In the paper [E. Jiménez-Fernández, J. Rodríguez-López, E. A. Sánchez-Pérez, Fuzzy Sets and Systems 406 (2021),66-81], a McShane-Whitney extension theorem is presented for real-valued fuzzy Lipschitz maps between fuzzy metric spaces. Specifically, the codomain space is considered as a so-called Euclidean fuzzy metric space However, while the function ϕ is only required to be increasing, some results of the paper implicitly assume that ϕ is invertible, even though this is not explicitly stated. We propose here an alternative possibility that only requires ϕ to be also left-continuous.
在论文中[d]。Jiménez-Fernández， J. Rodríguez-López， E. a . Sánchez-Pérez, Fuzzy Sets and Systems 406(2021),66-81]，给出了模糊度量空间间实值模糊Lipschitz映射的McShane-Whitney扩展定理。具体来说，上域空间被认为是一个所谓的欧几里得模糊度量空间（R, m φ,g,*）。然而，虽然函数φ只需要增加，但本文的一些结果隐含地假设φ是可逆的，即使这没有明确说明。我们在这里提出了另一种可能性，只要求φ也是左连续的。
{"title":"A revised and extended version of McShane-Whitney extensions for fuzzy Lipschitz maps","authors":"Eduardo Jiménez-Fernández , Jesús Rodríguez-López , Aurora Sánchez-Martín-Orozco , Enrique A. Sánchez-Pérez","doi":"10.1016/j.fss.2025.109731","DOIUrl":"10.1016/j.fss.2025.109731","url":null,"abstract":"<div><div>In the paper [E. Jiménez-Fernández, J. Rodríguez-López, E. A. Sánchez-Pérez, Fuzzy Sets and Systems 406 (2021),66-81], a McShane-Whitney extension theorem is presented for real-valued fuzzy Lipschitz maps between fuzzy metric spaces. Specifically, the codomain space is considered as a so-called Euclidean fuzzy metric space <span><math><mrow><mo>(</mo><mi>R</mi><mo>,</mo><msub><mi>M</mi><mrow><mi>ϕ</mi><mo>,</mo><mi>g</mi></mrow></msub><mo>,</mo><mo>*</mo><mo>)</mo><mo>.</mo></mrow></math></span> However, while the function <em>ϕ</em> is only required to be increasing, some results of the paper implicitly assume that <em>ϕ</em> is invertible, even though this is not explicitly stated. We propose here an alternative possibility that only requires <em>ϕ</em> to be also left-continuous.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"529 ","pages":"Article 109731"},"PeriodicalIF":2.7,"publicationDate":"2025-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145771992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-12DOI: 10.1016/j.fss.2025.109728
Tong Kang , Jun Li , Leifan Yan , Ran Wang
In this note, we present a necessary and sufficient condition of the coincidence for the concave integral and the pan-integral. The previous results on this topic are improved.
本文给出了凹积分与泛积分重合的一个充分必要条件。之前关于这个主题的结果得到了改进。
{"title":"A sufficient and necessary condition of coincidence of the concave integral and the pan-integral","authors":"Tong Kang , Jun Li , Leifan Yan , Ran Wang","doi":"10.1016/j.fss.2025.109728","DOIUrl":"10.1016/j.fss.2025.109728","url":null,"abstract":"<div><div>In this note, we present a necessary and sufficient condition of the coincidence for the concave integral and the pan-integral. The previous results on this topic are improved.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"528 ","pages":"Article 109728"},"PeriodicalIF":2.7,"publicationDate":"2025-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145790851","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-11DOI: 10.1016/j.fss.2025.109729
Keyi Xiao , Yong Su , Wenwen Zong
The modularity holds a significant position in fuzzy set theory, which is closely related to the distributivity and can be viewed as a restricted general associativity equation. In this paper, we study the modularity between semi-t-operators and bi-uninorms, and completely describe the structure of such a pair of functions without any additional assumptions. Several examples are also provided to illustrate the theoretical results.
{"title":"The modularity condition for semi-t-operators and Bi-uninorms","authors":"Keyi Xiao , Yong Su , Wenwen Zong","doi":"10.1016/j.fss.2025.109729","DOIUrl":"10.1016/j.fss.2025.109729","url":null,"abstract":"<div><div>The modularity holds a significant position in fuzzy set theory, which is closely related to the distributivity and can be viewed as a restricted general associativity equation. In this paper, we study the modularity between semi-t-operators and bi-uninorms, and completely describe the structure of such a pair of functions without any additional assumptions. Several examples are also provided to illustrate the theoretical results.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"528 ","pages":"Article 109729"},"PeriodicalIF":2.7,"publicationDate":"2025-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145790849","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-07DOI: 10.1016/j.fss.2025.109725
Yu Kong, Hua-Wen Liu
<div><div>In this paper, we mainly study the classes of uninorms whose neutral elements are middle-transitive and not in any non-trivial cycle, defined on bounded pseudo-ordered sets. Firstly, to generalize the definitions of <span><math><mrow><msub><mi>U</mi><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub></mrow></math></span> and <span><math><mrow><msub><mi>U</mi><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mrow></math></span> of uninorms on bounded lattices, we introduce the definitions of the classes <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msubsup></mrow></math></span> and <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msubsup></mrow></math></span> of uninorms on bounded pseudo-ordered sets. Secondly, we characterize the members of <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msubsup></mrow></math></span> on bounded pseudo-ordered sets using strong t-subnorms and uninorms. We characterize the members of <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msubsup></mrow></math></span> on bounded pseudo-ordered sets using strong t-superconorms and uninorms. In addition, we introduce the definitions of the subclass <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mo>*</mo></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msubsup></mrow></math></span> of <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msubsup></mrow></math></span> and the subclass <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi><mo>*</mo></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msubsup></mrow></math></span> of <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msubsup></mrow></math></span> on bounded pseudo-ordered sets, generalizing the definitions of <span><math><mrow><msub><mi>U</mi><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub></mrow></math></span> and <span><math><mrow><msub><mi>U</mi><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mrow></math></span> on bounded lattices. We introduce the definitions of <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mi>r</mi></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msubsup></mrow></math></span> and <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi><mi>r</mi></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msubsup></mrow></math></span> on bounded pseudo-ordered sets as the extensions of the definitions of <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow><mi>r</mi></msubs
{"title":"Several classes of uninorms on bounded pseudo-ordered sets","authors":"Yu Kong, Hua-Wen Liu","doi":"10.1016/j.fss.2025.109725","DOIUrl":"10.1016/j.fss.2025.109725","url":null,"abstract":"<div><div>In this paper, we mainly study the classes of uninorms whose neutral elements are middle-transitive and not in any non-trivial cycle, defined on bounded pseudo-ordered sets. Firstly, to generalize the definitions of <span><math><mrow><msub><mi>U</mi><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub></mrow></math></span> and <span><math><mrow><msub><mi>U</mi><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mrow></math></span> of uninorms on bounded lattices, we introduce the definitions of the classes <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msubsup></mrow></math></span> and <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msubsup></mrow></math></span> of uninorms on bounded pseudo-ordered sets. Secondly, we characterize the members of <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msubsup></mrow></math></span> on bounded pseudo-ordered sets using strong t-subnorms and uninorms. We characterize the members of <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msubsup></mrow></math></span> on bounded pseudo-ordered sets using strong t-superconorms and uninorms. In addition, we introduce the definitions of the subclass <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mo>*</mo></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msubsup></mrow></math></span> of <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msubsup></mrow></math></span> and the subclass <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi><mo>*</mo></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msubsup></mrow></math></span> of <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msubsup></mrow></math></span> on bounded pseudo-ordered sets, generalizing the definitions of <span><math><mrow><msub><mi>U</mi><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub></mrow></math></span> and <span><math><mrow><msub><mi>U</mi><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mrow></math></span> on bounded lattices. We introduce the definitions of <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mi>r</mi></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msubsup></mrow></math></span> and <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi><mi>r</mi></mrow><mrow><mi>p</mi><mi>s</mi></mrow></msubsup></mrow></math></span> on bounded pseudo-ordered sets as the extensions of the definitions of <span><math><mrow><msubsup><mrow><mi>U</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow><mi>r</mi></msubs","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"528 ","pages":"Article 109725"},"PeriodicalIF":2.7,"publicationDate":"2025-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145790850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-07DOI: 10.1016/j.fss.2025.109716
G. Caterina
We develop a moment-preserving defuzzifier based on the Cafagna-Caterina characterization theorem. Given the first n Fourier moments of a membership function f ∈ [0, 1] on S1, it returns a crisp function χn which is a union of n arcs and whose first n moments match those of f exactly. For , corresponding to mass and centroid, we give a closed-form solution for the arc endpoints, whereas for n > 2 we present two numerical methods to compute those arcs. As an application, we apply the method to California Independent System Operator (CAISO) net demand data, and show that the four-arc solution automatically produces a midday gap that captures the solar valley, with arc locations emerging from moment-matching constraints rather than threshold selection or manual specification.
{"title":"A moment-preserving spectral defuzzification on the circle: From theory to practice","authors":"G. Caterina","doi":"10.1016/j.fss.2025.109716","DOIUrl":"10.1016/j.fss.2025.109716","url":null,"abstract":"<div><div>We develop a moment-preserving defuzzifier based on the Cafagna-Caterina characterization theorem. Given the first <em>n</em> Fourier moments of a membership function <em>f</em> ∈ [0, 1] on <em>S</em><sup>1</sup>, it returns a crisp function <em>χ<sub>n</sub></em> which is a union of <em>n</em> arcs and whose first <em>n</em> moments match those of <em>f</em> exactly. For <span><math><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow></math></span>, corresponding to mass and centroid, we give a closed-form solution for the arc endpoints, whereas for <em>n</em> > 2 we present two numerical methods to compute those arcs. As an application, we apply the method to California Independent System Operator (CAISO) net demand data, and show that the four-arc solution automatically produces a midday gap that captures the solar valley, with arc locations emerging from moment-matching constraints rather than threshold selection or manual specification.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"528 ","pages":"Article 109716"},"PeriodicalIF":2.7,"publicationDate":"2025-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-03DOI: 10.1016/j.fss.2025.109717
Firas Kaabi
We introduce the Fuzzy Life-Cycle Gender Equality Index (FLGEI). It treats gender (in)equality as graded membership in a normative set of equal-opportunity life-cycle states. Indicators are aggregated with a non-additive Choquet integral to capture complementarities and limited substitutability across domains. FLGEI separates de jure rights (Women, Business and the Law) from de facto outcomes (household surveys, ILOSTAT, time-use) and defines their difference as an implementation gap. A delay-aversion axiom penalizes persistent late-stage shortfalls. Shapley values and interaction indices provide transparent domain-level attribution. We also describe how the index is identified and estimated, how uncertainty is quantified, and how it can be used in simple counterfactual simulations. A Tunisia-MENA pilot (2000-2024) shows that childcare access and safety constraints jointly limit labor equality even under strong legal frameworks. Relative to additive and crisp alternatives, FLGEI is more stable to small perturbations and improves budget-targeting performance.
{"title":"Beyond parity: A fuzzy life-cycle index of gender equality with legal–Practical implementation gaps","authors":"Firas Kaabi","doi":"10.1016/j.fss.2025.109717","DOIUrl":"10.1016/j.fss.2025.109717","url":null,"abstract":"<div><div>We introduce the <em>Fuzzy Life-Cycle Gender Equality Index</em> (<span>FLGEI</span>). It treats gender (in)equality as graded membership in a normative set of equal-opportunity life-cycle states. Indicators are aggregated with a non-additive Choquet integral to capture complementarities and limited substitutability across domains. <span>FLGEI</span> separates <em>de jure</em> rights (Women, Business and the Law) from <em>de facto</em> outcomes (household surveys, ILOSTAT, time-use) and defines their difference as an implementation gap. A delay-aversion axiom penalizes persistent late-stage shortfalls. Shapley values and interaction indices provide transparent domain-level attribution. We also describe how the index is identified and estimated, how uncertainty is quantified, and how it can be used in simple counterfactual simulations. A Tunisia-MENA pilot (2000-2024) shows that childcare access and safety constraints jointly limit labor equality even under strong legal frameworks. Relative to additive and crisp alternatives, <span>FLGEI</span> is more stable to small perturbations and improves budget-targeting performance.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"528 ","pages":"Article 109717"},"PeriodicalIF":2.7,"publicationDate":"2025-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737900","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The present study explores the issue of fixed-time control for nonlinear systems that are affected by stochastic perturbations. The issue of infinite gain is effectively addressed by employing a function defined by variable gain over time, and two novel theorems are established. In the first theorem, the scenario is analysed in which both the drift and diffusion terms in stochastic nonlinear systems are known, and this demonstrates that the system achieves fixed-time stability in probability. However, given the potential imperfections inherent in real-world systems, such as model uncertainties and external disturbances, the concept of practical mean-square fixed-time stability is further introduced in Theorem 2. Traditional approaches typically rely on parameter-dependent upper-bound functions to estimate the settling time, which lack flexibility and adaptability to varying system requirements. In contrast, the proposed fixed-time control strategy, which includes a prescribed upper-bound on the dwell time, provides greater flexibility. It enables users to specify the upper limit of the dwell time in accordance with practical requirements. Furthermore, the design of the state-feedback controller employs fuzzy logic systems to approximate the unknown drift and diffusion terms in stochastic nonlinear systems, and by integrating a time-varying gain function, it enables the arbitrary specification of the upper bound on the dwell time. To verify the validity of the prescribed control scheme, this paper presents two simulation case analyses. A comparison and analysis of the results with those obtained by alternative methods demonstrates the rationality and quality of the prescribed control mechanism.
{"title":"Fixed-time stability of unknown stochastic nonlinear systems: A new approach with prescribed upper bound","authors":"Yixuan Yuan , Liping Xie , Junsheng Zhao , Kanjian Zhang","doi":"10.1016/j.fss.2025.109707","DOIUrl":"10.1016/j.fss.2025.109707","url":null,"abstract":"<div><div>The present study explores the issue of fixed-time control for nonlinear systems that are affected by stochastic perturbations. The issue of infinite gain is effectively addressed by employing a function defined by variable gain over time, and two novel theorems are established. In the first theorem, the scenario is analysed in which both the drift and diffusion terms in stochastic nonlinear systems are known, and this demonstrates that the system achieves fixed-time stability in probability. However, given the potential imperfections inherent in real-world systems, such as model uncertainties and external disturbances, the concept of practical mean-square fixed-time stability is further introduced in Theorem 2. Traditional approaches typically rely on parameter-dependent upper-bound functions to estimate the settling time, which lack flexibility and adaptability to varying system requirements. In contrast, the proposed fixed-time control strategy, which includes a prescribed upper-bound on the dwell time, provides greater flexibility. It enables users to specify the upper limit of the dwell time in accordance with practical requirements. Furthermore, the design of the state-feedback controller employs fuzzy logic systems to approximate the unknown drift and diffusion terms in stochastic nonlinear systems, and by integrating a time-varying gain function, it enables the arbitrary specification of the upper bound on the dwell time. To verify the validity of the prescribed control scheme, this paper presents two simulation case analyses. A comparison and analysis of the results with those obtained by alternative methods demonstrates the rationality and quality of the prescribed control mechanism.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"528 ","pages":"Article 109707"},"PeriodicalIF":2.7,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-30DOI: 10.1016/j.fss.2025.109706
Roberto G. Aragón , Pascual Jara , Jesús Medina
Lattice-based sum provides a procedure to obtain posets and lattices from families of posets and lattices, respectively. Establishing sufficient conditions to ensure the lattice structure was the most significant challenge achieved in previous works. Next steps are to consider structures with general operators defined on the lattices of the family, introduce a sum of these operators on the obtained lattice-based sum and study the properties preserved by this new definition. We will prove that the natural definition preserve, in general, the monotonicity, associativity, commutativity, etc. This paper also introduces a new mechanism focused on preserving the left-continuity property of the operators defined on the lattices. This new approach also preserves the associativity and the infimum of non-empty subsets, and takes into account (infinite) complete lattices, unlike the previous works.
{"title":"General and left-continuous operators on lattice-based sums","authors":"Roberto G. Aragón , Pascual Jara , Jesús Medina","doi":"10.1016/j.fss.2025.109706","DOIUrl":"10.1016/j.fss.2025.109706","url":null,"abstract":"<div><div>Lattice-based sum provides a procedure to obtain posets and lattices from families of posets and lattices, respectively. Establishing sufficient conditions to ensure the lattice structure was the most significant challenge achieved in previous works. Next steps are to consider structures with general operators defined on the lattices of the family, introduce a sum of these operators on the obtained lattice-based sum and study the properties preserved by this new definition. We will prove that the natural definition preserve, in general, the monotonicity, associativity, commutativity, etc. This paper also introduces a new mechanism focused on preserving the left-continuity property of the operators defined on the lattices. This new approach also preserves the associativity and the infimum of non-empty subsets, and takes into account (infinite) complete lattices, unlike the previous works.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"528 ","pages":"Article 109706"},"PeriodicalIF":2.7,"publicationDate":"2025-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-29DOI: 10.1016/j.fss.2025.109699
L.C. Barros , M. Shahidi , E. Esmi , T. Allahviranloo
In this paper, we extend the concepts of differentiability and integrability to some classes of multivariable fuzzy functions, called A-linearly correlated bivariate fuzzy processes. We introduce key concepts such as the A-total and A-directional derivatives for A-linearly correlated bivariate fuzzy processes, along with a tangent plane interpretation and the A-Jacobian matrix. Their properties and relationships are also examined. Moreover, we present the second order Fréchet differentiability for A-linearly correlated bivariate fuzzy processes and define the A-Hessian matrix. Furthermore, we introduce the concept of the A-double integral and apply Fubini’s Theorem to A-linearly correlated bivariate fuzzy processes. We provide several examples demonstrating its practical applications.
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Pub Date : 2025-11-27DOI: 10.1016/j.fss.2025.109701
Aziz El Ghazouani, M’hamed Elomari , Said Melliani
This paper establishes rigorous mathematical foundations for fuzzy fractional delay integro-differential equations (FFDIDEs) involving the generalized Hukuhara ψ-Caputo fractional derivative – a previously unexplored combination in the literature. We address the critical theoretical gap in analyzing systems that simultaneously incorporate fuzzy uncertainty, memory effects (time delays), and non-local dynamics (integral terms). Through an innovative synthesis of Banach’s fixed point theorem and a novel monotone iterative technique, we prove: (1) existence and uniqueness of solutions under Lipschitz conditions (Theorems 2 and 3), (2) constructive approximation via monotone sequences converging uniformly to the solution (Lemma 2), and (3) continuous dependence on initial conditions with explicit stability bounds (Theorem 4).
Our framework systematically handles both d-increasing and d-decreasing solution cases through a unified ψ-Caputo operational calculus. The theoretical advances are validated through computational experiments demonstrating convergence rates for α ∈ (0, 1), with MATLAB simulations providing quantitative analysis of triangular fuzzy solutions (Example 1, Figures 1– 6). Beyond its theoretical contributions, this work enables new applications in fuzzy control systems with delays, fractional-order neural networks with uncertainty, and other complex systems requiring simultaneous treatment of non-locality and vagueness. The results fundamentally extend the existing fuzzy fractional calculus literature by establishing the first comprehensive solution theory for this important class of equations.
{"title":"Fuzzy fractional delay integro-differential equation with the generalized Hukuhara ψ-Caputo fractional derivative","authors":"Aziz El Ghazouani, M’hamed Elomari , Said Melliani","doi":"10.1016/j.fss.2025.109701","DOIUrl":"10.1016/j.fss.2025.109701","url":null,"abstract":"<div><div>This paper establishes rigorous mathematical foundations for fuzzy fractional delay integro-differential equations (FFDIDEs) involving the generalized Hukuhara <em>ψ</em>-Caputo fractional derivative – a previously unexplored combination in the literature. We address the critical theoretical gap in analyzing systems that simultaneously incorporate fuzzy uncertainty, memory effects (time delays), and non-local dynamics (integral terms). Through an innovative synthesis of Banach’s fixed point theorem and a novel monotone iterative technique, we prove: (1) existence and uniqueness of solutions under Lipschitz conditions (Theorems 2 and 3), (2) constructive approximation via monotone sequences converging uniformly to the solution (Lemma 2), and (3) continuous dependence on initial conditions with explicit stability bounds (Theorem 4).</div><div>Our framework systematically handles both <em>d</em>-increasing and <em>d</em>-decreasing solution cases through a unified <em>ψ</em>-Caputo operational calculus. The theoretical advances are validated through computational experiments demonstrating <span><math><mrow><mi>O</mi><mo>(</mo><msup><mi>h</mi><mrow><mn>1</mn><mo>+</mo><mi>α</mi></mrow></msup><mo>)</mo></mrow></math></span> convergence rates for <em>α</em> ∈ (0, 1), with MATLAB simulations providing quantitative analysis of triangular fuzzy solutions (Example 1, Figures 1– 6). Beyond its theoretical contributions, this work enables new applications in fuzzy control systems with delays, fractional-order neural networks with uncertainty, and other complex systems requiring simultaneous treatment of non-locality and vagueness. The results fundamentally extend the existing fuzzy fractional calculus literature by establishing the first comprehensive solution theory for this important class of equations.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"528 ","pages":"Article 109701"},"PeriodicalIF":2.7,"publicationDate":"2025-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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