Fuzzy Sets Theory and Its Applications

Topic Information

Dear Colleagues,

The concept of fuzzy sets introduced by L.A. Zadeh in 1965 tried to extend the classical set theory. It is well-known that a classical set corresponds to an indicator function whose values are only taken to be 0 and 1. With the aid of membership functions associated with a fuzzy set, each element in a set allows one to take any values between 0 and 1 that can be treated as the degree of membership. This kind of imprecision draws forth a multitude of applications. This topic will focus on the original research that reflects the theoretical developments and applicable results.

The topics of interest include but are not limited to the following:

• Foundation of fuzzy sets (fuzzy arithmetic operations, extension principle, possibility measures, etc.).
• Fuzzy mathematics (fuzzy topology, fuzzy real analysis, fuzzy integral and differential equations, fuzzy metric spaces, fuzzy algebra, etc.). 
• Fuzzy logics (many-valued logics, type-2 fuzzy logics, intuitionistic fuzzy logics, etc.). 
• Fuzzy statistical analysis (fuzzy random variables, fuzzy regression analysis, fuzzy reliability analysis, fuzzy times series, fuzzy Markov process, etc.). 
• Hybrid systems (fuzzy control, fuzzy neural networks, genetic fuzzy systems, fuzzy intelligent systems, fuzzy biomedical systems, fuzzy chaotic systems, fuzzy information systems, etc.). 
• Nature of computation (ant colony optimization, artificial immune systems, genetic algorithms, particle swarm intelligence, simulated annealing, tabu search, etc.). 
• Computational intelligence (artificial intelligence, approximate reasoning, expert systems, machine learning, support vector machines, robotics, image processing, pattern recognition, information retrieval, etc.). 
• Operations research and management sciences (fuzzy games theory, fuzzy inventory models, fuzzy queueing theory, fuzzy scheduling problems, fuzzy optimization, fuzzy decision making, fuzzy data mining, fuzzy clustering, stochastic optimization, financial derivatives, uncertainty modeling, etc.).

Prof. Dr. Hsien-Chung Wu
Prof. Dr. Ziye Zhang
Dr. Rongrong Yu
Topic Editors

Keywords

Participating Journals

Journal Name	Impact Factor	CiteScore	Launched Year	First Decision (median)	APC
Algorithms algorithms	2.1	4.5	2008	19.2 Days	CHF 1800	Submit
Axioms axioms	1.6	-	2012	21.7 Days	CHF 2400	Submit
Energies energies	3.2	7.3	2008	16.8 Days	CHF 2600	Submit
Mathematics mathematics	2.2	4.6	2013	17.3 Days	CHF 2600	Submit
Symmetry symmetry	2.2	5.3	2009	15.8 Days	CHF 2400	Submit

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Published Papers (11 papers)

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20 pages, 502 KB  

Open AccessArticle

Fuzzy Skew Maps: Preserving Robust Chaos Under Uncertainty with Applications to Cryptography

by Illych Alvarez, Antonio S. E. Chong, Jorge Chamba, Ximena Quiñonez and Ivy Peña

Mathematics 2026, 14(6), 1010; https://doi.org/10.3390/math14061010 - 17 Mar 2026

Viewed by 293

Abstract

We introduce fuzzy skew maps as a levelwise ($\alpha$-cut) extension of robustly chaotic skew transformations of S-unimodal maps to epistemically uncertain environments. Our central hypothesis is that the robust-chaos mechanism of the underlying skew family transfers to fuzzy parameter uncertainty [...] Read more.

We introduce fuzzy skew maps as a levelwise ($\alpha$-cut) extension of robustly chaotic skew transformations of S-unimodal maps to epistemically uncertain environments. Our central hypothesis is that the robust-chaos mechanism of the underlying skew family transfers to fuzzy parameter uncertainty in a set-based (not probabilistic) sense is as follows: for every $\alpha \in [0,1]$, the induced crisp family $\{F(·,q):q\in {\left[\tilde{q}\right]}_{\alpha }\}$ preserves the absence of periodic windows and maintains strictly positive Lyapunov exponents. This yields a precise notion of fuzzy robustness that is distinct from interval enclosures (pure bounds) and stochastic robustness (average-case guarantees). We also formalize fuzzy topological entropy via the extension principle and discuss its basic structural properties under mild continuity assumptions. For chaos-based image encryption, fuzzification provides an uncertainty-aware key representation and stabilizes cryptographic indicators across $\alpha$-cuts as follows: in our experiments, NPCR remains within $99.58$–$99.64\%$, UACI within $33.41$–$33.52\%$, and the cipher entropy is near 8 bits, while pixel correlation stays close to zero. These results support fuzzy skew maps as a robust primitive for secure information systems operating under parametric uncertainty. Full article

(This article belongs to the Topic Fuzzy Sets Theory and Its Applications)

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30 pages, 442 KB  

Open AccessArticle

A New Type of Soft Group: Soft Symmetric Difference Group with Group Theory Applications

by Aslıhan Sezgin, İbrahim Durak and Erdal Karaduman

Mathematics 2026, 14(6), 999; https://doi.org/10.3390/math14060999 - 16 Mar 2026

Viewed by 276

Abstract

In this paper, a new type of soft group called the soft symmetric difference group (SSD-group) is introduced and systematically developed. This structure is constructed by integrating soft set theory with group theory through the symmetric difference operation and set inclusion. Fundamental concepts [...] Read more.

In this paper, a new type of soft group called the soft symmetric difference group (SSD-group) is introduced and systematically developed. This structure is constructed by integrating soft set theory with group theory through the symmetric difference operation and set inclusion. Fundamental concepts such as characteristic soft symmetric difference groups, soft symmetric difference subgroups, normal soft symmetric difference subgroups, soft normalizers, and soft cosets are defined, and their essential algebraic properties are investigated. Several characterizations of soft normality are also established through these concepts. Various axiomatic results are obtained, providing necessary and sufficient conditions for a soft set to form an SSD-group. Furthermore, soft quotient (factor) groups of SSD-groups are introduced and their structural properties are examined in detail. The relationship between SSD-group theory and classical group theory is also established through several corresponding concepts. Illustrative examples are provided to demonstrate the applicability and internal consistency of the proposed framework. Overall, the results obtained in this study extend existing soft group structures and contribute to the development of algebraic theory within the context of soft sets, while also providing a foundation for further generalizations to other algebraic frameworks such as semigroups, rings, and fields. Full article

(This article belongs to the Topic Fuzzy Sets Theory and Its Applications)

19 pages, 364 KB  

Open AccessArticle

New Fuzzy Topologies via Ideals and Generalized Openness

by Ahu Açıkgöz

Mathematics 2026, 14(5), 904; https://doi.org/10.3390/math14050904 - 6 Mar 2026

Viewed by 235

Abstract

This paper introduces and investigates a new class of generalized open sets, called fuzzy $h_{\mathcal{I}}$-open sets, in fuzzy ideal topological spaces $(X,\tilde{\tau },\tilde{\mathcal{I}})$. We prove that the collection of all fuzzy $h_{\mathcal{I}}$ [...] Read more.

This paper introduces and investigates a new class of generalized open sets, called fuzzy $h_{\mathcal{I}}$-open sets, in fuzzy ideal topological spaces $(X,\tilde{\tau },\tilde{\mathcal{I}})$. We prove that the collection of all fuzzy $h_{\mathcal{I}}$-open sets forms a fuzzy topology ${\tilde{\tau }}^{h_{\mathcal{I}}}$ satisfying $\tilde{\tau }\subseteq {\tilde{\tau }}^{h_{\mathcal{I}}}$ and show that ${\tilde{\tau }}^{\ast }$ and ${\tilde{\tau }}^{h_{\mathcal{I}}}$ are in general incomparable, demonstrating that the $h_{\mathcal{I}}$-construction captures fundamentally different information from the ∗-topology. We establish precise conditions under which these topologies coincide and introduce a fuzzy $h_{\mathcal{I}}$-$T_1$ separation axiom. Furthermore, we develop a comprehensive hierarchy of generalizations—fuzzy $h{\alpha }_{\mathcal{I}}$-open, fuzzy $h{p}_{\mathcal{I}}$-open, fuzzy $h{s}_{\mathcal{I}}$-open, and fuzzy $h{\beta }_{\mathcal{I}}$-open sets—and prove that these classes are pairwise distinct through genuinely fuzzy (non-characteristic) examples. We introduce fuzzy $h_{\mathcal{I}}$-continuous and fuzzy $h_{\mathcal{I}}$-irresolute functions, providing six equivalent characterizations and a closed-set criterion via the ∗-interior operator. The framework is applied to a concrete multi-criteria decision-making problem, where the ideal filters negligible criteria and the $h_{\mathcal{I}}$-interior provides a refined ranking that demonstrably outperforms the original fuzzy topology. Full article

(This article belongs to the Topic Fuzzy Sets Theory and Its Applications)

26 pages, 364 KB  

Open AccessFeature PaperArticle

Novel Operational Properties and Inductive Relationships of TL-Fuzzy Filters and TL-Fuzzy Congruences in Residuated Lattices

by Xiaowu Zhou and Yingying An

Mathematics 2026, 14(3), 427; https://doi.org/10.3390/math14030427 - 26 Jan 2026

Viewed by 382

Abstract

Filters and congruences are fundamental concepts in residuated lattices for characterizing their structure. In this paper, using a complete lattice L as the truth-value set and based on a triangular norm T and its induced operator ${\nu }_T$, we investigate new operational [...] Read more.

Filters and congruences are fundamental concepts in residuated lattices for characterizing their structure. In this paper, using a complete lattice L as the truth-value set and based on a triangular norm T and its induced operator ${\nu }_T$, we investigate new operational properties of $TL$-fuzzy filters and $TL$-fuzzy congruences. We first define the ${\otimes }_T$ and ${\nu }_T$-operations on L-fuzzy sets and study their effects on $TL$-fuzzy filters. Next, we examine the congruence-preserving properties and operational rules of $TL$-fuzzy congruences under ${\ast }_T$ and ${\diamond }_{{\nu }_T}$-compositions. Finally, leveraging the correspondence between $TL$-fuzzy filters and $TL$-fuzzy congruences, we explore the interplay and internal relationships among their respective operations. Full article

(This article belongs to the Topic Fuzzy Sets Theory and Its Applications)

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24 pages, 419 KB  

Open AccessArticle

A Systematic Study on Distributivity of Threshold-Generated Implications over Uninorms

by Zhihong Yi

Axioms 2025, 14(11), 807; https://doi.org/10.3390/axioms14110807 - 30 Oct 2025

Viewed by 335

Abstract

The distributivity of implications over fuzzy operators is a desirable property for fuzzy systems and can be employed in the elimination of the explosion of if–then rules. In this paper, we try to explore the relationship between the distributivity over the uninorms-related fuzzy [...] Read more.

The distributivity of implications over fuzzy operators is a desirable property for fuzzy systems and can be employed in the elimination of the explosion of if–then rules. In this paper, we try to explore the relationship between the distributivity over the uninorms-related fuzzy connectives and the distributivity over uninorms in the threshold generation method, i.e., the distributive equations $I(u,U_1(v,w))=U_2(I(u,v),I(u,w))$ and $I(U_1(u,v),w))=U_2(I(u,w),I(v,w))$ with I being the threshold-generated implication. Consequently, we find that if the uninorms are restricted to special classes, then the distributivity property by the first equation can be preserved between the original and threshold-generated implications; under certain constraints on the threshold-generated implication, the distributivity property by the second equation becomes trivial. Full article

(This article belongs to the Topic Fuzzy Sets Theory and Its Applications)

22 pages, 14103 KB  

Open AccessArticle

The Fourier Regularization for Solving a Cauchy Problem for the Laplace Equation with Uncertainty

by Xiaoya Liu, Yiliang He and Hong Yang

Axioms 2025, 14(11), 805; https://doi.org/10.3390/axioms14110805 - 30 Oct 2025

Viewed by 655

Abstract

The Laplace equation is an important partial differential equation, typically used to describe the properties of steady-state distributions or passive fields in physical phenomena. Its Cauchy problem is one of the classic, serious, ill-posed problems, characterized by the fact that minor disturbances in [...] Read more.

The Laplace equation is an important partial differential equation, typically used to describe the properties of steady-state distributions or passive fields in physical phenomena. Its Cauchy problem is one of the classic, serious, ill-posed problems, characterized by the fact that minor disturbances in the data can lead to significant errors in the solution and lack stability. Secondly, the determination of the parameters of the classical Laplace equation is difficult to adapt to the requirements of complex applications. For this purpose, in this paper, the Laplace equation with uncertain parameters is defined, and the uncertainty is represented by fuzzy numbers. In the case of granular differentiability, it is transformed into a granular differential equation, proving its serious ill-posedness. To overcome the ill-posedness, the Fourier regularization method is used to stabilize the numerical solution, and the stability estimation and error analysis between the regularization solution and the exact solution are given. Finally, numerical examples are given to illustrate the effectiveness and practicability of this method. Full article

(This article belongs to the Topic Fuzzy Sets Theory and Its Applications)

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25 pages, 440 KB  

Open AccessArticle

An Exhaustive Analysis of the OR-Product of Soft Sets: A Symmetry Perspective

by Keziban Orbay, Metin Orbay and Aslıhan Sezgin

Symmetry 2025, 17(10), 1661; https://doi.org/10.3390/sym17101661 - 5 Oct 2025

Cited by 2 | Viewed by 549

Abstract

This paper provides a theoretical investigation of the OR-product (∨-product) in soft set theory, an operation of central importance for handling uncertainty in decision-making. A comprehensive algebraic analysis is carried out with respect to various types of subsets and equalities, with particular emphasis [...] Read more.

This paper provides a theoretical investigation of the OR-product (∨-product) in soft set theory, an operation of central importance for handling uncertainty in decision-making. A comprehensive algebraic analysis is carried out with respect to various types of subsets and equalities, with particular emphasis on M-subset and M-equality, which represent the strictest forms of subsethood and equality. This framework reveals intrinsic algebraic symmetries, particularly in commutativity, associativity, and idempotency, which enrich the structural understanding of soft set theory. In addition, certain missing results on OR-products in the literature are completed, and our findings are systematically compared with existing ones, ensuring a more rigorous theoretical framework. A central contribution of this study is the demonstration that the collection of all soft sets over a universe, equipped with a restricted/extended intersection and the OR-product, forms a commutative hemiring with identity under soft L-equality. This structural result situates the OR-product within one of the most fundamental algebraic frameworks, connecting soft set theory with broader areas of algebra. To illustrate its practical relevance, the int-uni decision-making method on the OR-product is applied to a pilot recruitment case, showing how theoretical insights can support fair and transparent multi-criteria decision-making under uncertainty. From an applied perspective, these findings embody a form of symmetry in decision-making, ensuring fairness and balanced evaluation among multiple decision-makers. By bridging abstract algebraic development with concrete decision-making applications, the results affirm the dual significance of the OR-product—strengthening the theoretical framework of soft set theory while also providing a viable methodology for applied decision-making contexts. Full article

(This article belongs to the Topic Fuzzy Sets Theory and Its Applications)

18 pages, 314 KB  

Open AccessArticle

A Type of Fuzzy Metric and Its Applications

by Peng Chen

Axioms 2025, 14(10), 744; https://doi.org/10.3390/axioms14100744 - 30 Sep 2025

Viewed by 495

Abstract

In this paper, we aim to investigate a type of lattice-valued fuzzy metric within the framework of L-topology. Firstly, we present a comprehensive construction theorem for this type of metric, utilizing the concept of L-quasi metric. Secondly, we provide an equivalent [...] Read more.

In this paper, we aim to investigate a type of lattice-valued fuzzy metric within the framework of L-topology. Firstly, we present a comprehensive construction theorem for this type of metric, utilizing the concept of L-quasi metric. Secondly, we provide an equivalent characterization through the use of C-nbd clusters, which are formed from all $B_r$: one of four types of basic spheres defined herein. Thirdly, recognizing that these four types of basic spheres serve as essential tools for characterizing various metrics, we meticulously examine the relationships among them and outline a series of topological properties associated with these metrics, which include their opening and closing characteristics, symmetrical property, and more. Finally, in addressing the corresponding symmetry problem between two types of basic spheres, namely $B_r\left(a\right)$ and $Q_r\left(a\right)$, we introduce a novel fuzzy p-metric and demonstrate tht the L-real line $\mathbf{R}\left(L\right)$ satisfies this fuzzy p-metric. Full article

(This article belongs to the Topic Fuzzy Sets Theory and Its Applications)

20 pages, 300 KB  

Open AccessArticle

Categories of L-Primals, L-Pre-Proximities, and L-Topologies

by Ahmed A. Ramadan and Anwar J. Fawakhreh

Axioms 2025, 14(7), 541; https://doi.org/10.3390/axioms14070541 - 18 Jul 2025

Cited by 2 | Viewed by 739

Abstract

This paper introduces and investigates the fundamental properties of L-primals, a generalization of the primal concept within the framework of L-fuzzy sets and complete lattices. Building upon the established theories of L-topological spaces and L-pre-proximity spaces, this research explores [...] Read more.

This paper introduces and investigates the fundamental properties of L-primals, a generalization of the primal concept within the framework of L-fuzzy sets and complete lattices. Building upon the established theories of L-topological spaces and L-pre-proximity spaces, this research explores the interrelations among these three generalized topological structures. The study establishes novel categorical links, demonstrating the existence of concrete functors between categories of L-primal spaces and L-pre-proximity spaces, as well as between categories of L-pre-proximity spaces and stratified L-primal spaces. Furthermore, the paper clarifies the existence of a concrete functor between the category of stratified L-primal spaces and the category of L-topological spaces, and vice versa, thereby establishing Galois correspondences between these categories. Theoretical findings are supported by illustrative examples, including applications within the contexts of information systems and medicine, demonstrating the practical aspects of the developed theory. Full article

(This article belongs to the Topic Fuzzy Sets Theory and Its Applications)

28 pages, 418 KB  

Open AccessArticle

Geometric Accumulation Operators of Dombi Weighted Trapezoidal-Valued Fermatean Fuzzy Numbers with Multi-Attribute Group Decision Making

by M. Kaviyarasu, J. Angel and Mohammed Alqahtani

Symmetry 2025, 17(7), 1114; https://doi.org/10.3390/sym17071114 - 10 Jul 2025

Cited by 2 | Viewed by 866

Abstract

Trapezoidal-valued fermatean fuzzy numbers (TpVFFNs) are essential for handling daily decision-making issues in the engineering and management fields. Accumulation processes on the set of TpVFFN are used to address decision-making problems described in this environment as necessary. The primary goal of this paper [...] Read more.

Trapezoidal-valued fermatean fuzzy numbers (TpVFFNs) are essential for handling daily decision-making issues in the engineering and management fields. Accumulation processes on the set of TpVFFN are used to address decision-making problems described in this environment as necessary. The primary goal of this paper is to provide the concept of Dombi t-norm $\left(D_{tn}\right)$- and Dombi t-conorm $\left(D_{tcn}\right)$-based accumulation operators on the class of TpVFFN, emphasizing how they behave symmetrically in aggregation processes to maintain consistency and fairness. To use s to illustrate mathematical circumstances, we first create a trapezoidal-valued fermatean fuzzy Dombi’s weighted geometric operator, hexagonal hybird geometric operator, fermatean fuzzy order weighted geometric operator. Second, we use a multi-attribute group decision-making (MAGDM) approach to compute the recommended accumulation operators. Finally, we demonstrate the potential practical application of the proposed decision-making problem related to the pink cab. Full article

(This article belongs to the Topic Fuzzy Sets Theory and Its Applications)

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30 pages, 2525 KB  

Open AccessArticle

A Dynamic Threat Assessment Method for Multi-Target Unmanned Aerial Vehicles at Multiple Time Points Based on Fuzzy Multi-Attribute Decision Making and Fuse Intention

by Qianru Niu, Shuangyin Ren, Wei Gao and Chunjiang Wang

Mathematics 2025, 13(10), 1663; https://doi.org/10.3390/math13101663 - 19 May 2025

Cited by 2 | Viewed by 1823

Abstract

In response to the threat assessment challenge posed by unmanned aerial vehicles (UAVs) in air defense operations, this paper proposes a dynamic assessment model grounded in fuzzy multi-attribute decision making. First, a three-dimensional evaluation index system is established, encompassing capability, opportunity, and intention. [...] Read more.

In response to the threat assessment challenge posed by unmanned aerial vehicles (UAVs) in air defense operations, this paper proposes a dynamic assessment model grounded in fuzzy multi-attribute decision making. First, a three-dimensional evaluation index system is established, encompassing capability, opportunity, and intention. Quantification functions for assessing the threat level of each attribute are then designed. To account for the temporal dynamics of the battlefield, an innovative fusion approach is developed, integrating inverse Poisson distribution time weights with subjective–objective comprehensive weighting, thereby establishing a dynamic variable weight fusion mechanism. Among these, the subjective weights are determined by integrating the intention probability matrix, effectively incorporating the intentions into the threat assessment process to reflect their dynamic changes and enhancing the overall evaluation accuracy. Leveraging the improved technique for order preference by similarity to ideal solution (TOPSIS), the model achieves threat prioritization. Experimental results demonstrate that this method significantly enhances the reliability of threat assessments in uncertain and dynamic battlefield environments, offering valuable support for air defense command and control systems. Full article

(This article belongs to the Topic Fuzzy Sets Theory and Its Applications)

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