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- W2973518990 abstract "Let $X$ be a nonempty set. Denote by $mathcal{F}^n_k$ the class of associative operations $Fcolon X^nto X$ 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 $mathcal{F}^n_1=cdots =mathcal{F}^n_{n-2}varsubsetneqmathcal{F}^n_{n-1}varsubsetneqmathcal{F}^n_n$. The class $mathcal{F}^n_1$ was recently characterized by Couceiro and Devillet [2], 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 $mathcal{F}^n_{n-1}setminusmathcal{F}^n_1$ 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 $n-1$." @default.
- W2973518990 created "2019-09-26" @default.
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- W2973518990 date "2019-09-23" @default.
- W2973518990 modified "2023-10-09" @default.
- W2973518990 title "Reducibility of $n$-ary semigroups: from quasitriviality towards idempotency" @default.
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