Fuzzy Sets and Systems最新文献

{"title":"Construction of uninorms on complete lattices by a monotone function and its pseudo-inverse","authors":"Zhenyu Xiu ,&nbsp;Xu Zheng","doi":"10.1016/j.fss.2026.109766","DOIUrl":"10.1016/j.fss.2026.109766","url":null,"abstract":"<div><div>In this paper, we primarily investigate methods for generating uninorms on complete lattices using monotone functions and their pseudo-inverses. First, we present a construction of a uninorm on a complete lattice based on a t-norm, a complete inf-homomorphism, and its pseudo-inverse. Next, we introduce a new method for generating a uninorm via a given uninorm, a complete inf-homomorphism, and its pseudo-inverse. Finally, we explore methods for constructing a uninorm on a complete lattice using a given uninorm together with an injective complete inf-homomorphism and its pseudo-inverse.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109766"},"PeriodicalIF":2.7,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981151","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}

{"title":"Solving fuzzy linear systems in Gaussian PDMF space","authors":"Chuang Zheng","doi":"10.1016/j.fss.2026.109772","DOIUrl":"10.1016/j.fss.2026.109772","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this paper, we solve the fuzzy linear systems in a fuzzy number space &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, namely the Gaussian probability density membership function (Gaussian-PDMF) space. The fuzzy linear systems include two types: the semi-fuzzy linear system (SFLS) and the fully-fuzzy linear system (FFLS). First, we solve the SFLS &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is a real-valued matrix, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is a fuzzy number vector, and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is the unknown fuzzy number vector. The elements of both &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; belong to &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. We present the Cramer’s rule to calculate the solution with square matrix &lt;em&gt;A&lt;/em&gt; and find out that its solution set is a &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; dimensional affine space with &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;em&gt;R&lt;/em&gt;(&lt;em&gt;A&lt;/em&gt;) being the rank of &lt;em&gt;A&lt;/em&gt;. The explicit form of the solution for RREF matrix &lt;em&gt;A&lt;/em&gt; is stated to ensure usability for modeling. Secondly, we solve the FFLS &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is a fuzzy matrix with all components in &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. We analyze its solution set and present the parametric form of solutions under the fuzzy RREF matrix. We then adapt Gaussian elimination method to fuzzy matrices and systems by restricting it to the unit group of ring &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, proving the equivalence of solution sets after elementary row operations. We also establish the connection between FFLS and SFLS by confining elements of &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; to a subset of &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; that forms a field. In the third part, two numerical examples are given to illustrated our method. All results in this paper are explicit since the Gaussian-PDMF space &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, to which the membership function of the fuzzy number belongs, possesses a complete algebraic structure. The proposed framework offers a feasible and systematical tool for solving the mathematical m","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109772"},"PeriodicalIF":2.7,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039481","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}

{"title":"Adaptive dynamic event-triggered control for IT-2 fuzzy delayed semi-Markov jump systems with different uncertain transition rates","authors":"Hao Qiu ,&nbsp;Huamin Wang ,&nbsp;Likui Wang ,&nbsp;Shiping Wen","doi":"10.1016/j.fss.2026.109773","DOIUrl":"10.1016/j.fss.2026.109773","url":null,"abstract":"<div><div>In practical industrial systems, we often encounter nonlinear uncertainties, parameter jumps, or time-delay phenomena that easily destroy the stability of the system. To stabilize these disturbances, we construct a more general closed-loop interval type-2 fuzzy delayed semi-Markov jump system (IT-2FD S-MJS) and investigate its stochastic stability in this article. Firstly, to optimize the performance of computational resources and data transmission, we introduce quantization techniques and novel adaptive dynamic data sampling-based event-triggered mechanisms, greatly reducing the number of triggers and improving the flexibility of adjustment. The framework’s discrete sampling nature inherently prevents Zeno behavior by eliminating the possibility of infinite triggers within finite time intervals. Then, by constructing boundary/general uncertain transition rates (BUTR/GUTR) and slack matrices, we derive sufficient conditions with less conservatism of stochastic stability for IT-2FD S-MJS. It should be noticed that the unknown transition information is modeled by BUTR/GUTR, and the conservatism of mismatched membership functions is reduced by introducing the slack matrices. Meanwhile, we obtain the corresponding gain parameters of the adaptive dynamic event-triggered quantization controller using linear matrix inequality (LMI) technology, with implementation details specified in Algorithms 1 and 2. Finally, we take the robotic arm and tunnel diode circuit as examples to verify the validity of the theorems.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109773"},"PeriodicalIF":2.7,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039542","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}

{"title":"Fuzzy multiset regular languages and their basic characterizations","authors":"Pavel Martinek","doi":"10.1016/j.fss.2026.109761","DOIUrl":"10.1016/j.fss.2026.109761","url":null,"abstract":"<div><div>The paper provides a survey of several ways how to describe fuzzy multiset regular languages, i.e., languages generated by fuzzy multiset regular grammars. These languages can also be characterized by means of fuzzy multiset finite automata (both in general and in reduced forms), fuzzy multiset regular expressions, and as fuzzy multiset languages which can be expressed in a semilinear form. Moreover, it is pointed out that a prevailing number of already published papers concerning fuzzy multiset finite automata is based on a wrong definition. It is also shown that the name ‘deterministic fuzzy multiset finite automaton’ is often used incorrectly for automata deserving adjective pseudodeterministic.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109761"},"PeriodicalIF":2.7,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981150","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}

{"title":"Cartesian closedness of the category of real-valued sets, I","authors":"Lili Shen,&nbsp;Jian Zhang","doi":"10.1016/j.fss.2026.109767","DOIUrl":"10.1016/j.fss.2026.109767","url":null,"abstract":"<div><div>Let [0, 1]_* be the unit interval [0,1] equipped with a continuous t-norm *. It is shown that the category of [0, 1]_*-sets is cartesian closed if, and only if, * is the minimum t-norm on [0,1].</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109767"},"PeriodicalIF":2.7,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039484","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}

{"title":"Exploring projective synchronization in discrete-time fractional-order fuzzy cellular neural networks with distributed delays","authors":"Kejin Li,&nbsp;Feifei Du","doi":"10.1016/j.fss.2026.109770","DOIUrl":"10.1016/j.fss.2026.109770","url":null,"abstract":"<div><div>The projective synchronization (PS) of discrete-time fractional-order fuzzy cellular neural networks (DFFCNNs) with distributed delays is investigated in this paper. First, based on the nabla fractional-order difference theory, a comparison principle suitable for fractional-order systems with variable coefficients and multiple time delays is established, and the sub-multiplicative law of the nabla Mittag-Leffler function is rigorously proved. Second, a discrete-time fractional-order Halanay inequality with arbitrary step size, variable coefficients, and multiple time-varying delays is introduced. Furthermore, leveraging the aforementioned inequality, a sufficient condition for the PS of DFFCNNs is derived. Finally, an example is presented to confirm the validity of the results.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109770"},"PeriodicalIF":2.7,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145963130","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}

{"title":"ψ-Type multistability of takagi-Sugeno fuzzy neural networks with general discontinuous activation functions","authors":"Yang Liu ,&nbsp;Zhen Wang ,&nbsp;Xia Huang ,&nbsp;Hao Shen","doi":"10.1016/j.fss.2026.109768","DOIUrl":"10.1016/j.fss.2026.109768","url":null,"abstract":"<div><div>This paper put forward a type of general discontinuous activation functions (AFs) and then investigate the <em>ψ</em>-type multistability of fuzzy neural networks (FNNs). Through determining some algebraic inequalities, it is shown that FNNs with such discontinuous AFs can produce <span><math><msup><mrow><mo>(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mi>n</mi></msup></math></span> equilibrium points (EPs), in which <span><math><msup><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mi>n</mi></msup></math></span> EPs are locally <em>ψ</em>-stable and located at points of continuity (POC) of the AFs. Here, <em>k</em> refers to the number of discontinuous points of the AFs. Depending on the choice of the function <em>ψ</em>(<em>t</em>), the obtained <span><math><msup><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mi>n</mi></msup></math></span> EPs in FNNs can exhibit different types of stability. FNNs with the designed AFs are able to possess larger number of locally <em>ψ</em>-stable EPs and total EPs compared with general continuous AFs. Therefore, when applied in associative memory, FNNs with the above discontinuous AFs are able to store more memory patterns. Besides, attraction basins (ABs) associated with the <em>ψ</em>-stable EPs in FNNs are estimated. The correctness of the obtained results are verified through three examples.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109768"},"PeriodicalIF":2.7,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981152","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}

{"title":"Some properties of two types of fuzzy rough sets on complete lattices constructed by means of overlap and grouping functions","authors":"Siyu Xu,&nbsp;Xiaodong Pan,&nbsp;Yexing Dan,&nbsp;Keyun Qin","doi":"10.1016/j.fss.2026.109769","DOIUrl":"10.1016/j.fss.2026.109769","url":null,"abstract":"<div><div>Recently, by using overlap and grouping functions, Han et al. introduced two novel types of fuzzy rough sets on complete lattices in an <em>L</em>-fuzzy approximation space (<em>U, V, R</em>), along with their application to three-way decisions. We refer to these two types of fuzzy rough sets as the 1^<em>st</em> type and the 2^<em>nd</em> type of fuzzy rough sets. It is worth noting that, in the case where (<em>U, V, R</em>) is an <em>L</em>-fuzzy approximation space, many properties of these two types of fuzzy rough sets established in <em>L</em>-fuzzy approximation spaces of the form (<em>U, R</em>) (i.e., <span><math><mrow><mi>U</mi><mo>=</mo><mi>V</mi></mrow></math></span>) are generally difficult to establish in this more general framework. Therefore, in this paper, we focus on exploring some properties of these two types of fuzzy rough sets under the restriction to an <em>L</em>-fuzzy approximation space (<em>U, R</em>), with particular emphasis on how they generate Alexandrov <em>L</em>-fuzzy topologies. In particular, regarding the 2^<em>nd</em> type of fuzzy rough sets, we deduce the behaviours of the upper and lower <em>L</em>-fuzzy rough approximation operators of an <em>L</em>-fuzzy approximation space (<em>U, R</em>) in the case of a family of <em>L</em>-fuzzy relations. Moreover, we explore the relationships among the pair of upper and lower <em>L</em>-fuzzy rough approximation operators proposed by Jiang and Hu in 2022 and the two pairs of upper and lower <em>L</em>-fuzzy rough approximation operators introduced in this study. Our investigations can be regarded as a contribution to enriching the theoretical framework of the two novel types of fuzzy rough sets by Han et al. in an <em>L</em>-fuzzy approximation space (<em>U, V, R</em>).</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109769"},"PeriodicalIF":2.7,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981153","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}

{"title":"Context-based sum via multi-adjoint bonds","authors":"Roberto G. Aragón,&nbsp;Jesús Medina,&nbsp;Samuel Molina-Ruiz","doi":"10.1016/j.fss.2026.109771","DOIUrl":"10.1016/j.fss.2026.109771","url":null,"abstract":"<div><div>In many situations is fundamental to use a procedure to aggregate information obtained from different sources (devices), such as when an edge computing system is used. Bonds were introduced in formal concept analysis as an aggregation method for linking different contexts (datasets) whilst preserving the information they contain. In this paper, we generalize the notion of bond to the multi-adjoint concept lattice framework, which is a fuzzy and flexible extension of formal concept analysis. Furthermore, we study several properties of multi-adjoint bonds defined by the constantly top or constantly bottom relations, with an emphasis on how they aggregate the information in the concept lattices.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109771"},"PeriodicalIF":2.7,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981154","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}

{"title":"Distributivity between S-uninorms and general overlap or general grouping functions","authors":"Jieqiong Shi ,&nbsp;Bin Zhao ,&nbsp;Bernard De Baets","doi":"10.1016/j.fss.2026.109777","DOIUrl":"10.1016/j.fss.2026.109777","url":null,"abstract":"<div><div>It is well known that the distributivity of one aggregation function over another is a desirable interaction between aggregation functions and has been continuously studied in the literature both for theoretical and practical reasons. Among the many classes of aggregation functions introduced over the past decades, the classes of uninorms and nullnorms stand out because of their potential applications in a broad variety of fields. Similarly, the classes of overlap and grouping functions have received ample attention. In this paper, on the one hand, we focus on the class of S-uninorms, a common generalization of nullnorms and conjunctive uninorms. On the other hand, we consider other more general classes of general overlap and general grouping functions. We continue and wrap up the investigation of the distributivity equation for the above-mentioned classes. In particular, we discuss the distributivity of S-uninorms (with an underlying uninorm belonging to <span><math><msub><mi>U</mi><mi>min</mi></msub></math></span>) over general overlap or general grouping functions, and vice versa. In both cases, we fully characterize the solutions by providing necessary and sufficient conditions.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109777"},"PeriodicalIF":2.7,"publicationDate":"2026-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039543","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}