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

TL;DR

This paper explores the structure of associative n-ary operations with certain element-repetition properties, showing how some can be reduced to simpler algebraic structures, extending previous results on quasitrivial operations.

Contribution

It characterizes the class of (n-1)-ary operations that are reducible, building on prior work on quasitrivial operations, and describes their algebraic decompositions.

Findings

Elements of _{n-1}_{1} are reducible to binary operations.

The paper provides explicit constructions of reductions using semigroups and Abelian groups.

Some n-ary operations are not reducible, highlighting structural differences.

Abstract

Let $X$ be a nonempty set. Denote by $\mathcal{F}^n_k$ the class of associative 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 $\mathcal{F}^n_1$ 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 $\mathcal{F}^n_1$ 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 $\mathcal{F}^n_{n-1}\setminus\mathcal{F}^n_1$ 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 $n-1$.