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Radko Mesiar - One of the best experts on this subject based on the ideXlab platform.
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Triangular norms: Basic notions and properties
Logical Algebraic Analytic and Probabilistic Aspects of Triangular Norms, 2020Co-Authors: Erich Peter Klement, Radko MesiarAbstract:Abstract The basic definitions concerning triangular norms and conorms are collected in this chapter. We also mention the most important algebraic and analytical properties a triangular norm may have. We discuss in detail the construction of triangular norms by means of additive and multiplicative generators and via ordinal sums, and we mention also some other construction methods. Finally we present the representation theorems of Continuous Archimedean triangular norms (via Continuous additive or multiplicative generators) and of Continuous triangular norms (as ordinal sums of Continuous Archimedean triangular norms).
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lipschitz continuity of triangular subnorms
Fuzzy Sets and Systems, 2014Co-Authors: Roberto Ghiselli Ricci, Radko Mesiar, Andrea MesiarovazemankovaAbstract:This paper deals with the Lipschitz property of triangular subnorms. Unlike the case of triangular norms, for these operations the problem is still open and presents an interesting variety of situations. We provide some characterization results by weakening the notion of convexity, introducing two generalized versions of convexity for real functions, called @a-lower convexity and sub-convexity. The @a-lower convex and sub-convex real mappings present characteristics quite different from the usual convex real mappings. We will discuss the link between such kind of functions and the generators, and their pseudo-inverse, of Continuous Archimedean triangular subnorms.
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Triangular norms. Position paper III: Continuous t-norms
Fuzzy Sets and Systems, 2004Co-Authors: Erich Peter Klement, Radko MesiarAbstract:This third and last part of a series of position papers on triangular norms (for Parts I and II see (E.P. Klement, R. Mesiar, E. Pap, Triangular norms, Position paper I: basic analytical and algebraic properties, Fuzzy Sets and Systems, in press; E.P. Klement, R. Mesiar, E. Pap, Triangular norms. Position paper II: general constructions and parameterized families, submitted for publication) presents the representation of Continuous Archimedean t-norms by means of additive generators, and the representation of Continuous t-norms by means of ordinal sums with Archimedean summands, both with full proofs. Finally some consequences of these representation theorems in the context of comparison and convergence of Continuous t-norms, and of the determination of Continuous t-norms by their diagonal sections are mentioned.
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Metrics and T-Equalities
Journal of Mathematical Analysis and Applications, 2002Co-Authors: Bernard De Baets, Radko MesiarAbstract:Abstract The relationship between metrics and T -equalities is investigated; the latter are a special case of T -equivalences, a natural generalization of the classical concept of an equivalence relation. It is shown that in the construction of metrics from T -equalities triangular norms with an additive generator play a key role. Conversely, in the construction of T -equalities from metrics this role is played by triangular norms with a Continuous additive generator or, equivalently, by Continuous Archimedean triangular norms. These results are then applied to the biresidual operator E T of a triangular norm T . It is shown that E T is a T -equality on [0, 1] if and only if T is left-Continuous. Furthermore, it is shown that to any left-Continuous triangular norm T there correspond two particular T -equalities on F ( X ), the class of fuzzy sets in a given universe X ; one of these T -equalities is obtained from the biresidual operator E T T by means of a natural extension procedure. These T -equalities then give rise to interesting metrics on F ( X ).
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Continuous generated associative aggregation operators
Fuzzy Sets and Systems, 2002Co-Authors: Tomasa Calvo, Radko MesiarAbstract:Continuous generated associative aggregation operators are discussed. The only asymmetric solutions are the trivial operators of the projection to the first or the last coordinate, respectively. The only nontrivial convenient aggregation operators are necessarily symmetric and they are either Continuous Archimedean t-norms and t-conorms, or generated nullnorms. The latter may combine any nilpotent t-conorm and any t-norm with any prescribed annihilator k ∈ ]0, 1[.
Milan Petrík - One of the best experts on this subject based on the ideXlab platform.
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Explicit formulas for generators of triangular norms
Publicationes Mathematicae Debrecen, 2020Co-Authors: Mirko Navara, Milan Petrík, Peter SarkociAbstract:Triangular norms are associative operations which represent conjunctions in fuzzy logic. They were also studied in the context of probabilistic metric spaces. It is known that each Continuous Archimedean triangular norm can be determined by additive and multiplicative generators. However, ¯nding a generator of a given triangular norm may be a di±cult task. The geometry of the generator does not seem to re°ect the properties of the triangular norm in an intuitive way. We show that this need not be the case for a large class of triangular norms which allow to reconstruct the generators from partial derivatives of triangular norms. This class is broad enough to cover all Continuous Archimedean triangular norms which we found in the literature.
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dominance on Continuous Archimedean triangular norms and generalized mulholland inequality
Fuzzy Sets and Systems, 2020Co-Authors: Milan PetríkAbstract:Abstract As a preceding result, it has been shown that the dominance relation is not transitive on the set of strict triangular norms. This result has been achieved thanks to new results on Mulholland inequality. Recently, Saminger-Platz, De Baets, and De Meyer have introduced the generalized Mulholland inequality which characterizes the dominance on all Continuous Archimedean triangular norms in an analogous way as does Mulholland inequality on the strict triangular norms. Based on these new results, the present paper shows that the dominance relation is not transitive on the set of nilpotent triangular norms and, consequently, on the set of Continuous Archimedean triangular norms. This result is achieved by introducing a new sufficient condition under which a given function solves the generalized Mulholland inequality and by showing that the set of the functions that solve the inequality is not closed with respect to compositions.
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IFSA-SCIS - On generalized mulholland inequality and dominance on nilpotent triangular norms
2017 Joint 17th World Congress of International Fuzzy Systems Association and 9th International Conference on Soft Computing and Intelligent Systems (, 2017Co-Authors: Milan PetríkAbstract:The paper is focused on the generalized Mulholland inequality and its connection with the dominance relation defined on the set of Continuous Archimedean triangular norms. A counter-example is provided showing that the set of the solutions of the generalized Mulholland inequality is not closed with respect to compositions and that the dominance relation is not transitive on the set of nilpotent triangular norms.
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Associativity of triangular norms characterized by the geometry of their level sets
Fuzzy Sets and Systems, 2012Co-Authors: Milan Petrík, Peter SarkociAbstract:Associativity of triangular norms is an algebraic property which, unlike for example their commutativity, is usually understood as hardly visually interpretable. This problem has been studied intensively in the last decade and, as a result, geometric symmetries of triangular norms with involutive level sets have been revealed. The presented paper intends to introduce a different approach which gives more general results. The inspiration is taken from web geometry, a branch of differential geometry, and its concept of Reidemeister closure condition which is known to provide a geometric characterization of associativity of loops. The paper shows that this concept can be adopted successfully for triangular norms so that it characterizes their associativity in a similar way. Moreover, the offered adaptation preserves the beneficial transparency and simplicity of the Reidemeister closure condition. This way, a visual characterization of the associativity, based on the geometry of the level sets, is provided for general, Continuous, and Continuous Archimedean triangular norms.
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ISMVL - Reconstruction of Additive Generators from Partial Derivatives of Continuous Archimedean t-Norms
2010 40th IEEE International Symposium on Multiple-Valued Logic, 2010Co-Authors: Mirko Navara, Milan Petrík, Peter SarkociAbstract:The paper shows a direct correspondence between the first partial derivatives of a Continuous Archimedean triangular norm and the first derivatives of its additive generator. An explicit formula for the additive generator is obtained. Application of the result is demonstrated on the problem of convex combinations of strict triangular norms.
Bernard De Baets - One of the best experts on this subject based on the ideXlab platform.
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Bell-type inequalities for parametric families of triangular norms
Kybernetika, 2020Co-Authors: Saskia Janssens, Bernard De Baets, Hans De MeyerAbstract:In recent work we have shown that the reformulation of the classical Bell inequalities into the context of fuzzy probability calculus leads to related inequalities on the commutative conjunctor used for modelling pointwise fuzzy set intersection. Also, an important role has been attributed to commutative quasi-copulas. In this paper, we consider these new Bell-type inequalities for Continuous t-norms. Our contribution is twofold: first, we prove that ordinal sums preserve these Bell-type inequalities; second, for the most important parametric families of Continuous Archimedean t-norms and each of the inequalities, we identify the parameter values such that the corresponding t-norms satisfy the inequality considered.
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EUSFLAT Conf. - The Domination Relation Between Continuous T-Norms
2020Co-Authors: Susanne Saminger, Bernard De Baets, Hans De MeyerAbstract:The present contribution deals with domination in the framework of Continuous t-norms. Basic properties and recent results for Continuous Archimedean and Continuous ordinal sum t-norms are presented and discussed. The domination property within several families of t-norms is mentioned.
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Differential inequality conditions for dominance between Continuous Archimedean t-norms
Mathematical Inequalities & Applications, 2020Co-Authors: Susanne Saminger-platz, Bernard De Baets, Hans De MeyerAbstract:Dominance between triangular norms (t-norms) is a versatile relationship. For Continuous Archimedean t-norms, dominance can be verified by checking one of many sufficient conditions derived from a generalization of the Mulholland inequality. These conditions pertain to various convexity properties of compositions of additive generators and their inverses. In this paper, assuming differentiability of these additive generators, we propose equivalent sufficient conditions that can be expressed as inequalities involving derivatives of the additive generators, avoiding the need of composing them. We demonstrate the powerfulness of the results by the straightforward rediscovery of dominance relationships in the Schweizer-Sklar t-norm family, as well as by unveiling some formerly unknown dominance relationships in the Sugeno-Weber t-norm family. Finally, we illustrate that the results can also be applied to members of different parametric families of t-norm.
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On the construction of fuzzy betweenness relations from metrics
Fuzzy Sets and Systems, 2020Co-Authors: Hua-peng Zhang, Raúl Pérez-fernández, Bernard De BaetsAbstract:Abstract We consider the problem of constructing a fuzzy betweenness relation from a metric. More precisely, given a Continuous Archimedean triangular norm, we present two construction methods for a fuzzy betweenness relation from a metric by making use of the pseudo-inverse of either a Continuous additive generator or a Continuous multiplicative generator of the triangular norm. In case the metric is bounded and given a 1-Lipschitz Continuous triangular norm, we present a third construction method for a fuzzy betweenness relation from a metric by making use of the residual implication of the triangular norm. Since the Łukasiewicz and product triangular norms are both Continuous Archimedean and 1-Lipschitz Continuous, all three construction methods may be used. Interestingly, the construction method based on the residual implication is proved to coincide with that based on a Continuous additive generator for the Łukasiewicz triangular norm and with that based on a Continuous multiplicative generator for the product triangular norm. We end by noting that all three construction methods result in a fuzzy prebetweenness relation when considering a pseudometric instead of a metric.
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Ordinal sums of triangular norms on a bounded lattice
Fuzzy Sets and Systems, 2020Co-Authors: Yao Ouyang, Hua-peng Zhang, Bernard De BaetsAbstract:Abstract The ordinal sum construction provides a very effective way to generate a new triangular norm on the real unit interval from existing ones. One of the most prominent theorems concerning the ordinal sum of triangular norms on the real unit interval states that a triangular norm is Continuous if and only if it is uniquely representable as an ordinal sum of Continuous Archimedean triangular norms. However, the ordinal sum of triangular norms on subintervals of a bounded lattice is not always a triangular norm (even if only one summand is involved), if one just extends the ordinal sum construction to a bounded lattice in a naive way. In the present paper, appropriately dealing with those elements that are incomparable with the endpoints of the given subintervals, we propose an alternative definition of ordinal sum of countably many (finite or countably infinite) triangular norms on subintervals of a complete lattice, where the endpoints of the subintervals constitute a chain. The completeness requirement for the lattice is not needed when considering finitely many triangular norms. The newly proposed ordinal sum is shown to be always a triangular norm. Several illustrative examples are given.
Shuming Wang - One of the best experts on this subject based on the ideXlab platform.
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Modeling renewal processes in fuzzy decision system
Applied Mathematical Modelling, 2015Co-Authors: Shuming WangAbstract:Abstract Under expected value of fuzzy variable and Continuous Archimedean triangular norms, this paper discusses a renewal process and a renewal reward process for T-independent L - R fuzzy variables in fuzzy decision systems. First, a renewal process with T-independent L - R fuzzy interarrival times is discussed, some limit theorems on renewal variable, average renewal time, and long-term renewal rate in (fuzzy) measure are obtained, and a fuzzy elementary renewal theorem is proved for the limit of the long-term expected renewal rate. Second, a renewal reward process with T-independent L - R fuzzy interarrival times and rewards is discussed, a limit theorem on reward rate in (fuzzy) measure is derived, and a fuzzy renewal reward theorem is proved for the limit value of expected reward rate. Finally, the comparison with stochastic counterparts shows an interesting and reasonable homology in convergence mode and limit value between the results obtained in fuzzy renewal processes and the corresponding results in stochastic renewal processes, though they build on two essentially different mathematical cornerstones, possibility theory and probability theory, respectively.
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Some properties of T-independent fuzzy variables
Mathematical and Computer Modelling, 2011Co-Authors: Shuming Wang, Junzo WatadaAbstract:T-independence of fuzzy variables is a more general concept than the classical independence. The objective of this study is to deal with some new properties of T-independent fuzzy variables. First of all, for any general t-norm, some criteria of T-independence are discussed for fuzzy variables under possibility, necessity and credibility measures. Subsequently, on the basis of left Continuous t-norms, some formulas are derived on the ''max'' and ''min'' operations of the T-independent fuzzy variables in possibility distribution and in expectation. Finally, making use of Continuous Archimedean t-norms, several convergence properties are discussed for T-independent fuzzy variables in credibility and in expectation, respectively, and some laws of large numbers are proved as well.
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T-norm-based limit theorems for fuzzy random variables
Journal of Intelligent and Fuzzy Systems, 2010Co-Authors: Shuming Wang, Junzo WatadaAbstract:The objective of this paper is to derive some limit theorems of fuzzy random variables under the extension principle associated with Continuous Archimedean triangular norms (t-norms). First of all, some convergence theorems for the sum of fuzzy random variables in chance measure and expected value are proved respectively based on the arithmetics of Continuous Archimedean triangular norms. Then, a law of large numbers for fuzzy random variables is established by using the obtained convergence theorems. The results of the derived law of large numbers can degenerate to the strong laws of large numbers for random variables and fuzzy variables, respectively.
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Fuzzy random renewal reward process and its applications
Information Sciences, 2009Co-Authors: Shuming Wang, Junzo WatadaAbstract:This paper studies a renewal reward process with fuzzy random interarrival times and rewards under the @?-independence associated with any Continuous Archimedean t-norm @?. The interarrival times and rewards of the renewal reward process are assumed to be positive fuzzy random variables whose fuzzy realizations are @?-independent fuzzy variables. Under these conditions, some limit theorems in mean chance measure are derived for fuzzy random renewal rewards. In the sequel, a fuzzy random renewal reward theorem is proved for the long-run expected reward per unit time of the renewal reward process. The renewal reward theorem obtained in this paper can degenerate to that of stochastic renewal theory. Finally, some application examples are provided to illustrate the utility of the result.
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Fuzzy random renewal process with queueing applications
Computers & Mathematics With Applications, 2009Co-Authors: Shuming Wang, Junzo WatadaAbstract:Using extension principle associated with a class of Continuous Archimedean triangular norms, this paper studies a fuzzy random renewal process in which the interarrival times are assumed to be independent and identically distributed fuzzy random variables. Some limit theorems in chance measure and in expected value for the sum of fuzzy random variables are proved on the basis of the Continuous Archimedean triangular norm based arithmetics. Furthermore, we discuss the fuzzy random renewal process based on the obtained limit theorems, and derive a fuzzy random elementary renewal theorem for the long-run expected renewal rate. The renewal theorem obtained in this paper can degenerate to the corresponding classical result in stochastic renewal process. Finally, two case studies of queueing systems are provided to illustrate the application of the fuzzy random elementary renewal theorem.
Junzo Watada - One of the best experts on this subject based on the ideXlab platform.
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Some properties of T-independent fuzzy variables
Mathematical and Computer Modelling, 2011Co-Authors: Shuming Wang, Junzo WatadaAbstract:T-independence of fuzzy variables is a more general concept than the classical independence. The objective of this study is to deal with some new properties of T-independent fuzzy variables. First of all, for any general t-norm, some criteria of T-independence are discussed for fuzzy variables under possibility, necessity and credibility measures. Subsequently, on the basis of left Continuous t-norms, some formulas are derived on the ''max'' and ''min'' operations of the T-independent fuzzy variables in possibility distribution and in expectation. Finally, making use of Continuous Archimedean t-norms, several convergence properties are discussed for T-independent fuzzy variables in credibility and in expectation, respectively, and some laws of large numbers are proved as well.
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T-norm-based limit theorems for fuzzy random variables
Journal of Intelligent and Fuzzy Systems, 2010Co-Authors: Shuming Wang, Junzo WatadaAbstract:The objective of this paper is to derive some limit theorems of fuzzy random variables under the extension principle associated with Continuous Archimedean triangular norms (t-norms). First of all, some convergence theorems for the sum of fuzzy random variables in chance measure and expected value are proved respectively based on the arithmetics of Continuous Archimedean triangular norms. Then, a law of large numbers for fuzzy random variables is established by using the obtained convergence theorems. The results of the derived law of large numbers can degenerate to the strong laws of large numbers for random variables and fuzzy variables, respectively.
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Fuzzy random renewal reward process and its applications
Information Sciences, 2009Co-Authors: Shuming Wang, Junzo WatadaAbstract:This paper studies a renewal reward process with fuzzy random interarrival times and rewards under the @?-independence associated with any Continuous Archimedean t-norm @?. The interarrival times and rewards of the renewal reward process are assumed to be positive fuzzy random variables whose fuzzy realizations are @?-independent fuzzy variables. Under these conditions, some limit theorems in mean chance measure are derived for fuzzy random renewal rewards. In the sequel, a fuzzy random renewal reward theorem is proved for the long-run expected reward per unit time of the renewal reward process. The renewal reward theorem obtained in this paper can degenerate to that of stochastic renewal theory. Finally, some application examples are provided to illustrate the utility of the result.
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Fuzzy random renewal process with queueing applications
Computers & Mathematics With Applications, 2009Co-Authors: Shuming Wang, Junzo WatadaAbstract:Using extension principle associated with a class of Continuous Archimedean triangular norms, this paper studies a fuzzy random renewal process in which the interarrival times are assumed to be independent and identically distributed fuzzy random variables. Some limit theorems in chance measure and in expected value for the sum of fuzzy random variables are proved on the basis of the Continuous Archimedean triangular norm based arithmetics. Furthermore, we discuss the fuzzy random renewal process based on the obtained limit theorems, and derive a fuzzy random elementary renewal theorem for the long-run expected renewal rate. The renewal theorem obtained in this paper can degenerate to the corresponding classical result in stochastic renewal process. Finally, two case studies of queueing systems are provided to illustrate the application of the fuzzy random elementary renewal theorem.
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On laws of large numbers for L-R fuzzy variables
2008 International Conference on Machine Learning and Cybernetics, 2008Co-Authors: Shuming Wang, Junzo WatadaAbstract:In this study, we discuss the laws of large numbers for T-independent L-R fuzzy variables based on Continuous Archimedean t-norm and expected value of fuzzy variable. First, by using Continuous Archimedean t-norm, we derive several convergent properties of sum of L-R fuzzy variables in credibility measure and in expected value, respectively. Then, on the basis of the obtained convergent properties, we establish some laws of large numbers for T-independent L-R fuzzy variables.