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A classification of polynomial functions satisfying the Jacobi identity over integral domains Marichal, Jean-Luc ; in Aequationes Mathematicae (in press) 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: 18 (2 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: 68 (15 UL)Strongly barycentrically associative and preassociative functions Teheux, Bruno ; Marichal, Jean-Luc Scientific Conference (2016, November 08) Detailed reference viewed: 23 (4 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: 85 (13 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: 84 (17 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: 173 (6 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: 42 (9 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: 51 (12 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: 101 (10 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: 36 (4 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: 49 (12 UL)A characterization of barycentrically preassociative functions Marichal, Jean-Luc ; Teheux, Bruno in Results in Mathematics (2016), 69(1), 245-256 We provide a characterization of the variadic functions which are barycentrically preassociative as compositions of length-preserving associative string functions with one-to-one unary maps. We also ... [more ▼] We provide a characterization of the variadic functions which are barycentrically preassociative as compositions of length-preserving associative string functions with one-to-one unary maps. We also discuss some consequences of this characterization. [less ▲] Detailed reference viewed: 66 (26 UL)Agrégation des valeurs médianes et fonctions compatibles pour la comparaison ; Marichal, Jean-Luc ; Teheux, Bruno in Lefèvre, Erice; Hadjali, Allel (Eds.) Rencontres francophones sur la logique floue et ses applications 2015 (2015, November) Nous caractérisons les fonctions d'agrégation qui préservent la valeur médiane dans le cas où l'opération médiane est conservative. Nous commençons par rappeler les notions d'algèbre et de demi-treillis ... [more ▼] Nous caractérisons les fonctions d'agrégation qui préservent la valeur médiane dans le cas où l'opération médiane est conservative. Nous commençons par rappeler les notions d'algèbre et de demi-treillis médian en les introduisant à partir de la notion de valeur médiane sur les réels. Nous obtenons également une double caractérisation des algèbres médianes conservatives en termes de sous-structures interdites et de représentations par des chaînes [less ▲] Detailed reference viewed: 45 (10 UL)A classification of barycentrically associative polynomial functions Marichal, Jean-Luc ; ; in Aequationes Mathematicae (2015), 89(5), 1281-1291 We describe the class of polynomial functions which are barycentrically associative over an infinite commutative integral domain. Detailed reference viewed: 62 (12 UL)Median Preserving Aggregation Functions ; Marichal, Jean-Luc ; Teheux, Bruno in Baczyński, Michal; De Baets, Bernard; Mesiar, Radko (Eds.) Proceedings of the 8th Int. Summer School on Aggregation Operators (2015, July) A median algebra is a ternary algebra that satisfies every equation satisfied by the median terms of distributive lattices. We present a characterization theorem for aggregation functions over ... [more ▼] A median algebra is a ternary algebra that satisfies every equation satisfied by the median terms of distributive lattices. We present a characterization theorem for aggregation functions over conservative median algebras. In doing so, we give a characterization of conservative median algebras by means of forbidden substructures and by providing their representation as chains. [less ▲] Detailed reference viewed: 55 (3 UL)Generalizations and variants of associativity for aggregation functions (2-hour tutorial) Marichal, Jean-Luc ; Teheux, Bruno in Baczyński, Michal; De Baets, Bernard; Mesiar, Radko (Eds.) Proceedings of the 8th Int. Summer School on Aggregation Operators (2015, July) Detailed reference viewed: 49 (17 UL)Generalizations and variants of associativity for variadic functions: a survey (invited talk) Marichal, Jean-Luc Scientific Conference (2015, June 17) See attached PDF file Detailed reference viewed: 36 (7 UL)Preassociative aggregation functions Marichal, Jean-Luc ; Teheux, Bruno in Fuzzy Sets & Systems (2015), 268 The classical property of associativity is very often considered in aggregation function theory and fuzzy logic. In this paper we provide axiomatizations of various classes of preassociative functions ... [more ▼] The classical property of associativity is very often considered in aggregation function theory and fuzzy logic. In this paper we provide axiomatizations of various classes of preassociative functions, where preassociativity is a generalization of associativity recently introduced by the authors. These axiomatizations are based on existing characterizations of some noteworthy classes of associative operations, such as the class of Aczélian semigroups and the class of t-norms. [less ▲] Detailed reference viewed: 76 (12 UL)Algorithms and formulas for conversion between system signatures and reliability functions Marichal, Jean-Luc in Journal of Applied Probability (2015), 52(2), 490-507 The concept of signature is a useful tool in the analysis of semicoherent systems with continuous and i.i.d. component lifetimes, especially for the comparison of different system designs and the ... [more ▼] The concept of signature is a useful tool in the analysis of semicoherent systems with continuous and i.i.d. component lifetimes, especially for the comparison of different system designs and the computation of the system reliability. For such systems, we provide conversion formulas between the signature and the reliability function through the corresponding vector of dominations and we derive efficient algorithms for the computation of any of these concepts from the other. We also show how the signature can be easily computed from the reliability function via basic manipulations such as differentiation, coefficient extraction, and integration. [less ▲] Detailed reference viewed: 97 (18 UL)Barycentrically associative and preassociative functions Marichal, Jean-Luc ; Teheux, Bruno in Acta Mathematica Hungarica (2015), 145(2), 468-488 We investigate the barycentric associativity property for functions with indefinite arities and discuss the more general property of barycentric preassociativity, a generalization of barycentric ... [more ▼] We investigate the barycentric associativity property for functions with indefinite arities and discuss the more general property of barycentric preassociativity, a generalization of 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 barycentrically preassociative functions. [less ▲] Detailed reference viewed: 69 (16 UL) |
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