Pub Date : 2024-11-08DOI: 10.1016/j.fss.2024.109180
Yang Wu, Lianjun Hu, Qi Chen, Yong Zhang, Libing Wu
This paper focuses on the problem of dynamic event-triggered fault-tolerant control (FTC) for nonaffine systems with unmeasurable state and asymmetric error constrain. At first, an adaptive failure compensation observer is designed to calculate unpredictable state of the nonaffine systems. Then, a dynamic event triggering strategy (DETS) is constructed to adjust the control signal and make it more flexible in transmission. Next, a new error-dependent conversion function (EDCF) is designed to confine tracing error, which can ensure that the designed Lyapunov function is always positive definite, and extend the symmetric constraint boundary to the asymmetric constraint boundary. The tracking performance and stability of closed-loop systems are proven through Lyapunov stability analysis. Finally, the availability of this method is verified by simulation experiments.
{"title":"Dynamic event-triggered fault-tolerant control for nonaffine systems with asymmetric error constraint","authors":"Yang Wu, Lianjun Hu, Qi Chen, Yong Zhang, Libing Wu","doi":"10.1016/j.fss.2024.109180","DOIUrl":"10.1016/j.fss.2024.109180","url":null,"abstract":"<div><div>This paper focuses on the problem of dynamic event-triggered fault-tolerant control (FTC) for nonaffine systems with unmeasurable state and asymmetric error constrain. At first, an adaptive failure compensation observer is designed to calculate unpredictable state of the nonaffine systems. Then, a dynamic event triggering strategy (DETS) is constructed to adjust the control signal and make it more flexible in transmission. Next, a new error-dependent conversion function (EDCF) is designed to confine tracing error, which can ensure that the designed Lyapunov function is always positive definite, and extend the symmetric constraint boundary to the asymmetric constraint boundary. The tracking performance and stability of closed-loop systems are proven through Lyapunov stability analysis. Finally, the availability of this method is verified by simulation experiments.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"499 ","pages":"Article 109180"},"PeriodicalIF":3.2,"publicationDate":"2024-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142663943","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 : 2024-11-08DOI: 10.1016/j.fss.2024.109188
Bin Pang , Xiu-Yun Wu , Bernard De Baets
Ternary fuzzy relations, and fuzzy betweenness relations in particular, are witnessing increasing attention in recent years. A key reason is that axiomatic properties of ternary fuzzy relations seem to be ideally suited to capture geometric characteristics of the abstract notion of betweenness. In this paper, we introduce several new properties of ternary fuzzy relations, including the Peano property, the Pasch property and the sand-glass property, that can be qualified as geometric properties. We investigate their interrelationships as well as their connections with various types of fuzzy betweenness relations. Additionally, in the context of our study of the Pasch property and the sand-glass property, we introduce the convexity property of ternary fuzzy relations by taking inspiration from the solid theoretical basis of the theory of fuzzy convex structures.
{"title":"Geometric properties of ternary fuzzy relations","authors":"Bin Pang , Xiu-Yun Wu , Bernard De Baets","doi":"10.1016/j.fss.2024.109188","DOIUrl":"10.1016/j.fss.2024.109188","url":null,"abstract":"<div><div>Ternary fuzzy relations, and fuzzy betweenness relations in particular, are witnessing increasing attention in recent years. A key reason is that <em>axiomatic properties</em> of ternary fuzzy relations seem to be ideally suited to capture <em>geometric characteristics</em> of the abstract notion of betweenness. In this paper, we introduce several new properties of ternary fuzzy relations, including the Peano property, the Pasch property and the sand-glass property, that can be qualified as <em>geometric properties</em>. We investigate their interrelationships as well as their connections with various types of fuzzy betweenness relations. Additionally, in the context of our study of the Pasch property and the sand-glass property, we introduce the convexity property of ternary fuzzy relations by taking inspiration from the solid theoretical basis of the theory of fuzzy convex structures.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"499 ","pages":"Article 109188"},"PeriodicalIF":3.2,"publicationDate":"2024-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142663945","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 : 2024-11-07DOI: 10.1016/j.fss.2024.109165
Ali Dastbaravarde, Ali Dolati
A popular measure of association is the tail dependence coefficient, which measures the strength of dependence in the tail of a bivariate distribution. In this paper, we introduce the concept of quantile dependence, which extends the idea of tail dependence and can be used to measure the dependence of specific quantiles of two random variables in a particular area of the distribution's domain. We analyze the characteristics of the proposed quantile dependence coefficient and provide various examples to demonstrate our findings.
{"title":"Quantile dependence: A generalization of upper and lower tail dependence","authors":"Ali Dastbaravarde, Ali Dolati","doi":"10.1016/j.fss.2024.109165","DOIUrl":"10.1016/j.fss.2024.109165","url":null,"abstract":"<div><div>A popular measure of association is the tail dependence coefficient, which measures the strength of dependence in the tail of a bivariate distribution. In this paper, we introduce the concept of quantile dependence, which extends the idea of tail dependence and can be used to measure the dependence of specific quantiles of two random variables in a particular area of the distribution's domain. We analyze the characteristics of the proposed quantile dependence coefficient and provide various examples to demonstrate our findings.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"499 ","pages":"Article 109165"},"PeriodicalIF":3.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142663940","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 : 2024-11-07DOI: 10.1016/j.fss.2024.109169
Wojciech Sałabun
This paper introduces Asymmetric Interval Numbers , a novel type of interval numbers that combines the straightforward identification characteristic of classical interval numbers with the advanced capability for modeling uncertainty found in fuzzy sets, all while maintaining simplicity. incorporate the expected value within the interval, providing a more accurate representation of the uncertainty of the data compared to the classical interval numbers. We define basic arithmetic operations for , discuss their properties, and provide proofs. Additionally, we present theorems on symmetry and asymmetry for fundamental binary and unary operations, enhancing the mathematical framework for . Several illustrative examples demonstrate the practical application of . Although challenges remain, such as exploring performance with different distributions and reducing overestimation, present a promising advancement in interval arithmetic. This study underscores the practical and theoretical implications of , paving the way for further research and application in diverse scientific and industrial contexts.
本文介绍了非对称区间数 (AIN)，这是一种新型区间数，它结合了经典区间数的直接识别特性和模糊集的先进不确定性建模能力，同时又保持了简洁性。AIN 包含区间内的期望值，与经典区间数相比，能更准确地表示数据的不确定性。我们定义了 AIN 的基本算术运算，讨论了它们的特性，并提供了证明。此外，我们还提出了基本二元和一元运算的对称性和不对称性定理，从而加强了 AIN 的数学框架。几个示例展示了 AIN 的实际应用。尽管仍存在一些挑战，如探索不同分布的性能和减少高估，但 AINs 是区间运算中一个很有前途的进步。本研究强调了 AINs 的实践和理论意义，为进一步研究 AINs 并将其应用于各种科学和工业领域铺平了道路。
{"title":"Asymmetric interval numbers: A new approach to modeling uncertainty","authors":"Wojciech Sałabun","doi":"10.1016/j.fss.2024.109169","DOIUrl":"10.1016/j.fss.2024.109169","url":null,"abstract":"<div><div>This paper introduces Asymmetric Interval Numbers <span><math><mo>(</mo><mi>A</mi><mi>I</mi><mi>N</mi><mi>s</mi><mo>)</mo></math></span>, a novel type of interval numbers that combines the straightforward identification characteristic of classical interval numbers with the advanced capability for modeling uncertainty found in fuzzy sets, all while maintaining simplicity. <span><math><mi>A</mi><mi>I</mi><mi>N</mi><mi>s</mi></math></span> incorporate the expected value within the interval, providing a more accurate representation of the uncertainty of the data compared to the classical interval numbers. We define basic arithmetic operations for <span><math><mi>A</mi><mi>I</mi><mi>N</mi><mi>s</mi></math></span>, discuss their properties, and provide proofs. Additionally, we present theorems on symmetry and asymmetry for fundamental binary and unary operations, enhancing the mathematical framework for <span><math><mi>A</mi><mi>I</mi><mi>N</mi><mi>s</mi></math></span>. Several illustrative examples demonstrate the practical application of <span><math><mi>A</mi><mi>I</mi><mi>N</mi><mi>s</mi></math></span>. Although challenges remain, such as exploring performance with different distributions and reducing overestimation, <span><math><mi>A</mi><mi>I</mi><mi>N</mi><mi>s</mi></math></span> present a promising advancement in interval arithmetic. This study underscores the practical and theoretical implications of <span><math><mi>A</mi><mi>I</mi><mi>N</mi><mi>s</mi></math></span>, paving the way for further research and application in diverse scientific and industrial contexts.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"499 ","pages":"Article 109169"},"PeriodicalIF":3.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142663942","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 : 2024-11-07DOI: 10.1016/j.fss.2024.109187
József Dombi , Tamás Jónás
In fuzzy logic, most of the implication operators are based on generalizations of the classical, material implication. That is, these implications are defined as the disjunction of the negated value of the first argument and the value of the second argument, while the underlying disjunction operators are associative triangular conorms. In our study, we concentrate on how a class of implication operators, called the preference implication operators, can be used in approximate reasoning. Using this implication operator family, we present a novel, Modus Ponens-like approximate reasoning method, in which we have two premises: (1) a statement and (2) a preference implication with an antecedent of this statement. Here, we show how the continuous logical value of the consequent of the preference implication can be derived from the continuous logical values of the premises. We point out that this novel approximate reasoning method is strongly connected with the so-called aggregative operator, which is a representable uninorm. Next, we also present a threshold value-based generalization of the Modus Ponens syllogism and demonstrate that the Modus Tollens syllogism can be generalized in the same way. Lastly, we provide an illustrative example.
{"title":"Approximate reasoning based on the preference implication","authors":"József Dombi , Tamás Jónás","doi":"10.1016/j.fss.2024.109187","DOIUrl":"10.1016/j.fss.2024.109187","url":null,"abstract":"<div><div>In fuzzy logic, most of the implication operators are based on generalizations of the classical, material implication. That is, these implications are defined as the disjunction of the negated value of the first argument and the value of the second argument, while the underlying disjunction operators are associative triangular conorms. In our study, we concentrate on how a class of implication operators, called the preference implication operators, can be used in approximate reasoning. Using this implication operator family, we present a novel, Modus Ponens-like approximate reasoning method, in which we have two premises: (1) a statement and (2) a preference implication with an antecedent of this statement. Here, we show how the continuous logical value of the consequent of the preference implication can be derived from the continuous logical values of the premises. We point out that this novel approximate reasoning method is strongly connected with the so-called aggregative operator, which is a representable uninorm. Next, we also present a threshold value-based generalization of the Modus Ponens syllogism and demonstrate that the Modus Tollens syllogism can be generalized in the same way. Lastly, we provide an illustrative example.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"499 ","pages":"Article 109187"},"PeriodicalIF":3.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142663941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-05DOI: 10.1016/j.fss.2024.109178
Tong Kang , Radko Mesiar , Yao Ouyang , Jun Li
In this note, we provide an example to show the weak (M)-property is really weaker than the (M)-property for some finite monotone measure defined on infinite space, and hence answer an open problem which was proposed in the paper (Li et al. (2023) [9]). We prove that if a monotone measure μ is autocontinuous, then the weak (M)-property and the (M)-property of μ are equivalent. We propose the concept of (C-P)-property of monotone measures and show a set of sufficient and necessary conditions that the Choquet integral coincides with the pan-integral. We further study the relationships between the Choquet integral and the pan-integral in the setting of the ordered pairs of monotone measures, and obtain some interesting properties.
{"title":"Some notes on the coincidence of the Choquet integral and the pan-integral","authors":"Tong Kang , Radko Mesiar , Yao Ouyang , Jun Li","doi":"10.1016/j.fss.2024.109178","DOIUrl":"10.1016/j.fss.2024.109178","url":null,"abstract":"<div><div>In this note, we provide an example to show the weak (M)-property is really weaker than the (M)-property for some finite monotone measure defined on infinite space, and hence answer an open problem which was proposed in the paper (Li et al. (2023) <span><span>[9]</span></span>). We prove that if a monotone measure <em>μ</em> is autocontinuous, then the weak (M)-property and the (M)-property of <em>μ</em> are equivalent. We propose the concept of <em>(C-P)-property</em> of monotone measures and show a set of sufficient and necessary conditions that the Choquet integral coincides with the pan-integral. We further study the relationships between the Choquet integral and the pan-integral in the setting of the ordered pairs of monotone measures, and obtain some interesting properties.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"499 ","pages":"Article 109178"},"PeriodicalIF":3.2,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142664022","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 : 2024-11-05DOI: 10.1016/j.fss.2024.109177
Bilel Selmi
The main aim of this paper is to present a comprehensive exploration of general multifractal dimensions of Borel probability measures, with a specific emphasis on the diverse methodologies available for their estimation or computation. We prove that these general dimensions enable us to compare a probability measure ϑ with the general multifractal Hausdorff or packing measure. Also, we aim to measure the degree of regularity or singularity of ϑ, and we examine the correlations between various approaches to defining the general multifractal exact dimension of Borel probability measures, using fractal analysis of the essential supports of these measures as the foundation.
{"title":"General multifractal dimensions of measures","authors":"Bilel Selmi","doi":"10.1016/j.fss.2024.109177","DOIUrl":"10.1016/j.fss.2024.109177","url":null,"abstract":"<div><div>The main aim of this paper is to present a comprehensive exploration of general multifractal dimensions of Borel probability measures, with a specific emphasis on the diverse methodologies available for their estimation or computation. We prove that these general dimensions enable us to compare a probability measure <em>ϑ</em> with the general multifractal Hausdorff or packing measure. Also, we aim to measure the degree of regularity or singularity of <em>ϑ</em>, and we examine the correlations between various approaches to defining the general multifractal exact dimension of Borel probability measures, using fractal analysis of the essential supports of these measures as the foundation.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"499 ","pages":"Article 109177"},"PeriodicalIF":3.2,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142593729","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 : 2024-11-04DOI: 10.1016/j.fss.2024.109174
Carmen Torres-Blanc , Jesus Martinez-Mateo , Susana Cubillo , Luis Magdalena , Francisco Javier Talavera , Jorge Elorza
In this work we present a novel axiomatic framework for subsethood measures in type-2 fuzzy sets. It differs from previous approaches in two key ways. First, the degree of membership is not simply a fuzzy set as considered in other papers, but rather a label of the variable Truth, more in line with Zadeh's original idea. Secondly, the concept of subsethood is approached in terms of its relationship with cardinality. Additionally, illustrative examples of such measures are provided.
在这项研究中，我们提出了一个新颖的公理框架，用于研究 2 型模糊集合中的子实体度量。它在两个关键方面不同于以往的方法。首先，成员度并不像其他论文所考虑的那样只是一个模糊集，而是变量 Truth 的一个标签，这更符合 Zadeh 最初的想法。其次，子实体的概念是根据其与万有引力的关系来处理的。此外，还提供了此类度量的示例。
{"title":"Subsethood measures based on cardinality of type-2 fuzzy sets","authors":"Carmen Torres-Blanc , Jesus Martinez-Mateo , Susana Cubillo , Luis Magdalena , Francisco Javier Talavera , Jorge Elorza","doi":"10.1016/j.fss.2024.109174","DOIUrl":"10.1016/j.fss.2024.109174","url":null,"abstract":"<div><div>In this work we present a novel axiomatic framework for subsethood measures in type-2 fuzzy sets. It differs from previous approaches in two key ways. First, the degree of membership is not simply a fuzzy set as considered in other papers, but rather a label of the variable <em>Truth</em>, more in line with Zadeh's original idea. Secondly, the concept of subsethood is approached in terms of its relationship with cardinality. Additionally, illustrative examples of such measures are provided.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"499 ","pages":"Article 109174"},"PeriodicalIF":3.2,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142593727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-04DOI: 10.1016/j.fss.2024.109175
Yongchao Wang , Bin Pang , Fu-Gui Shi
Based on a complete residuated lattice L, we propose a new approach to fuzzification of coarse structures. Firstly, we introduce the concepts of L-entourages and L-(quasi, semi)-coarse structures and study the maps between L-coarse spaces. Secondly, we provide some examples of L-coarse structures from the aspects of L-metrics, L-relations and L-hyperstructures. Thirdly, we introduce the notion of L-ball structures to characterize L-coarse structures. Finally, we establish the relationship among L-entourage spaces, L-quasi-coarse spaces, L-semi-coarse spaces and L-coarse spaces in a categorical viewpoint.
基于完整的残差网格 L，我们提出了一种模糊化粗结构的新方法。首先，我们介绍了 L-梯度和 L-（准、半）粗结构的概念，并研究了 L-粗空间之间的映射。其次，我们从 L-度量、L-关系和 L-超结构等方面提供了一些 L-粗结构的例子。第三，我们引入 L 球结构的概念来表征 L 粗结构。最后，我们从分类的角度建立了 L-鼓励空间、L-准粗大空间、L-半粗大空间和 L-粗大空间之间的关系。
{"title":"Lattice-valued coarse structures","authors":"Yongchao Wang , Bin Pang , Fu-Gui Shi","doi":"10.1016/j.fss.2024.109175","DOIUrl":"10.1016/j.fss.2024.109175","url":null,"abstract":"<div><div>Based on a complete residuated lattice <em>L</em>, we propose a new approach to fuzzification of coarse structures. Firstly, we introduce the concepts of <em>L</em>-entourages and <em>L</em>-(quasi, semi)-coarse structures and study the maps between <em>L</em>-coarse spaces. Secondly, we provide some examples of <em>L</em>-coarse structures from the aspects of <em>L</em>-metrics, <em>L</em>-relations and <em>L</em>-hyperstructures. Thirdly, we introduce the notion of <em>L</em>-ball structures to characterize <em>L</em>-coarse structures. Finally, we establish the relationship among <em>L</em>-entourage spaces, <em>L</em>-quasi-coarse spaces, <em>L</em>-semi-coarse spaces and <em>L</em>-coarse spaces in a categorical viewpoint.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"499 ","pages":"Article 109175"},"PeriodicalIF":3.2,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142593728","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 : 2024-10-31DOI: 10.1016/j.fss.2024.109172
Wenwen Zong , Yong Su , Radko Mesiar
This paper focuses on t-norms which have some additive generators. We show that a t-norm T has some additive generators if and only if the fully ordered semigroup induced by T can be embedded into some fully ordered semigroup induced by either the product or the Lukasiewicz t-norm .
本文重点研究具有一些可加生成器的 t-norms 。我们证明，当且仅当 T 所诱导的完全有序半群可以嵌入到乘积 TP 或卢卡西维茨 t-norm TL 所诱导的完全有序半群中时，t-norm T 才有一些可加生成器。
{"title":"A note on t-norms having additive generators","authors":"Wenwen Zong , Yong Su , Radko Mesiar","doi":"10.1016/j.fss.2024.109172","DOIUrl":"10.1016/j.fss.2024.109172","url":null,"abstract":"<div><div>This paper focuses on t-norms which have some additive generators. We show that a t-norm <em>T</em> has some additive generators if and only if the fully ordered semigroup induced by <em>T</em> can be embedded into some fully ordered semigroup induced by either the product <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> or the Lukasiewicz t-norm <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>L</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"499 ","pages":"Article 109172"},"PeriodicalIF":3.2,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142578848","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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