Pub Date : 2026-03-15Epub Date: 2025-11-24DOI: 10.1016/j.fss.2025.109704
Changzhong Wang , Haiyang Zhao , Shuang An
The fuzzy rough set theory has shown considerable promise in feature selection. However, traditional methods encounter significant challenges when dealing with datasets characterized by large class density differences and noise contamination. To address these issues, this paper introduces a directed distance-based soft neighborhood fuzzy rough set model (DSNFRS) for feature selection, designed to improve both its effectiveness and robustness. To account for class density differences, the model utilizes class variances to comprehensively capture the intricate relationships between samples. To mitigate the effects of data noise, it integrates the concepts of directed distance, soft neighborhoods, and fuzzy rough sets, and introduces a novel pair of fuzzy rough approximation operators that better characterize data uncertainty. This approach effectively filters out noise and outliers, enhancing the stability and accuracy of fuzzy rough feature selection. Experimental evaluations across 15 datasets demonstrate that the proposed algorithm outperforms most existing feature selection methods. The DSNFRS model offers an efficient and robust solution for feature selection in multi-density and noise data environments.
{"title":"Feature selection based on fuzzy rough sets with directed soft neighborhood","authors":"Changzhong Wang , Haiyang Zhao , Shuang An","doi":"10.1016/j.fss.2025.109704","DOIUrl":"10.1016/j.fss.2025.109704","url":null,"abstract":"<div><div>The fuzzy rough set theory has shown considerable promise in feature selection. However, traditional methods encounter significant challenges when dealing with datasets characterized by large class density differences and noise contamination. To address these issues, this paper introduces a directed distance-based soft neighborhood fuzzy rough set model (DSNFRS) for feature selection, designed to improve both its effectiveness and robustness. To account for class density differences, the model utilizes class variances to comprehensively capture the intricate relationships between samples. To mitigate the effects of data noise, it integrates the concepts of directed distance, soft neighborhoods, and fuzzy rough sets, and introduces a novel pair of fuzzy rough approximation operators that better characterize data uncertainty. This approach effectively filters out noise and outliers, enhancing the stability and accuracy of fuzzy rough feature selection. Experimental evaluations across 15 datasets demonstrate that the proposed algorithm outperforms most existing feature selection methods. The DSNFRS model offers an efficient and robust solution for feature selection in multi-density and noise data environments.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"527 ","pages":"Article 109704"},"PeriodicalIF":2.7,"publicationDate":"2026-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145625151","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 : 2026-03-15Epub Date: 2025-11-19DOI: 10.1016/j.fss.2025.109684
Alessandro Gallo, Francesca Adele Giambona
Economic insecurity has gained increasing attention over the last decade, particularly in terms of its measurement and how it affects everyday life. This paper contributes to the literature on measurement by proposing a new individual-level, multidimensional index based on a fuzzy sets approach. The fuzzy logic moves beyond the classic binary framework of set theory, which classifies elements strictly as 0 or 1. In the fuzzy sets approach, each set is defined by a membership function that indicates the degree to which each element belongs to the set. This flexibility makes it particularly well suited for capturing complex socio-economic conditions such as economic insecurity. The proposed measure incorporates a range of economic insecurity indicators and offers some advantages. First, it produces an individual score that can be easily aggregated for geographical and socio-demographic comparisons. Second, the methodology allows for precise estimation of the variance, which is useful for assessing the reliability of aggregate estimates. The new index is applied to the Italian context using the most recent EU-SILC data. Aggregate estimates by region and socio-demographic group are derived and compared. Results indicate that the well-known North-South gradient persists and that economic insecurity is higher among the most disadvantaged sub-populations. In particular, individuals with low educational attainment and those who are unemployed or inactive experience the highest levels of economic insecurity.
{"title":"Measuring economic insecurity using a fuzzy sets approach","authors":"Alessandro Gallo, Francesca Adele Giambona","doi":"10.1016/j.fss.2025.109684","DOIUrl":"10.1016/j.fss.2025.109684","url":null,"abstract":"<div><div>Economic insecurity has gained increasing attention over the last decade, particularly in terms of its measurement and how it affects everyday life. This paper contributes to the literature on measurement by proposing a new individual-level, multidimensional index based on a fuzzy sets approach. The fuzzy logic moves beyond the classic binary framework of set theory, which classifies elements strictly as 0 or 1. In the fuzzy sets approach, each set is defined by a membership function that indicates the degree to which each element belongs to the set. This flexibility makes it particularly well suited for capturing complex socio-economic conditions such as economic insecurity. The proposed measure incorporates a range of economic insecurity indicators and offers some advantages. First, it produces an individual score that can be easily aggregated for geographical and socio-demographic comparisons. Second, the methodology allows for precise estimation of the variance, which is useful for assessing the reliability of aggregate estimates. The new index is applied to the Italian context using the most recent EU-SILC data. Aggregate estimates by region and socio-demographic group are derived and compared. Results indicate that the well-known North-South gradient persists and that economic insecurity is higher among the most disadvantaged sub-populations. In particular, individuals with low educational attainment and those who are unemployed or inactive experience the highest levels of economic insecurity.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"527 ","pages":"Article 109684"},"PeriodicalIF":2.7,"publicationDate":"2026-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145584369","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 : 2026-03-15Epub Date: 2025-11-26DOI: 10.1016/j.fss.2025.109702
Pingke Li
The consistency of max-min equations is a crucial issue to be concerned when modelling with fuzzy relations in approximate reasoning. An inconsistent system of max-min equations may be perturbed slightly to restore the consistency. This paper tackles the inconsistency resolving problem by means of Chebyshev approximation, i.e., minimizing the maximum absolute deviation in both the coefficient matrix and right-hand side vector. It demonstrates that the minimum deviation level for consistency may be obtained by a polynomial-time direct search method. A Chebyshev approximation is constructed accordingly with the number of modified elements in the coefficient matrix as few as possible.
{"title":"A note on Chebyshev approximation to an inconsistent system of max-min equations","authors":"Pingke Li","doi":"10.1016/j.fss.2025.109702","DOIUrl":"10.1016/j.fss.2025.109702","url":null,"abstract":"<div><div>The consistency of max-min equations is a crucial issue to be concerned when modelling with fuzzy relations in approximate reasoning. An inconsistent system of max-min equations may be perturbed slightly to restore the consistency. This paper tackles the inconsistency resolving problem by means of Chebyshev approximation, i.e., minimizing the maximum absolute deviation in both the coefficient matrix and right-hand side vector. It demonstrates that the minimum deviation level for consistency may be obtained by a polynomial-time direct search method. A Chebyshev approximation is constructed accordingly with the number of modified elements in the coefficient matrix as few as possible.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"527 ","pages":"Article 109702"},"PeriodicalIF":2.7,"publicationDate":"2026-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145694062","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 : 2026-03-15Epub Date: 2025-11-19DOI: 10.1016/j.fss.2025.109673
Bo Xu , Changzhong Wang , Shuang An , Yang Huang
Fuzzy rough set theory offers an effective approach for feature selection; however, traditional methods lack an adaptive learning mechanism to adjust feature weights, making it difficult to accurately measure the contribution of each feature to classification. To address this issue, this paper introduces a novel dynamic optimization feature selection method based on maximum likelihood estimation. The method leverages the fuzzy similarity relation strategy from fuzzy rough sets to handle data uncertainty, while employing maximum likelihood estimation to assess feature importance. Specifically, the proposed model treats class labels as observed data and sample features as hidden variables, evaluating the classification ability of features by constructing a maximum likelihood function. Feature weights and class variances are integrated into the fuzzy similarity relation, and they are dynamically adjusted in accordance with the data characteristics through collaborative optimization. The inclusion degrees of samples are utilized to derive the empirical estimation of the conditional probability of classes relative to features. Finally, maximum likelihood estimation is applied to optimize the weighted features, assess their impact on the target variable, and select those that best explain the variation of the target variable. In this way, the model combines the strengths of fuzzy similarity relations in addressing uncertainty and the power of maximum likelihood estimation in parameter estimation, significantly enhancing the accuracy and robustness of feature selection. The experimental results show that the proposed algorithm has significant advantages over mainstream comparison methods on 18 benchmark data sets and provides a novel solution for feature selection in the field of uncertain data.
{"title":"Feature selection driven by maximum likelihood estimation and fuzzy similarity relation learning","authors":"Bo Xu , Changzhong Wang , Shuang An , Yang Huang","doi":"10.1016/j.fss.2025.109673","DOIUrl":"10.1016/j.fss.2025.109673","url":null,"abstract":"<div><div>Fuzzy rough set theory offers an effective approach for feature selection; however, traditional methods lack an adaptive learning mechanism to adjust feature weights, making it difficult to accurately measure the contribution of each feature to classification. To address this issue, this paper introduces a novel dynamic optimization feature selection method based on maximum likelihood estimation. The method leverages the fuzzy similarity relation strategy from fuzzy rough sets to handle data uncertainty, while employing maximum likelihood estimation to assess feature importance. Specifically, the proposed model treats class labels as observed data and sample features as hidden variables, evaluating the classification ability of features by constructing a maximum likelihood function. Feature weights and class variances are integrated into the fuzzy similarity relation, and they are dynamically adjusted in accordance with the data characteristics through collaborative optimization. The inclusion degrees of samples are utilized to derive the empirical estimation of the conditional probability of classes relative to features. Finally, maximum likelihood estimation is applied to optimize the weighted features, assess their impact on the target variable, and select those that best explain the variation of the target variable. In this way, the model combines the strengths of fuzzy similarity relations in addressing uncertainty and the power of maximum likelihood estimation in parameter estimation, significantly enhancing the accuracy and robustness of feature selection. The experimental results show that the proposed algorithm has significant advantages over mainstream comparison methods on 18 benchmark data sets and provides a novel solution for feature selection in the field of uncertain data.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"527 ","pages":"Article 109673"},"PeriodicalIF":2.7,"publicationDate":"2026-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145625045","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}
In this paper, the impact of nonlinear components is studied for cooperative load transportation systems with any number of quadrotors and a single slung load suspended by ropes. The main goal is to control and estimate constraints caused by the nonlinear term of the load transportation system. A novel distributed control strategy is proposed for cooperative systems based on adaptive fuzzy wavelet networks (AFWNs). Distributed AFWNs are employed to compensate for nonlinear effects. Another result is the expansion of the system’s attraction region for the initial state values. Also, by employing an integral term in the control law, the formation error of the agents converges to zero. These expansions allow the system to significantly improve its robustness to disturbances. The simulation results illustrate that the proposed method can keep the agents in desired formation and guide the load in right direction.
{"title":"Adaptive fuzzy wavelet network control for nonlinear cooperative load transportation systems","authors":"Matin Fadavi , Majdeddin Najafi , Farid Sheikholeslam","doi":"10.1016/j.fss.2025.109681","DOIUrl":"10.1016/j.fss.2025.109681","url":null,"abstract":"<div><div>In this paper, the impact of nonlinear components is studied for cooperative load transportation systems with any number of quadrotors and a single slung load suspended by ropes. The main goal is to control and estimate constraints caused by the nonlinear term of the load transportation system. A novel distributed control strategy is proposed for cooperative systems based on adaptive fuzzy wavelet networks (AFWNs). Distributed AFWNs are employed to compensate for nonlinear effects. Another result is the expansion of the system’s attraction region for the initial state values. Also, by employing an integral term in the control law, the formation error of the agents converges to zero. These expansions allow the system to significantly improve its robustness to disturbances. The simulation results illustrate that the proposed method can keep the agents in desired formation and guide the load in right direction.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"527 ","pages":"Article 109681"},"PeriodicalIF":2.7,"publicationDate":"2026-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145625044","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 : 2026-03-15Epub Date: 2025-11-21DOI: 10.1016/j.fss.2025.109685
Nicolás Madrid, Manuel Ojeda-Aciego
We analyze the use of the composition of mappings as a fuzzy conjunction between indexes of inclusion. Instead of the general approach of the φ-index of inclusion, we consider a fresh approach that computes the φ-index of inclusion when restricted to a join-subsemilattice of indexes of inclusion. Under this restriction, we identify a certain join-subsemilattice which has a biresiduated structure when composition is interpreted as conjunction. The main consequence of this biresiduated structure is a representation theorem of biresiduated lattices on the unit interval in terms of the composition and subsets of indexes of inclusion.
{"title":"Composition as a fuzzy conjunction between indexes of inclusion","authors":"Nicolás Madrid, Manuel Ojeda-Aciego","doi":"10.1016/j.fss.2025.109685","DOIUrl":"10.1016/j.fss.2025.109685","url":null,"abstract":"<div><div>We analyze the use of the composition of mappings as a fuzzy conjunction between indexes of inclusion. Instead of the general approach of the φ-index of inclusion, we consider a fresh approach that computes the φ-index of inclusion when restricted to a join-subsemilattice of indexes of inclusion. Under this restriction, we identify a certain join-subsemilattice which has a biresiduated structure when composition is interpreted as conjunction. The main consequence of this biresiduated structure is a representation theorem of biresiduated lattices on the unit interval in terms of the composition and subsets of indexes of inclusion.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"527 ","pages":"Article 109685"},"PeriodicalIF":2.7,"publicationDate":"2026-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145694061","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 : 2026-03-01Epub Date: 2025-11-14DOI: 10.1016/j.fss.2025.109679
M.D.M. Bibiloni-Femenias , O. Valero
In the literature there are two different approaches that extend the classical crisp notion of equivalence relation to the fuzzy framework. On the one hand, one can find the notion of indistinguishability operator and a few of its generalizations. These can be understood as a kind of measurement of the degree of similarity or indistinguishability between objects. On the other hand, fuzzy (quasi-)metrics measure such a degree with respect to a parameter. The study of both types of the aforesaid notions has been carried out independently without any connection between them. As a consequence, the notion of modular indistinguishability operator has been introduced recently. Such a notion unifies under the same framework both aforesaid similarity concepts. In this paper, we explore the aggregation problem for modular indistinguishability operators and for several generalizations. Hence we introduce the notions of modular fuzzy pre-order, modular fuzzy partial order and modular equality and we characterize the functions that are able to fuse all these different types of modular similarities. The aforementioned characterizations are stated in terms of triangular triplets or related notions, monotony and dominance. In contrast to the non-modular case, the class of those functions that merge modular fuzzy pre-orders (modular fuzzy partial orders) is shown to match the class of modular indistinguishability operators (modular equalities). Furthermore, the relationships between the non-modular aggregation problem, the modular one and the fuzzy metric aggregation problem are explored and the differences between them are clarified by means of appropriate examples.
{"title":"Modular indistinguishability: The aggregation problem","authors":"M.D.M. Bibiloni-Femenias , O. Valero","doi":"10.1016/j.fss.2025.109679","DOIUrl":"10.1016/j.fss.2025.109679","url":null,"abstract":"<div><div>In the literature there are two different approaches that extend the classical crisp notion of equivalence relation to the fuzzy framework. On the one hand, one can find the notion of indistinguishability operator and a few of its generalizations. These can be understood as a kind of measurement of the degree of similarity or indistinguishability between objects. On the other hand, fuzzy (quasi-)metrics measure such a degree with respect to a parameter. The study of both types of the aforesaid notions has been carried out independently without any connection between them. As a consequence, the notion of modular indistinguishability operator has been introduced recently. Such a notion unifies under the same framework both aforesaid similarity concepts. In this paper, we explore the aggregation problem for modular indistinguishability operators and for several generalizations. Hence we introduce the notions of modular fuzzy pre-order, modular fuzzy partial order and modular equality and we characterize the functions that are able to fuse all these different types of modular similarities. The aforementioned characterizations are stated in terms of triangular triplets or related notions, monotony and dominance. In contrast to the non-modular case, the class of those functions that merge modular fuzzy pre-orders (modular fuzzy partial orders) is shown to match the class of modular indistinguishability operators (modular equalities). Furthermore, the relationships between the non-modular aggregation problem, the modular one and the fuzzy metric aggregation problem are explored and the differences between them are clarified by means of appropriate examples.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"526 ","pages":"Article 109679"},"PeriodicalIF":2.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145580333","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 : 2026-03-01Epub Date: 2025-11-04DOI: 10.1016/j.fss.2025.109665
Jiaying Wang , Zhehuang Huang , Zhifeng Weng , Jinjin Li
As a typical multi-granularity data analysis model, multi-scale decision systems have received widespread attention from researchers in recent years. However, most multi-scale models struggle to handle continuous data and fail to accurately characterize the differences between samples in complex scenes. Moreover, there is a lack of investigation on fuzzy multi-scale uncertainty measures, as well as their application in dimension reduction. Motivated by these issues, we put forth a new multi-scale fuzzy relation decision system and investigate the uncertainty measures for fuzzy relation families at different scales. To this end, δ-fuzzy similarity relationship is presented to characterize the correlation of target objects. Fuzzy scale entropy is then proposed to reflect the distinguishing ability of fuzzy relation families with different scales. Some variants of the uncertainty measure, such as joint fuzzy scale entropy, conditional fuzzy scale entropy, and mutual fuzzy scale entropy, are then presented to reveal the relationship between the distinguishing ability of feature subsets. Finally, a knowledge reduction algorithm for multi-scale fuzzy relation decision systems is developed from the perspective of maintaining the distinguishing ability. Extensive experiments on 16 public datasets exhibit that our model can effectively reduce redundant features from different scales, and demonstrates competitive classification performance compared with four state-of-the-art dimension reduction algorithms.
{"title":"Feature subset selection using fuzzy scale entropy-Based uncertainty measures for multi-scale fuzzy relation decision systems","authors":"Jiaying Wang , Zhehuang Huang , Zhifeng Weng , Jinjin Li","doi":"10.1016/j.fss.2025.109665","DOIUrl":"10.1016/j.fss.2025.109665","url":null,"abstract":"<div><div>As a typical multi-granularity data analysis model, multi-scale decision systems have received widespread attention from researchers in recent years. However, most multi-scale models struggle to handle continuous data and fail to accurately characterize the differences between samples in complex scenes. Moreover, there is a lack of investigation on fuzzy multi-scale uncertainty measures, as well as their application in dimension reduction. Motivated by these issues, we put forth a new multi-scale fuzzy relation decision system and investigate the uncertainty measures for fuzzy relation families at different scales. To this end, <em>δ</em>-fuzzy similarity relationship is presented to characterize the correlation of target objects. Fuzzy scale entropy is then proposed to reflect the distinguishing ability of fuzzy relation families with different scales. Some variants of the uncertainty measure, such as joint fuzzy scale entropy, conditional fuzzy scale entropy, and mutual fuzzy scale entropy, are then presented to reveal the relationship between the distinguishing ability of feature subsets. Finally, a knowledge reduction algorithm for multi-scale fuzzy relation decision systems is developed from the perspective of maintaining the distinguishing ability. Extensive experiments on 16 public datasets exhibit that our model can effectively reduce redundant features from different scales, and demonstrates competitive classification performance compared with four state-of-the-art dimension reduction algorithms.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"526 ","pages":"Article 109665"},"PeriodicalIF":2.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145529490","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 : 2026-03-01Epub Date: 2025-11-13DOI: 10.1016/j.fss.2025.109680
Raquel Fernandez-Peralta , Andrea Mesiarová-Zemánková
Fuzzy implication functions constitute fundamental operators in fuzzy logic systems, extending classical conditionals to manage uncertainty in logical inference. Among the extensive families of these operators, generalizations of the classical material implication have received considerable theoretical attention, particularly (S, N)-implications constructed from t-conorms and fuzzy negations, and their further generalizations to (U, N)-implications using disjunctive uninorms. Prior work has established characterization theorems for these families under the assumption that the fuzzy negation N is continuous, ensuring uniqueness of representation. In this paper, we disprove this last fact for (U, N)-implications and we show that they do not necessarily possess a unique representation, even if the fuzzy negation is continuous. Further, we provide a comprehensive study of uniqueness conditions for both uninorms with continuous and non-continuous underlying functions. Our results offer important theoretical insights into the structural properties of these operators.
{"title":"On the non-uniqueness of representation of (U, N)-implications","authors":"Raquel Fernandez-Peralta , Andrea Mesiarová-Zemánková","doi":"10.1016/j.fss.2025.109680","DOIUrl":"10.1016/j.fss.2025.109680","url":null,"abstract":"<div><div>Fuzzy implication functions constitute fundamental operators in fuzzy logic systems, extending classical conditionals to manage uncertainty in logical inference. Among the extensive families of these operators, generalizations of the classical material implication have received considerable theoretical attention, particularly (<em>S, N</em>)-implications constructed from t-conorms and fuzzy negations, and their further generalizations to (<em>U, N</em>)-implications using disjunctive uninorms. Prior work has established characterization theorems for these families under the assumption that the fuzzy negation <em>N</em> is continuous, ensuring uniqueness of representation. In this paper, we disprove this last fact for (<em>U, N</em>)-implications and we show that they do not necessarily possess a unique representation, even if the fuzzy negation is continuous. Further, we provide a comprehensive study of uniqueness conditions for both uninorms with continuous and non-continuous underlying functions. Our results offer important theoretical insights into the structural properties of these operators.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"526 ","pages":"Article 109680"},"PeriodicalIF":2.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145580334","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 : 2026-03-01Epub Date: 2025-11-14DOI: 10.1016/j.fss.2025.109678
Jih-Jeng Huang , Chin-Yi Chen
We introduce a unified framework extending the classical Choquet integral by incorporating Stieltjes-type accumulation functions and dual set-functions. This construction, termed the dual Choquet–Stieltjes (DCS) integral, broadens non-additive integral theory, allowing simultaneous treatment of threshold-dependent behaviors and asymmetric interactions. We prove fundamental properties including well-definedness, monotonicity, and comonotonic additivity under precisely specified conditions. We establish convergence theorems (monotone convergence, Fatou’s lemma, dominated convergence) with complete proofs, and demonstrate applications in decision-making. Our framework generalizes existing extensions under a single, coherent approach that maintains theoretical properties while enhancing modeling flexibility. Through parameter recovery studies, we demonstrate the theoretical soundness of our approach and identify scenarios where the full DCS framework is necessary to capture complex interdependencies and threshold effects.
{"title":"The choquet–Stieltjes integral with dual set-Functions: a unified theory and applications","authors":"Jih-Jeng Huang , Chin-Yi Chen","doi":"10.1016/j.fss.2025.109678","DOIUrl":"10.1016/j.fss.2025.109678","url":null,"abstract":"<div><div>We introduce a unified framework extending the classical Choquet integral by incorporating Stieltjes-type accumulation functions and dual set-functions. This construction, termed the dual Choquet–Stieltjes (DCS) integral, broadens non-additive integral theory, allowing simultaneous treatment of threshold-dependent behaviors and asymmetric interactions. We prove fundamental properties including well-definedness, monotonicity, and comonotonic additivity under precisely specified conditions. We establish convergence theorems (monotone convergence, Fatou’s lemma, dominated convergence) with complete proofs, and demonstrate applications in decision-making. Our framework generalizes existing extensions under a single, coherent approach that maintains theoretical properties while enhancing modeling flexibility. Through parameter recovery studies, we demonstrate the theoretical soundness of our approach and identify scenarios where the full DCS framework is necessary to capture complex interdependencies and threshold effects.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"526 ","pages":"Article 109678"},"PeriodicalIF":2.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145580336","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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