20 research outputs found

    Reducibility of n-ary semigroups: from quasitriviality towards idempotency

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    Let XX be a nonempty set. Denote by Fkn\mathcal{F}^n_k the class of associative operations F ⁣:XnXF\colon X^n\to X satisfying the condition F(x1,,xn){x1,,xn}F(x_1,\ldots,x_n)\in\{x_1,\ldots,x_n\} whenever at least kk of the elements x1,,xnx_1,\ldots,x_n are equal to each other. The elements of F1n\mathcal{F}^n_1 are said to be quasitrivial and those of Fnn\mathcal{F}^n_n are said to be idempotent. We show that F1n==Fn2nFn1nFnn\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 XX for the last inclusions to be strict. The class F1n\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 Fnn\mathcal{F}^n_n are not reducible. In this paper, we characterize the class Fn1nF1n\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 n1n-1

    Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020)

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    International audienceOriginating in arithmetics and logic, the theory of ordered sets is now a field of combinatorics that is intimately linked to graph theory, universal algebra and multiple-valued logic, and that has a wide range of classical applications such as formal calculus, classification, decision aid and social choice.This international conference “Algebras, graphs and ordered set” (ALGOS) brings together specialists in the theory of graphs, relational structures and ordered sets, topics that are omnipresent in artificial intelligence and in knowledge discovery, and with concrete applications in biomedical sciences, security, social networks and e-learning systems. One of the goals of this event is to provide a common ground for mathematicians and computer scientists to meet, to present their latest results, and to discuss original applications in related scientific fields. On this basis, we hope for fruitful exchanges that can motivate multidisciplinary projects.The first edition of ALgebras, Graphs and Ordered Sets (ALGOS 2020) has a particular motivation, namely, an opportunity to honour Maurice Pouzet on his 75th birthday! For this reason, we have particularly welcomed submissions in areas related to Maurice’s many scientific interests:• Lattices and ordered sets• Combinatorics and graph theory• Set theory and theory of relations• Universal algebra and multiple valued logic• Applications: formal calculus, knowledge discovery, biomedical sciences, decision aid and social choice, security, social networks, web semantics..

    On the structure of symmetric nn-ary bands

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    We study the class of symmetric nn-ary bands. These are nn-ary semigroups (X,F)(X,F) such that FF is invariant under the action of permutations and idempotent, i.e., satisfies F(x,,x)=xF(x,\ldots,x)=x for all xXx\in X. We first provide a structure theorem for these symmetric nn-ary bands that extends the classical (strong) semilattice decomposition of certain classes of bands. We introduce the concept of strong nn-ary semilattice of nn-ary semigroups and we show that the symmetric nn-ary bands are exactly the strong nn-ary semilattices of nn-ary extensions of Abelian groups whose exponents divide n1n-1. Finally, we use the structure theorem to obtain necessary and sufficient conditions for a symmetric nn-ary band to be reducible to a semigroup

    Decomposition schemes for symmetric n-ary bands

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    We extend the classical (strong) semilattice decomposition scheme of certain classes of semigroups to the class of idempotent symmetric n-ary semigroups (i.e. symmetric n-ary bands) where n \geq 2 is an integer. More precisely, we show that these semigroups are exactly the strong n-ary semilattices of n-ary extensions of Abelian groups whose exponents divide n-1. We then use this main result to obtain necessary and sufficient conditions for a symmetric n-ary band to be reducible to a semigroup

    On idempotent n-ary semigroups

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    This thesis, which consists of two parts, focuses on characterizations and descriptions of classes of idempotent n-ary semigroups where n >= 2 is an integer. Part I is devoted to the study of various classes of idempotent semigroups and their link with certain concepts stemming from social choice theory. In Part II, we provide constructive descriptions of various classes of idempotent n-ary semigroups. More precisely, after recalling and studying the concepts of single-peakedness and rectangular semigroups in Chapters 1 and 2, respectively, in Chapter 3 we provide characterizations of the classes of idempotent semigroups and totally ordered idempotent semigroups, in which the latter two concepts play a central role. Then in Chapter 4 we particularize the latter characterizations to the classes of quasitrivial semigroups and totally ordered quasitrivial semigroups. We then generalize these results to the class of quasitrivial n-ary semigroups in Chapter 5. Chapter 6 is devoted to characterizations of several classes of idempotent n-ary semigroups satisfying quasitriviality on certain subsets of the domain. Finally, Chapter 7 focuses on characterizations of the class of symmetric idempotent n-ary semigroups. Throughout this thesis, we also provide several enumeration results which led to new integer sequences that are now recorded in The On-Line Encyclopedia of Integer Sequences (OEIS). For instance, one of these enumeration results led to a new definition of the Catalan numbers

    On quasitrivial semigroups

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    Characterizations of quasitrivial symmetric nondecreasing associative operations

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    We provide a description of the class of n-ary operations on an arbitrary chain that are quasitrivial, symmetric, nondecreasing, and associative. We also prove that associativity can be replaced with bisymmetry in the definition of this class. Finally we investigate the special situation where the chain is finite

    Reducibity of n-ary semigroups: from quasitriviality towards idempotency

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    Let XX be a nonempty set. Denote by Fkn\mathcal{F}^n_k the class of associative operations F ⁣:XnXF\colon X^n\to X satisfying the condition F(x1,,xn){x1,,xn}F(x_1,\ldots,x_n)\in\{x_1,\ldots,x_n\} whenever at least kk of the elements x1,,xnx_1,\ldots,x_n are equal to each other. The elements of F1n\mathcal{F}^n_1 are said to be quasitrivial and those of Fnn\mathcal{F}^n_n are said to be idempotent. We show that F1n==Fn2nFn1nFnn\mathcal{F}^n_1=\cdots =\mathcal{F}^n_{n-2}\varsubsetneq\mathcal{F}^n_{n-1}\varsubsetneq\mathcal{F}^n_n. The class F1n\mathcal{F}^n_1 was recently characterized by Couceiro and Devillet, who showed that its elements are reducible to binary associative operations. However, some elements of Fnn\mathcal{F}^n_n are not reducible. In this paper, we characterize the class Fn1nF1n\mathcal{F}^n_{n-1}\setminus\mathcal{F}^n_1 and show that its elements are reducible. In particular, we show that each of these elements is an extension of an nn-ary Abelian group operation whose exponent divides n1n-1

    Every quasitrivial n-ary semigroup is reducible to a semigroup

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    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

    Visual characterization of associative quasitrivial nondecreasing functions on finite chains

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    In this paper we provide visual characterization of associative quasitrivial nondecreasing operations on finite chains. We also provide a characterization of bisymmetric quasitrivial nondecreasing binary operations on finite chains. Finally, we estimate the number of functions belonging to the previous classes
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