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Enumerating quasitrivial semigroups Devillet, Jimmy ; ; Marichal, Jean-Luc Presentation (2017, October 03) We investigate the class of binary associative and quasitrivial operations on a given finite set. Here quasitriviality (also known as conserva-tiveness) means that the operation always outputs one of its ... [more ▼] We investigate the class of binary associative and quasitrivial operations on a given finite set. Here quasitriviality (also known as conserva-tiveness) means that the operation always outputs one of its input values. We also examine the special situations where the operations are commutative and nondecreasing. In the latter case, these operations reduce to discrete uninorms, which are discrete fuzzy connectives that play an important role in fuzzy logic. As we will see nondecreasing, associative and quasitrivial operations are chara-cterized in terms of total and weak orderings through the so-called single-peakedness property introduced in social choice theory by Duncan Black. This will enable visual interpretaions of the above mentioned algebraic properties. Motivated by these results, we will also address a number of counting issues: we enumerate all binary associative and quasitrivial operations on a given finite set as well as of those operations that are commutative, are nondecreasing, have neutral and/or annihilator elements. As we will see, these considerations lead to several, previously unknown, integer sequences. [less ▲] Detailed reference viewed: 75 (16 UL)Sur les uninormes discrètes idempotentes ; Devillet, Jimmy ; Marichal, Jean-Luc in Couceiro, Miguel; Devillet, Jimmy; Marichal, Jean-Luc (Eds.) LFA 2017 - Rencontres francophones sur la logique floue et ses applications (2017, October) In this paper we provide two axiomatizations of the class of idempotent discrete uninorms as conservative binary operations, where an operation is conservative if it always outputs one of its input values ... [more ▼] In this paper we provide two axiomatizations of the class of idempotent discrete uninorms as conservative binary operations, where an operation is conservative if it always outputs one of its input values. More precisely we first show that the idempotent discrete uninorms are exactly those operations that are conservative, symmetric, and nondecreasing. Then we show that, in this characterization, symmetry can be replaced with both bisymmetry and existence of a neutral element. [less ▲] Detailed reference viewed: 71 (8 UL)Integer sequence #A292933 Marichal, Jean-Luc Diverse speeches and writings (2017) Number of associative and quasitrivial binary operations on {1,...,n} that have neutral elements. Also: Number of associative and quasitrivial binary operations on {1,...,n} that have annihilator elements. Detailed reference viewed: 50 (6 UL)Integer sequence #A292934 Marichal, Jean-Luc Diverse speeches and writings (2017) Number of associative and quasitrivial binary operations on {1,...,n} that have both neutral and annihilator elements. Detailed reference viewed: 45 (5 UL)Integer sequence #A292932 Marichal, Jean-Luc Diverse speeches and writings (2017) Number of associative and quasitrivial binary operations on {1,...,n}. Convention a(0)=1. Detailed reference viewed: 58 (12 UL)A classification of polynomial functions satisfying the Jacobi identity over integral domains Marichal, Jean-Luc ; in Aequationes Mathematicae (2017), 91(4), 601-618 The Jacobi identity is one of the properties that are used to define the concept of Lie algebra and in this context is closely related to associativity. In this paper we provide a complete description of ... [more ▼] The Jacobi identity is one of the properties that are used to define the concept of Lie algebra and in this context is closely related to associativity. In this paper we provide a complete description of all bivariate polynomials that satisfy the Jacobi identity over infinite integral domains. Although this description depends on the characteristic of the domain, it turns out that all these polynomials are of degree at most one in each indeterminate. [less ▲] Detailed reference viewed: 150 (12 UL)Probability signatures of multistate systems made up of two-state components Marichal, Jean-Luc ; ; et al Scientific Conference (2017, July) The structure signature of a system made up of $n$ components having continuous and i.i.d. lifetimes was defined in the eighties by Samaniego as the $n$-tuple whose $k$-th coordinate is the probability ... [more ▼] The structure signature of a system made up of $n$ components having continuous and i.i.d. lifetimes was defined in the eighties by Samaniego as the $n$-tuple whose $k$-th coordinate is the probability that the $k$-th component failure causes the system to fail. More recently, a bivariate version of this concept was considered as follows. The joint structure signature of a pair of systems built on a common set of components having continuous and i.i.d. lifetimes is a square matrix of order $n$ whose $(k,l)$-entry is the probability that the $k$-th failure causes the first system to fail and the $l$-th failure causes the second system to fail. This concept was successfully used to derive a signature-based decomposition of the joint reliability of the two systems. In this talk we will show an explicit formula to compute the joint structure signature of two or more systems and extend this formula to the general non-i.i.d. case, assuming only that the distribution of the component lifetimes has no ties. Then we will discuss a condition on this distribution for the joint reliability of the systems to have a signature-based decomposition. Finally we will show how these results can be applied to the investigation of the reliability and signature of multistate systems made up of two-state components. [less ▲] Detailed reference viewed: 60 (3 UL)On conservative and associative operations on finite chains Devillet, Jimmy ; ; Marichal, Jean-Luc Scientific Conference (2017, June 16) See attached file Detailed reference viewed: 84 (24 UL)Recent results on conservative and symmetric n-ary semigroups Kiss, Gergely ; Devillet, Jimmy ; Marichal, Jean-Luc Scientific Conference (2017, June 16) See attached file Detailed reference viewed: 81 (17 UL)On idempotent discrete uninorms ; Devillet, Jimmy ; Marichal, Jean-Luc in De Baets, Bernard; Torra, Vicenç; Mesiar, Radko (Eds.) Aggregation Functions in Theory and in Practice (2017, June) In this paper we provide two axiomatizations of the class of idempotent discrete uninorms as conservative binary operations, where an operation is conservative if it always outputs one of its input values ... [more ▼] In this paper we provide two axiomatizations of the class of idempotent discrete uninorms as conservative binary operations, where an operation is conservative if it always outputs one of its input values. More precisely we first show that the idempotent discrete uninorms are exactly those operations that are conservative, symmetric, and nondecreasing in each variable. Then we show that, in this characterization, symmetry can be replaced with both bisymmetry and existence of a neutral element. [less ▲] Detailed reference viewed: 172 (36 UL)On the generalized associativity equation Marichal, Jean-Luc ; Teheux, Bruno in Aequationes Mathematicae (2017), 91(2), 265-277 The so-called generalized associativity functional equation G(J(x,y),z) = H(x,K(y,z)) has been investigated under various assumptions, for instance when the unknown functions G, H, J, and K are real ... [more ▼] The so-called generalized associativity functional equation G(J(x,y),z) = H(x,K(y,z)) has been investigated under various assumptions, for instance when the unknown functions G, H, J, and K are real, continuous, and strictly monotonic in each variable. In this note we investigate the following related problem: given the functions J and K, find every function F that can be written in the form F(x,y,z) = G(J(x,y),z) = H(x,K(y,z)) for some functions G and H. We show how this problem can be solved when any of the inner functions J and K has the same range as one of its sections. [less ▲] Detailed reference viewed: 186 (28 UL)Strongly barycentrically associative and preassociative functions Teheux, Bruno ; Marichal, Jean-Luc Scientific Conference (2016, November 08) Detailed reference viewed: 108 (7 UL)Relaxations of associativity and preassociativity for variadic functions ; Marichal, Jean-Luc ; Teheux, Bruno in Fuzzy Sets & Systems (2016), 299 In this paper we consider two properties of variadic functions, namely associativity and preassociativity, that are pertaining to several data and language processing tasks. We propose parameterized ... [more ▼] In this paper we consider two properties of variadic functions, namely associativity and preassociativity, that are pertaining to several data and language processing tasks. We propose parameterized relaxations of these properties and provide their descriptions in terms of factorization results. We also give an example where these parameterized notions give rise to natural hierarchies of functions and indicate their potential use in measuring the degrees of associativeness and preassociativeness. We illustrate these results by several examples and constructions and discuss some open problems that lead to further directions of research. [less ▲] Detailed reference viewed: 173 (21 UL)Structure functions and minimal path sets Marichal, Jean-Luc in IEEE Transactions on Reliability (2016), 65(2), 763-768 In this short note we give and discuss a general multilinear expression of the structure function of an arbitrary semicoherent system in terms of its minimal path and cut sets. We also examine the link ... [more ▼] In this short note we give and discuss a general multilinear expression of the structure function of an arbitrary semicoherent system in terms of its minimal path and cut sets. We also examine the link between the number of minimal path and cut sets consisting of one or two components and the concept of structure signature of the system. [less ▲] Detailed reference viewed: 165 (20 UL)International Symposium on Aggregation and Structures (ISAS 2016) - Book of abstracts Kiss, Gergely ; Marichal, Jean-Luc ; Teheux, Bruno Book published by NA (2016) Detailed reference viewed: 276 (15 UL)A characterisation of associative idempotent nondecreasing functions with neutral elements Kiss, Gergely ; ; Marichal, Jean-Luc et al Scientific Conference (2016, June) Detailed reference viewed: 78 (14 UL)Strongly barycentrically associative and preassociative functions Marichal, Jean-Luc ; Teheux, Bruno in Journal of Mathematical Analysis and Applications (2016), 437(1), 181-193 We study the property of strong barycentric associativity, a stronger version of barycentric associativity for functions with indefinite arities. We introduce and discuss the more general property of ... [more ▼] We study the property of strong barycentric associativity, a stronger version of barycentric associativity for functions with indefinite arities. We introduce and discuss the more general property of strong barycentric preassociativity, a generalization of strong barycentric associativity which does not involve any composition of functions. We also provide a generalization of Kolmogoroff-Nagumo's characterization of the quasi-arithmetic mean functions to strongly barycentrically preassociative functions. [less ▲] Detailed reference viewed: 127 (14 UL)Conservative median algebras and semilattices ; Marichal, Jean-Luc ; Teheux, Bruno in Order : A Journal on the Theory of Ordered Sets and its Applications (2016), 33(1), 121-132 We characterize conservative median algebras and semilattices by means of forbidden substructures and by providing their representation as chains. Moreover, using a dual equivalence between median ... [more ▼] We characterize conservative median algebras and semilattices by means of forbidden substructures and by providing their representation as chains. Moreover, using a dual equivalence between median algebras and certain topological structures, we obtain descriptions of the median-preserving mappings between products of finitely many chains. [less ▲] Detailed reference viewed: 174 (11 UL)The mathematics behind the property of associativity Marichal, Jean-Luc ; Teheux, Bruno in De Baets, Bernard; Mesiar, Radko; Saminger-Platz, Susanne (Eds.) et al 36th Linz Seminar on Fuzzy Set Theory (LINZ 2016) - Functional Equations and Inequalities (2016, February) The well-known equation of associativity for binary operations may be naturally generalized to variadic operations. In this talk, we illustrate different approaches that can be considered to study this ... [more ▼] The well-known equation of associativity for binary operations may be naturally generalized to variadic operations. In this talk, we illustrate different approaches that can be considered to study this extension of associativity, as well as some of its generalizations and variants, including barycentric associativity and preassociativity. [less ▲] Detailed reference viewed: 81 (7 UL)An extension of the concept of distance as functions of several variables Kiss, Gergely ; Marichal, Jean-Luc ; Teheux, Bruno in De Baets, Bernard; Mesiar, Radko; Saminger-Platz, Susanne (Eds.) et al 36th Linz Seminar on Fuzzy Set Theory (LINZ 2016) - Functional Equations and Inequalities (2016, February) Extensions of the concept of distance to more than two elements have been recently proposed in the literature to measure to which extent the elements of a set are spread out. Such extensions may be ... [more ▼] Extensions of the concept of distance to more than two elements have been recently proposed in the literature to measure to which extent the elements of a set are spread out. Such extensions may be particularly useful to define dispersion measures for instance in statistics or data analysis. In this note we provide and discuss an extension of the concept of distance, called n-distance, as functions of n variables. The key feature of this extension is a natural generalization of the triangle inequality. We also provide some examples of n-distances that involve geometric and graph theoretic constructions. [less ▲] Detailed reference viewed: 106 (12 UL) |
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