On maximal common divisors in Puiseux monoids

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.