# On associative operations on commutative integral domains

## TL;DR

This paper characterizes associative multilinear polynomial functions over commutative integral domains, extending previous results and offering a new proof for the classification of two-element n-semigroups.

## Contribution

It generalizes existing classifications to broader domains and introduces a novel proof technique for two-element n-semigroups.

## Key findings

- Extended classification to infinite integral domains.
- Provided a new proof for Andres's classification.
- Enhanced understanding of associative polynomial functions.

## Abstract

We describe the associative multilinear polynomial functions over commutative integral domains. This extends Marichal and Mathonet's result on infinite integral domains and provides a new proof of Andres's classification of two-element $n$-semigroups.