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Reducibility of $n$-ary Semigroups : from Quasitriviality Towards Idempotency

Miguel Couceiro 1 Jimmy Devillet 2 Jean-Luc Marichal 2 Pierre Mathonet 3
1 ORPAILLEUR - Knowledge representation, reasonning
Inria Nancy - Grand Est, LORIA - NLPKD - Department of Natural Language Processing & Knowledge Discovery
Abstract : 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, 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 $n-1$.
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https://hal.archives-ouvertes.fr/hal-03023830
Contributor : Miguel Couceiro <>
Submitted on : Wednesday, November 25, 2020 - 3:26:44 PM
Last modification on : Tuesday, January 12, 2021 - 11:26:08 AM

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Miguel Couceiro, Jimmy Devillet, Jean-Luc Marichal, Pierre Mathonet. Reducibility of $n$-ary Semigroups : from Quasitriviality Towards Idempotency. Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Springer Verlag, 2021, ⟨10.1007/s13366-020-00551-2⟩. ⟨hal-03023830⟩

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