Reducibility of n-ary semigroups: from quasitriviality towards idempotency
نویسندگان
چکیده
Let $X$ be a nonempty set. Denote by $\mathcal{F}^n_k$ the class of associative operations $F\colon X^n\to X$ satisfying condition $F(x_1,\ldots,x_n)\in\{x_1,\ldots,x_n\}$ whenever at least $k$ elements $x_1,\ldots,x_n$ are equal to each other. The $\mathcal{F}^n_1$ said quasitrivial and those $\mathcal{F}^n_n$ idempotent. We show that $\mathcal{F}^n_1=\cdots =\mathcal{F}^n_{n-2}\subseteq\mathcal{F}^n_{n-1}\subseteq\mathcal{F}^n_n$ we give conditions on set for last inclusions strict. was recently characterized Couceiro Devillet, who showed its reducible binary operations. However, some not reducible. In this paper, characterize $\mathcal{F}^n_{n-1}\setminus\mathcal{F}^n_1$ full description corresponding reductions how them is built from semigroup an Abelian group whose exponent divides $n-1$.
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ژورنال
عنوان ژورنال: Beiträge Zur Algebra Und Geometrie / Contributions To Algebra And Geometry
سال: 2021
ISSN: ['2191-0383', '0138-4821']
DOI: https://doi.org/10.1007/s13366-020-00551-2