Reducibility of $n$-ary semigroups: from quasitriviality towards idempotency

Abstract

Let X be a nonempty set. Denote by F n k the class of associative operations F ∶ X n → X satisfying the condition F (x 1 , . . . , xn) ∈ {x 1 , . . . , xn} whenever at least k of the elements x 1 , . . . , xn are equal to each other. The elements of F n 1 are said to be quasitrivial and those of F n n are said to be idempotent. We show thatThe class 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 F n n are not reducible. In this paper, we characterize the class F n n−1 ∖ 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.