Reference : Reducibility of n-ary semigroups: from quasitriviality towards idempotency
Scientific journals : Article
Physical, chemical, mathematical & earth Sciences : Mathematics
Engineering, computing & technology : Computer science
Computational Sciences
Reducibility of n-ary semigroups: from quasitriviality towards idempotency
Couceiro, Miguel mailto [LORIA, CNRS - Inria Nancy Grand Est - Université de Lorraine]
Devillet, Jimmy mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit > ; Ecole Privée Fieldgen, Luxembourg, Luxembourg]
Marichal, Jean-Luc mailto [University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)]
Mathonet, Pierre mailto [University of Liège, Department of Mathematics, Liège, Belgium]
Beiträge zur Algebra und Geometrie
Yes (verified by ORBilu)
[en] Semigroup ; polyadic semigroup ; Abelian group ; reducibility ; quasitriviality ; idempotency
[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 $n-1$.
University of Luxembourg - UL
Researchers ; Professionals ; Students

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