Reference : Reducibility of n-ary semigroups: from quasitriviality towards idempotency
 Document type : Scientific journals : Article Discipline(s) : Physical, chemical, mathematical & earth Sciences : MathematicsEngineering, computing & technology : Computer science Focus Areas : Computational Sciences To cite this reference: http://hdl.handle.net/10993/40481
 Title : Reducibility of n-ary semigroups: from quasitriviality towards idempotency Language : English Author, co-author : Couceiro, Miguel [LORIA, CNRS - Inria Nancy Grand Est - Université de Lorraine] Devillet, Jimmy [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit > ; Ecole Privée Fieldgen, Luxembourg, Luxembourg] Marichal, Jean-Luc [University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)] Mathonet, Pierre [University of Liège, Department of Mathematics, Liège, Belgium] Publication date : Mar-2022 Journal title : Beiträge zur Algebra und Geometrie Publisher : Springer Volume : 63 Issue/season : 1 Pages : 149-166 Peer reviewed : Yes (verified by ORBilu) Audience : International ISSN : 0138-4821 e-ISSN : 2191-0383 City : Berlin Country : Germany Keywords : [en] Semigroup ; polyadic semigroup ; Abelian group ; reducibility ; quasitriviality ; idempotency Abstract : [en] 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 $n-1$. Funders : University of Luxembourg - UL Target : Researchers ; Professionals ; Students Permalink : http://hdl.handle.net/10993/40481 DOI : 10.1007/s13366-020-00551-2 Other URL : http://arxiv.org/abs/1909.10412

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