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Reducibity of n-ary semigroups: from quasitriviality towards idempotency

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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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Dates and versions

hal-02293908 , version 1 (22-09-2019)

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  • HAL Id : hal-02293908 , version 1

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Miguel Couceiro, Jimmy Devillet, Jean-Luc Marichal, Pierre Mathonet. Reducibity of n-ary semigroups: from quasitriviality towards idempotency. 2019. ⟨hal-02293908⟩
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