Operations preserving equivalence relations

TL;DR

This paper provides a concise proof characterizing unary functions on integers that preserve all additive congruences, framing the result within universal algebra.

Contribution

It offers a shorter, more straightforward proof of a known characterization, integrating the result into the context of universal algebra.

Findings

Unary functions on integers preserving all additive congruences are characterized

The proof is simplified and made more direct

The result is contextualized within universal algebra

Abstract

In 2014, C\'egielski, Grigorieff and Guessarian characterized unary self-maps on the set $\mathbb{Z}$ of integers which preserve all congruences of the additive group. In this note, we propose a shorter and straigthforward proof. We replace this result in the frame of universal algebra.