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Let $X$ be a nonempty set. Denote by $\mathcal{F}^n_k$ the class ofassociative 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 $\mathcal{F}^n_1$ aresaid to be quasitrivial and those of $\mathcal{F}^n_n$ are said to beidempotent. We show that $\mathcal{F}^n_1=\cdots=\mathcal{F}^n_{n-2}\subseteq\mathcal{F}^n_{n-1}\subseteq\mathcal{F}^n_n$ andwe give conditions on the set $X$ for the last inclusions to be strict. Theclass $\mathcal{F}^n_1$ 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 $\mathcal{F}^n_{n-1}\setminus\mathcal{F}^n_1$ andshow that its elements are reducible. We give a full description of thecorresponding reductions and show how each of them is built from a quasitrivialsemigroup and an Abelian group whose exponent divides $n-1$.

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