On maximal common divisors in Puiseux monoids

TL;DR

This paper explores the existence of maximal common divisors in Puiseux monoids, linking their presence to atomicity and chain conditions, thereby advancing understanding of their algebraic structure.

Contribution

It provides new results on when maximal common divisors exist in rank-1 torsion-free monoids and connects these to atomicity and chain conditions.

Findings

Maximal common divisors exist under certain conditions in Puiseux monoids.

Existence of maximal common divisors relates to atomicity of the monoid.

Connections established between maximal common divisors and ascending chain conditions.

Abstract

Let $M$ be a commutative monoid. An element $d \in M$ is called a maximal common divisor of a nonempty subset $S$ of $M$ if $d$ is a common divisor of $S$ in $M$ and the only common divisors in $M$ of the set $\big\{ \frac{s}d : s \in S \big\}$ are the units of $M$. In this paper, we investigate the existence of maximal common divisors in rank-$1$ torsion-free commutative monoids, also known as Puiseux monoids. We also establish some connections between the existence of maximal common divisors and both atomicity and the ascending chain condition on principal ideals for the monoids we investigate here.