# On properties of square-free elements in commutative cancellative monoids

## TL;DR

This paper investigates the properties of square-free elements in commutative cancellative monoids, exploring factorizations, atomicity, and gcd properties, and characterizes submonoids with specific square-free element behaviors.

## Contribution

It provides a full characterization of submonoids of factorial monoids where all square-free elements are also square-free in the larger monoid, and explores conditions for atoms to be square-free.

## Key findings

- Characterization of submonoids where square-free elements are preserved
- Conditions under which atoms are square-free in submonoids
- Relationships between factorial properties and square-free elements

## Abstract

We discuss various square-free factorizations in monoids in the context of: atomicity, ascending chain condition for principal ideals, decomposition, and a greatest common divisor property. Moreover, we obtain a full characterization of submonoids of factorial monoids in which all square-free elements of a submonoid are square-free in a monoid. We also present factorial properties implying that all atoms of a submonoid are square-free in a monoid.