On square-free and radical factorizations and existence of some divisors

TL;DR

This paper explores different types of factorizations related to square-free and radical elements within monoids, examining how these properties interact with atomicity, chain conditions, and gcd-related properties.

Contribution

It provides new insights into the relationships between various factorization properties and divisor existence in monoids, expanding understanding of their algebraic structure.

Findings

Characterization of square-free and radical factorizations

Conditions for the existence of certain divisors in monoids

Connections between factorization properties and gcd conditions

Abstract

We discuss various square-free and radical factorizations and existence of some divisors in monoids in the context of: atomicity, ascending chain condition for principal ideals, a pre-Schreier property, a greatest common divisor property and a greatest common divisor for sets property.