# Every quasitrivial n-ary semigroup is reducible to a semigroup

## TL;DR

This paper proves that all quasitrivial n-ary semigroups can be reduced to binary semigroups, providing conditions for uniqueness, especially for symmetric cases, and enumerates their sizes on finite sets, revealing new integer sequences.

## Contribution

It establishes a reduction framework for quasitrivial n-ary semigroups to binary semigroups and characterizes the conditions for uniqueness, including symmetric cases.

## Key findings

- Reduction of quasitrivial n-ary semigroups to binary semigroups
- Necessary and sufficient conditions for reduction uniqueness
- Enumeration of classes on finite sets and discovery of new integer sequences

## Abstract

We show that every quasitrivial n-ary semigroup is reducible to a binary semigroup, and we provide necessary and sufficient conditions for such a reduction to be unique. These results are then refined in the case of symmetric n-ary semigroups. We also explicitly determine the sizes of these classes when the semigroups are defined on finite sets. As a byproduct of these enumerations, we obtain several new integer sequences.