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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: 160 (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: 77 (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: 99 (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: 124 (31 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: 45 (1 UL)Modal definability based on Łukasiewicz validity relations Teheux, Bruno in Studia Logica (2016), 104(2), 343-363 Detailed reference viewed: 83 (8 UL)Generalized Sugeno Integrals ; ; et al in Information Processing and Management of Uncertainty in Knowledge-Based Systems, Part I (2016) Detailed reference viewed: 61 (1 UL)A modal logic for games with lies Teheux, Bruno Presentation (2015, December 01) Detailed reference viewed: 54 (3 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: 79 (11 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: 67 (17 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: 74 (3 UL)Characterizing variants of qualitative Sugeno integrals in a totally ordered Heyting algebra ; ; et al in Alonso, José M.; Bustince, Humberto; Reformat, Marek (Eds.) Proceedings of the 2015 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and Technology (2015, July) Sugeno integral is one of the basic aggregation operations on a qualitative scale where only minimum and maximum, as well as order-reversing maps are allowed. Recently some variants of this aggregation ... [more ▼] Sugeno integral is one of the basic aggregation operations on a qualitative scale where only minimum and maximum, as well as order-reversing maps are allowed. Recently some variants of this aggregation operation, named soft and drastic integrals, have been introduced in a previous work by three of the authors. In these operations, importance weights play the role of tolerance thresholds enabling full satisfaction if ratings pass them. These new aggregation operations use residuated implications, hence need a slightly richer structure, and are part of larger family of qualitative aggregations. Based on some properties laid bare in a previous work, this paper proposes characterisation theorems for four variants of Sugeno integrals. These results pave the way to decision-theoretic axiomatizations of these natural qualitative aggregations. [less ▲] Detailed reference viewed: 88 (4 UL)Strongly barycentrically associative and preassociative functions Teheux, Bruno Scientific Conference (2015, June 17) Detailed reference viewed: 81 (5 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: 236 (17 UL)Clones of Pivotally Decomposable Functions ; Teheux, Bruno in Multiple-Valued Logic (ISMVL), 2015 IEEE International Symposium on (2015, May 20) This paper is a contribution to the understanding of the relation between pivotal decompositions of operations on a set and clones on the same set. Detailed reference viewed: 85 (4 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: 136 (19 UL)Associative string functions ; Marichal, Jean-Luc ; Teheux, Bruno in Asian-European Journal of Mathematics (2014), 7(4), 145005918 We introduce the concept of associativity for string functions, where a string function is a unary operation on the set of strings over a given alphabet. We discuss this new property and describe certain ... [more ▼] We introduce the concept of associativity for string functions, where a string function is a unary operation on the set of strings over a given alphabet. We discuss this new property and describe certain classes of associative string functions. We also characterize the recently introduced preassociative functions as compositions of associative string functions with injective unary maps. Finally, we provide descriptions of the classes of associative and preassociative functions which depend only on the length of the input. [less ▲] Detailed reference viewed: 119 (23 UL)A modal logic to reason about degrees of coalitional effectivity ; Teheux, Bruno in Lauber, Jimmy; Bisgambiglia, Paul-Antoine (Eds.) Rencontres francophones sur la logique floue et ses applications 2014 (2014, November) We generalize the notion of alpha-effectivity in order to express the degree with which a coalition can enforce a fuzzy-set of outcomes. Then, we introduce a modal extension of Łukasiewicz n+1-valued ... [more ▼] We generalize the notion of alpha-effectivity in order to express the degree with which a coalition can enforce a fuzzy-set of outcomes. Then, we introduce a modal extension of Łukasiewicz n+1-valued logic together with a complete many-valued neighborhood semantics in order to encode the properties of these many-valued effectivity functions. [less ▲] Detailed reference viewed: 65 (8 UL)Graded modal logic to reason about the power of attributes in multicriteria decision making Teheux, Bruno Scientific Conference (2014, October 29) Detailed reference viewed: 40 (3 UL)Associative and preassociative functions Marichal, Jean-Luc ; Teheux, Bruno in Semigroup Forum (2014), 89(2), 431-442 We investigate the associativity property for functions of multiple arities and introduce and discuss the more general property of preassociativity, a generalization of associativity which does not ... [more ▼] We investigate the associativity property for functions of multiple arities and introduce and discuss the more general property of preassociativity, a generalization of associativity which does not involve any composition of functions. [less ▲] Detailed reference viewed: 170 (23 UL) |
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