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Reducibility of n-ary semigroups: from quasitriviality towards idempotency ; Devillet, Jimmy ; Marichal, Jean-Luc et al in Beiträge zur Algebra und Geometrie (in press) Let $X$ be a nonempty set. Denote by $\mathcal{F}^n_k$ the class of associative operations $F\colon X^n\to X$ satisfying the condition $F(x_1,\ldots,x_n)\in\{x_1,\ldots,x_n\}$ whenever at least $k$ of the ... [more ▼] Let $X$ be a nonempty set. Denote by $\mathcal{F}^n_k$ the class of associative operations $F\colon X^n\to X$ satisfying the condition $F(x_1,\ldots,x_n)\in\{x_1,\ldots,x_n\}$ whenever at least $k$ of the elements $x_1,\ldots,x_n$ are equal to each other. The elements of $\mathcal{F}^n_1$ are said to be quasitrivial and those of $\mathcal{F}^n_n$ are said to be idempotent. We show that $\mathcal{F}^n_1=\cdots =\mathcal{F}^n_{n-2}\subseteq\mathcal{F}^n_{n-1}\subseteq\mathcal{F}^n_n$ and we give conditions on the set $X$ for the last inclusions to be strict. The class $\mathcal{F}^n_1$ was recently characterized by Couceiro and Devillet \cite{CouDev}, who showed that its elements are reducible to binary associative operations. However, some elements of $\mathcal{F}^n_n$ are not reducible. In this paper, we characterize the class $\mathcal{F}^n_{n-1}\setminus\mathcal{F}^n_1$ and show that its elements are reducible. We give a full description of the corresponding reductions and show how each of them is built from a quasitrivial semigroup and an Abelian group whose exponent divides $n-1$. [less ▲] Detailed reference viewed: 118 (17 UL)Every quasitrivial n-ary semigroup is reducible to a semigroup ; Devillet, Jimmy in Algebra Universalis (2019), 80(4), We show that every quasitrivial n-ary semigroup is reducible to a binary semigroup, and we provide necessary and sufficient conditions for such a reduction to be unique. These results are then refined in ... [more ▼] We show that every quasitrivial n-ary semigroup is reducible to a binary semigroup, and we provide necessary and sufficient conditions for such a reduction to be unique. These results are then refined in the case of symmetric n-ary semigroups. We also explicitly determine the sizes of these classes when the semigroups are defined on finite sets. As a byproduct of these enumerations, we obtain several new integer sequences. [less ▲] Detailed reference viewed: 272 (61 UL)Characterizations and enumerations of classes of quasitrivial n-ary semigroups Devillet, Jimmy ; Scientific Conference (2019, June 23) Detailed reference viewed: 90 (5 UL)Quasitrivial semigroups: characterizations and enumerations ; Devillet, Jimmy ; Marichal, Jean-Luc in Semigroup Forum (2019), 98(3), 472498 We investigate the class of quasitrivial semigroups and provide various characterizations of the subclass of quasitrivial and commutative semigroups as well as the subclass of quasitrivial and order ... [more ▼] We investigate the class of quasitrivial semigroups and provide various characterizations of the subclass of quasitrivial and commutative semigroups as well as the subclass of quasitrivial and order-preserving semigroups. We also determine explicitly the sizes of these classes when the semigroups are defined on finite sets. As a byproduct of these enumerations, we obtain several new integer sequences. [less ▲] Detailed reference viewed: 341 (102 UL)Single-peakedness in aggregation function theory Devillet, Jimmy ; ; Marichal, Jean-Luc Presentation (2019, May 14) Due to their great importance in data fusion, aggregation functions have been extensively investigated for a few decades. Among these functions, fuzzy connectives (such as uninorms) play an important role ... [more ▼] Due to their great importance in data fusion, aggregation functions have been extensively investigated for a few decades. Among these functions, fuzzy connectives (such as uninorms) play an important role in fuzzy logic. We establish a remarkable connection between a family of associative aggregation functions, which includes the class of idempotent uninorms, and the concepts of single-peakedness and single-plateaudness, introduced in social choice theory by D. Black. Finally, we enumerate those orders when the underlying set is finite. [less ▲] Detailed reference viewed: 96 (10 UL)Associative and quasitrivial operations on finite sets: characterizations and enumeration ; Devillet, Jimmy ; Marichal, Jean-Luc Scientific Conference (2018, July 02) We investigate the class of binary associative and quasitrivial operations on a given finite set. Here the quasitriviality property (also known as conservativeness) means that the operation always outputs ... [more ▼] We investigate the class of binary associative and quasitrivial operations on a given finite set. Here the quasitriviality property (also known as conservativeness) 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 which cases the operations reduce to discrete uninorms (which are discrete fuzzy connectives playing an important role in fuzzy logic). Interestingly, associative and quasitrivial operations that are nondecreasing are characterized in terms of total and weak orderings through the so-called single-peakedness property introduced in social choice theory by Duncan Black. We also address and solve a number of enumeration issues: we count the number of binary associative and quasitrivial operations on a given finite set as well as the number of those operations that are commutative and/or nondecreasing. [less ▲] Detailed reference viewed: 75 (5 UL)Clones of pivotally decomposable operations ; Teheux, Bruno Scientific Conference (2018, June) We investigate the clones of operations that are pivotally decomposable. Detailed reference viewed: 44 (2 UL)Characterizations of idempotent discrete uninorms ; Devillet, Jimmy ; Marichal, Jean-Luc in Fuzzy Sets & Systems (2018), 334 In this paper we provide an axiomatic characterization of the idempotent discrete uninorms by means of three conditions only: conservativeness, symmetry, and nondecreasing monotonicity. We also provide an ... [more ▼] In this paper we provide an axiomatic characterization of the idempotent discrete uninorms by means of three conditions only: conservativeness, symmetry, and nondecreasing monotonicity. We also provide an alternative characterization involving the bisymmetry property. Finally, we provide a graphical characterization of these operations in terms of their contour plots, and we mention a few open questions for further research. [less ▲] Detailed reference viewed: 262 (52 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: 112 (22 UL)Associative and quasitrivial operations on finite sets (invited lecture) Marichal, Jean-Luc ; ; Devillet, Jimmy Scientific Conference (2017, November 10) Detailed reference viewed: 76 (13 UL)On quasitrivial and associative operations Devillet, Jimmy ; ; Marichal, Jean-Luc Presentation (2017, October 25) Detailed reference viewed: 69 (13 UL)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: 78 (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: 74 (8 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: 89 (24 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: 196 (36 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: 183 (21 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: 182 (11 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: 88 (11 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: 85 (3 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: 102 (4 UL) |
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