Reference : Reducibility of n-ary semigroups: from quasitriviality towards idempotency
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http://hdl.handle.net/10993/40481
Reducibility of n-ary semigroups: from quasitriviality towards idempotency
English
Couceiro, Miguel []
Devillet, Jimmy mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Marichal, Jean-Luc mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Mathonet, Pierre []
23-Sep-2019
13
No
[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 $\mathcal{F}^n_1=\cdots =\mathcal{F}^n_{n-2}\varsubsetneq\mathcal{F}^n_{n-1}\varsubsetneq\mathcal{F}^n_n$.
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. In particular, we show that each of these elements is an extension of an $n$-ary Abelian group operation whose exponent divides $n-1$.
University of Luxembourg - UL
Researchers ; Professionals ; Students
http://hdl.handle.net/10993/40481
https://hal.inria.fr/hal-02293908/document

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