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

Let $X$ be a nonempty set. Denote by ${\mathcal{F}}_k^n$ the class of associative operations $F\text{\hspace{0.17em}⁣}:X^n\to X$ satisfying the condition $F(x_1,\dots ,x_n)\in \{x_1,\dots ,x_n\}$ whenever at least $k$ of the elements $x_1,\dots ,x_n$ are equal to each other. The elements of ${\mathcal{F}}_1^n$ 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}$^n_{n-2}\subseteq$\mathcal{F}$^n_{n-1}\subseteq$\mathcal{F}$^n_n$ and we give conditions on the set $X$ for the last inclusions to be strict. The class ${\mathcal{F}}_1^n$ 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 ${\mathcal{F}}_{n-1}^n\setminus {\mathcal{F}}_1^n$ and show that its elements are reducible. We give a full description of the corresponding reductions and show how each of them is built from a quasitrivial semigroup and an Abelian group whose exponent divides $n-1$.