - lower L - fuzzy rough approximation operator is defined the lower approximation operator in the (G O , O) - fuzzy rough set on complete lattices utilizing Q L - implications, as well as, the upper approximation operator in (G O , O) - fuzzy rough set on complete lattices is filled by the O - upper L - fuzzy rough approximation operator which is formulated by Jiang and Hu in (G , O) - fuzzy rough set. Second, we talk about some basic properties of the (G O , O) - fuzzy rough set on complete lattices; moreover, the discussion of these basic properties mainly focus on G O - lower L - fuzzy approximation operator. Third, it is considered the characterization of the (G O , O) - fuzzy approximation operator by making use of various classes of L - fuzzy relations. Finally, we study the topological properties of (G O , O) - fuzzy rough set. [ABSTRACT FROM AUTHOR]

Abstract Fuzzy rough set models are useful tools for dealing with fuzzy and real-valued data. They have been used in many real-world applications. In this paper, we investigate the fuzzy rough set model based on triangular norms and fuzzy implications. First, we extend some results in the published literature by removing the condition that is the continuity of triangular norms, and obtain more general conclusion about fuzzy upper approximation operators. Then, for the fuzzy neighborhood and the fuzzy lower approximation operator based on fuzzy implications, we investigate their characterization with each other. Finally, we establish the relationships between fuzzy rough sets and fuzzy topology. In this work, researches on the properties of fuzzy rough sets based on triangular norms which need not be continuous provide generalization results for fuzzy rough set theory from viewpoint of mathematics.

Based on the concept of Cauchy pair Τ-filters, we develop an axiomatic theory of completeness for non-symmetric spaces, such as Τ-quasi-uniform (limit) spaces or L-metric spaces. We show that the category of Τ-quasi-Cauchy spaces is topological and Cartesian closed and we construct a finest completion for a non-complete Τ-quasi-Cauchy space. In the special case of symmetry, Τ-quasi-Cauchy spaces can be identified with Τ-Cauchy spaces.

This paper presents novel concepts in fuzzy topologies, namely q -rung orthopair picture fuzzy (q -ROPF) topology and q -rung orthopair picture fuzzy point (q -ROPFP). These concepts extended the existing notions in fuzzy topologies. We introduced a more relaxed form of continuity, called q ε -ROPF continuity, which allows for a flexible analysis of functions between q -ROPF topological spaces by incorporating an error bound ε. We define the q -neighborhood system for a q -ROPF point in a q -ROPF topological space and investigate the convergence of sequences of such points. Our results shed light on the behavior of functions and sequences in these spaces, as well as the conditions for uniqueness of limits within the realm of supports. This research has practical implications in control systems, providing valuable insights into the stability and robustness of the control strategies in the presence of small changes represented by the epsilon value. [ABSTRACT FROM AUTHOR]

fter the introduction of the concept of fuzzy sets by Zadeh, several researches were conducted on the generalizations of the notion of fuzzy sets. There are many viewpoints on the notion of metric space in fuzzy topology. One of the most important problems in fuzzy topology is obtaining an appropriate concept of fuzzy metric space. This problem has been investigated by many authors from different points of view. Atanassov gives the concept of the intuitionistic fuzzy set as a generalization of the fuzzy set. Park introduced the notion of intuitionistic fuzzy metric space as a natural generalization of fuzzy metric spaces due to George and Veeramani. This paper introduces the concept of intuitionistic fuzzy modular space. Afterward, a Hausdorff topology induced by a δ-homogeneous intuitionistic fuzzy modular is defined and some related topological properties are also examined. After giving the fundamental definitions and the necessary examples, we introduce the definitions of intuitionistic fuzzy boundedness, intuitionistic fuzzy compactness, and intuitionistic fuzzy convergence, and obtain several preservation properties and some characterizations concerning them. Also, we investigate the relationship between an intuitionistic fuzzy modular and an intuitionistic fuzzy metric. Finally, we prove some known results of metric spaces including Baires theorem and the Uniform limit theorem for intuitionistic fuzzy modular spaces. [ABSTRACT FROM AUTHOR]

A λ-subset, or a [0,λ]-valued fuzzy subset, is a mapping from a nonempty set to the interval [0,λ]. In this paper, we use the notion of hemimetrics, a kind of distance functions, as the basic structure to define and study fuzzy rough set model of λ-subsets by using the usual addition and subtraction of real numbers. We define a pair of fuzzy upper/lower approximation operators and investigate their properties and interrelations. These two operators have nice logical descriptions by using the related Lukasiewicz logical systems. We show that upper definable sets, lower definable sets and definable sets are equivalent, and they form an Alexandrov fuzzy topology. A processing of a λ-subset via fuzzy upper/lower approximation operators can actually considered as a processing of the related image, and thus has potential applications in image processing. [ABSTRACT FROM AUTHOR]

In this paper, we study the concept of interval-valued fuzzy set on the family SS X , E of all soft sets over X with the set of parameters E and examine its basic properties. Later, we define the concept of interval-valued fuzzy topology (cotopology) τ on SS X , E . We obtain that each interval-valued fuzzy topology is a descending family of soft topologies. In addition, we study some topological structures such as interval-valued fuzzy neighborhood system of a soft point, base and subbase of τ and investigate some relationships among them. Finally, we give some concepts such as direct sum, open mapping and continuous mapping and consider connections between them. A few examples support the presented results. [ABSTRACT FROM AUTHOR]

Modern human civilization has been plagued with a host of ecological challenges. The rapid pace of global warming, rising sea level leading to loss of habitats and unprecedented flood causing havoc in both rural and city lives and other problems have still found no permanent solutions. In fact, the best ever scientific and technological interventions have failed miserably to mitigate these issues. Traditional and innovative ideas have been proved inadequate to address the issue of flood that has a global presence in varied form or severity. Hence there are many scopes for research and development of mathematical models or interventions to offer solution to this issue of global and local concern. So it is a most appealing branch of study to develop feasible solutions to these problems. In this paper we have considered one of the most burning problems, the flood problem and try to develop the best feasible solution this problem by using Fuzzy Soft Set theory which was introduced by Molodtsov in the year 1999. We have developed a sequence of key factors responsible for the flood problem as follows. {Population explosion, Deforestation, Global Warming, Big dam, Embankment, Blockage, Heavy rainfall} From this sequence we found that population explosion is the main cause for flood problem. Therefore, it is high time to control the population through a worldwide common policy. [ABSTRACT FROM AUTHOR]

Many real-world problems face strenuous in making decisions. Many theories have evolved for dealing with such problems. The present paper deals with Fuzzy Binary Soft Sets and their applications to Multi Criteria Decision Making (MCDM) problems. Then introduced an expanded matrix representation of Fuzzy Binary Soft Sets, an extended resultant matrix, and operator, and an algorithm to solve a proposed problem. [ABSTRACT FROM AUTHOR]

The boundary of a fuzzy set is an important topic that is defined using the concepts of fuzzy topology. The inherent vagueness in the fuzzy sets allows researchers to construct various definitions of the fuzzy boundary. The fuzzy boundary is not only crucial for the theoretical purpose, it is also used to express uncertain geographical objects in geographical information systems (GIS). Specifically, the fuzzy boundary is used for constructing algebraic models such as fuzzy 9 and 16-intersection matrices, which evaluate the topological relation between two uncertain geographical objects. Considering that, this paper analyzes the concept of the fuzzy boundary of a set in the fuzzy topological space. Next, several algebraic models where the fuzzy boundary has been used for analyzing the spatial objects are discussed. [ABSTRACT FROM AUTHOR]

P.M. Pu and Y.M. Liu extended Moore-Smith’s convergence of nets to fuzzy topology and Y.M. Liu provided analogous results to J. Kelley’s classical characterization theorem of net convergence by introducing the notion of fuzzy convergence classes. In a previous paper, the authors of this study provided modified versions of this characterization by using an alternative notion of convergence of fuzzy nets, introduced by B.M.U. Afsan, named fuzzy net ideal convergence. Our main scope here is to generalize and simplify the preceding results. Specifically, we insert the concept of a fuzzy function ideal convergence class, L, on a non-empty set, X, consisting of triads (f,e,I), where f is a function from a non-empty set, D, to the set FP(X) of fuzzy points in X, which we call fuzzy function, e∈FP(X), and I is a proper ideal on D, and we provide necessary and sufficient conditions to establish the existence of a unique fuzzy topology, δ, on X, such that (f,e,I)∈L iff fI-converges to e, relative to the fuzzy topology δ.

Following the generalization of Moore-Smith convergence of nets to fuzzy topological spaces which was given in Pu Pao-Ming, Liu Ying-Ming, Fuzzy topology I. Neighborhood structure of a fuzzy point and Moore-Smith convergence, J. Math. Anal. Appl., 76 (1980), 571-599, a characterization theorem between fuzzy topologies and fuzzy convergence classes was introduced in Ying-Ming Liu, On fuzzy convergence classes, Fuzzy Sets and Systems, 30 (1989), 47-51. Our goal in this paper is to provide modified versions of this characterization. Specifically, we will introduce the concept of fuzzy semi-convergence class to give an alternative characterization of fuzzy topology, in relation to the ordinary convergence of fuzzy nets, and then we will introduce the concept of fuzzy ideal convergence class to obtain analogous results, in relation to the ideal convergence of fuzzy nets. [ABSTRACT FROM AUTHOR]

Binary relations, coverings and neighborhood systems/operators are useful tools to study fuzzy rough set theory. In this paper, we use the notion of real-valued hemimetric, a weak version of the standard metric, as the basic structure to define and study fuzzy rough sets by using the usual addition and subtraction of real numbers. We define a pair of fuzzy upper and lower rough approximation operators and investigate their properties and interrelations. These two operators have nice logical descriptions by using the Lukasiewicz logical system. It is shown that upper definable fuzzy subsets, lower definable fuzzy subsets and Lipschitz fuzzy subsets are the same thing in this model. Definable fuzzy subsets are exactly the upper sets with respect to the induced fuzzy preorder, which form a stratified Alexandrov fuzzy topology. A comparison between hemimetric-based fuzzy rough sets and fuzzy preorder-based fuzzy rough sets has been made. Results show that the former can be considered as a real-valued extension of the latter. At the end, an application of the hemimetric-based rough set model to contour extraction of digital surfaces is proposed. [ABSTRACT FROM AUTHOR]

The concept of an Intuitionistic Fuzzy Modal Topological Structure (IFMTS) was introduced previously, and some of its properties were studied. So far, there are two different IFMTSs based on the classical intuitionistic fuzzy operations: "union" (∪) and "intersection" (∩). In the present paper, two new IFMTSs are developed. They are based on new intuitionistic fuzzy topological operators from closure and interior types, introduced here for the first time, and on the two standard intuitionistic fuzzy modal operators □ and ◊. Some basic properties of the new IFMTSs are discussed. The newly presented IFMTSs could be considered as a basis for the next research on the IFMTSs. Some ideas for the future development of the IFMTS theory and open problems are formulated, related to the existence of other intuitionistic fuzzy operations that can generate new intuitionistic fuzzy topological operators and, respectively, new IFMTSs. [ABSTRACT FROM AUTHOR]

In this work, sums of fuzzy soft topological spaces are defined with free union of a pairwise disjoint non-empty family of fuzzy soft topological spaces. Firstly, we give general information of fuzzy soft topological spaces. Then, we define free union of fuzzy soft topological spaces and disjoint union topology of fuzzy soft topological spaces. We call the free union of a pairwise disjoint non-empty family of fuzzy soft topological spaces the sum of fuzzy soft topological spaces. We show what are the interchangeable properties between fuzzy soft topological spaces and the sum of fuzzy soft topological spaces. For example, there are fuzzy soft interior, fuzzy soft closure, fuzzy soft limit points. Also, we provide some properties showing the relationships between fuzzy soft topological spaces and their sums. Some of these are fuzzy soft base, fuzzy soft sequences, fuzzy soft connected-disconnected, fuzzy soft compact spaces. Also, part of the research for this article is work on fuzzy soft convergence on fuzzy soft topological sum. With this paper, some results, theorems and definitions for fuzzy soft topological sum have been acquired with the help of results, theorems and definitions given in previous studies about fuzzy soft topological spaces. [ABSTRACT FROM AUTHOR]

In this paper, an Intelligent Energy Distribution System (IEDS) has been designed by using the Supervisory Control and Data Acquisition (SCADA) system, such as LabVIEW. The main purpose of this IEDS has been to optimize, according to the needs of the laboratory, the utilization of the energy produced by renewable sources in the installation and then distribute it rationally automatically. Thus, in the recommended approach, a Multi-Input Multi-Output (MIMO) Fuzzy Logic Controller (FLC) has been joined with a Fuzzy Logic Tuning Proportional, Integral, Derivative (FLTPID) controller to give a new typology of the controller, which is named fuzzy PID smart controller FPIDSC. Therefore, this paper has aimed to design a robust power management system based on a new strategy of control, to manipulate a hybrid system management. As a comparison result between three types of controllers, the On/Off controller, the FLC controller, and the proposed controller FPIDSC, the last one has given a decrease in the use of the grid by 6,7304% compared to the FLC controller, and 18,8934% compared to the On/Off controller. Furthermore, a, decrease energy loses by solar energy of 17% compared to the FLC controller and 41,325% compared to the On/Off controller. Based on the results obtained, the proposed controller has shown its efficiency compared to the controllers used in this study. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel notion of modular hemimetrics and use it as the basic structure to define a pair of fuzzy upper and lower approximation operators by using the usual addition and subtraction of real numbers. Then we investigate properties and interrelations of the two approximation operators. We demonstrate that upper definable sets, lower definable sets and definable sets are equivalent. Definable sets form an Alexandrov fuzzy topology such that the upper and lower approximation operators are the closure and the interior operators respectively. Finally, we consider an application of the modular hemimetrics based fuzzy rough sets to the acquisition of gray gradient images in order to show how the fuzzy rough set model based on modular hemimetrics can be successfully applied to solve image processing problems, such as image coding and decoding problems and image morphological processing. [ABSTRACT FROM AUTHOR]

In this article, mixed fuzzy topology and its topological properties have been studied. Mixed fuzzy topology is defined with the help of quasi-coincidence and closure of a fuzzy set in one of the fuzzy topologies. Thus, a new fuzzy topology is generated from the given two fuzzy topologies. This new fuzzy topology may or may not contain the topological properties of the parent topologies. This study identifies some topological properties that are carried to the mixed fuzzy topology from the given parent fuzzy topologies and some other properties which are not carried to the mixed fuzzy topology. Here a base for mixed fuzzy topology from the bases of the given parent topologies is constructed. Considering the regularity of one of the parent topologies mixed fuzzy topology is investigated. Hausdorff 's properties of mixed fuzzy topological spaces are also discussed. It is now of general interest to know which properties are carried to the mixed topology and which are not. A few of these are being tried to answer here in this paper. [ABSTRACT FROM AUTHOR]

This research provides a new control technique for mitigating conducted electromagnetic interference (EMI) in a buck converter designed for solar applications. Indeed, hard-switching direct current to direct current (DC-DC) converters, commonly used in industrial applications, pose a significant risk to the surrounding environment regarding electromagnetic compatibility (EMC). Usually, the fast-switching phase induces abrupt changes in current and voltage, which adds to substantial electromagnetic interference in both conducted and radiated modes and excessive auditory noise. An architecture based on the duality of soft-switching topology and fuzzy logic control technology is developed to address these issues. On the one hand, resonant circuit topologies are used to induce switches to achieve soft switching conditions, which subsequently lessen the effects of EMI. On the other hand, the adoption of fuzzy logic control technology is interesting since it can reduce electrical stresses during switching. Furthermore, the simulation results show that zero current switching (ZCS) soft-switching closed-loop fuzzy logic converters outperform typical open-loop converters and softswitching closed-loop converters with proportional integral (PI) control in terms of EMC requirements. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce the concept of intuitionistic fuzzy hypersoft to pology. Certain properties of intuitionistic fuzzy hypersoft (IFH) to pology like IFH basis, IFH subspace, IFH interior and IFH cloure are investigated. Furthermore, the multicriteria decision making (MCDM) algorithms with aggregation operators based on IFH topology are developed. In Algorithm 1 and Algorithm 2, MCDM problem is applied for IFH sets and IFH topology, respectively. Any real-life implementations of the proposed MCDM algorithms are demonstrated by numerical illustrations. [ABSTRACT FROM AUTHOR]

A Fermatean fuzzy set (FFS) is a reliable method for representing uncertainty in "multi-criteria decision-making" (MCDM). This research seeks to examine the topological properties of FFSs and to establish the notion of "Fermatean fuzzy topology" (FFT). An FFT is the generalisation of existing fuzzy topologies. Several aspects of FFT are examined and various novel concepts are proposed, which include Fermatean fuzzy α -continuity between FFTSs and Fermatean fuzzy connectedness. To deal multiple challenges in sustainable supply chain management, a Fermatean fuzzy "combinative distance-based assessment" (CODAS) method was developed. The proposed FF CODAS technique involves various key features for MCDM. Firstly, a known reputation vector or equal expert weights is determined based on the reputation, experience and qualifications of the experts. Secondly, the Fermatean fuzzy direct rating approach is used to establish the relative relevance of criteria based on the expert group's evaluation preferences. Thirdly, the Fermatean fuzzy CODAS approach is used to construct alternative orderings based on their assessment scores. Finally, an application is developed to show the benefit of the suggested supplier selection approach. Additionally, the symmetry of an optimal decision in application is carried out by a comparison analysis of the suggested models with some existing models. [ABSTRACT FROM AUTHOR]