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A generalization of the concept of distance based on the simplex inequality Kiss, Gergely ; Marichal, Jean-Luc ; Teheux, Bruno in Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry (in press) We introduce and discuss the concept of \emph{$n$-distance}, a generalization to $n$ elements of the classical notion of distance obtained by replacing the triangle inequality with the so-called simplex ... [more ▼] We introduce and discuss the concept of \emph{$n$-distance}, a generalization to $n$ elements of the classical notion of distance obtained by replacing the triangle inequality with the so-called simplex inequality \[ d(x_1, \ldots, x_n)~\leq~K\, \sum_{i=1}^n d(x_1, \ldots, x_n)_i^z{\,}, \qquad x_1, \ldots, x_n, z \in X, \] where $K=1$. Here $d(x_1,\ldots,x_n)_i^z$ is obtained from the function $d(x_1,\ldots,x_n)$ by setting its $i$th variable to $z$. We provide several examples of $n$-distances, and for each of them we investigate the infimum of the set of real numbers $K\in\left]0,1\right]$ for which the inequality above holds. We also introduce a generalization of the concept of $n$-distance obtained by replacing in the simplex inequality the sum function with an arbitrary symmetric function. [less ▲] Detailed reference viewed: 37 (13 UL)Pivotal decomposition schemes inducing clones of operations ; Teheux, Bruno in Beitraege zur Algebra und Geometrie = Contributions to Algebra and Geometry (2018), 59(1), 25-40 We study pivotal decomposition schemes and investigate classes of pivotally decomposable operations. We provide sufficient conditions on pivotal operations that guarantee that the corresponding classes of ... [more ▼] We study pivotal decomposition schemes and investigate classes of pivotally decomposable operations. We provide sufficient conditions on pivotal operations that guarantee that the corresponding classes of pivotally decomposable operations are clones, and show that under certain assumptions these conditions are also necessary. In the latter case, the pivotal operation together with the constant operations generate the corresponding clone. [less ▲] Detailed reference viewed: 43 (15 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: 113 (25 UL)Generalized qualitative Sugeno integrals ; ; et al in Information Sciences (2017), 415-416 Sugeno integrals are aggregation operations involving a criterion weighting scheme based on the use of set functions called capacities or fuzzy measures. In this paper, we define generalized versions of ... [more ▼] Sugeno integrals are aggregation operations involving a criterion weighting scheme based on the use of set functions called capacities or fuzzy measures. In this paper, we define generalized versions of Sugeno integrals on totally ordered bounded chains, by extending the operation that combines the value of the capacity on each subset of criteria and the value of the utility function over elements of the subset. We show that the generalized concept of Sugeno integral splits into two functionals, one based on a general multiple-valued conjunction (we call integral) and one based on a general multiple-valued implication (we call cointegral). These fuzzy conjunction and implication connectives are related via a so-called semiduality property, involving an involutive negation. Sugeno integrals correspond to the case when the fuzzy conjunction is the minimum and the fuzzy implication is Kleene-Dienes implication, in which case integrals and cointegrals coincide. In this paper, we consider a very general class of fuzzy conjunction operations on a finite setting, that reduce to Boolean conjunctions on extreme values of the bounded chain, and are non-decreasing in each place, and the corresponding general class of implications (their semiduals). The merit of these new aggregation operators is to go beyond pure lattice polynomials, thus enhancing the expressive power of qualitative aggregation functions, especially as to the way an importance weight can affect a local rating of an object to be chosen. [less ▲] Detailed reference viewed: 20 (3 UL)Modal extensions of Ł_n-valued logics, coalgebraically ; Teheux, Bruno ; Scientific Conference (2017) Detailed reference viewed: 11 (2 UL)Modal Extensions of Łukasiewicz Logic for Modeling Coalitional Power Teheux, Bruno ; in Journal of Logic & Computation (2017), 27(1), 129-154 Modal logics for reasoning about the power of coalitions capture the notion of effectivity functions associated with game forms. The main goal of coalition logics is to provide formal tools for modeling ... [more ▼] Modal logics for reasoning about the power of coalitions capture the notion of effectivity functions associated with game forms. The main goal of coalition logics is to provide formal tools for modeling the dynamics of a game frame whose states may correspond to different game forms. The two classes of effectivity functions studied are the families of playable and truly playable effectivity functions, respectively. In this paper we generalize the concept of effectivity function beyond the yes/no truth scale. This enables us to describe the situations in which the coalitions assess their effectivity in degrees, based on functions over the outcomes taking values in a finite Łukasiewicz chain. Then we introduce two modal extensions of Łukasiewicz finite-valued logic together with many-valued neighborhood semantics in order to encode the properties of many-valued effectivity functions associated with game forms. As our main results we prove completeness theorems for the two newly introduced modal logics. [less ▲] Detailed reference viewed: 87 (14 UL)Strongly barycentrically associative and preassociative functions Teheux, Bruno ; Marichal, Jean-Luc Scientific Conference (2016, November 08) Detailed reference viewed: 37 (6 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: 100 (17 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: 61 (14 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: 210 (10 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: 59 (12 UL)Automates, mots et décision Teheux, Bruno Presentation (2016, April 15) Detailed reference viewed: 24 (0 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: 113 (10 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: 67 (12 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: 53 (4 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: 80 (26 UL)Median algebra with a retraction: an example of variety closed under natural extension Teheux, Bruno Presentation (2016, January 14) The notion of natural extension can be defined for any algebra in an IRF-prevariety. In this talk, we introduce a way to extend functions between algebras to multifunctions between their natural ... [more ▼] The notion of natural extension can be defined for any algebra in an IRF-prevariety. In this talk, we introduce a way to extend functions between algebras to multifunctions between their natural extensions, and we give sufficient conditions under which we can assume that these multifunctions are functions. We illustrate these constructions for the variety of median algebras, and we prove that retractions over median algebras are preserved by natural extensions. [less ▲] Detailed reference viewed: 28 (0 UL)Modal definability based on Łukasiewicz validity relations Teheux, Bruno in Studia Logica (2016), 104(2), 343-363 Detailed reference viewed: 27 (8 UL)Generalized Sugeno Integrals ; ; et al in Information Processing and Management of Uncertainty in Knowledge-Based Systems, Part I (2016) Detailed reference viewed: 20 (1 UL)A modal logic for games with lies Teheux, Bruno Presentation (2015, December 01) Detailed reference viewed: 27 (2 UL) |
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