Atomicity and Density of Puiseux Monoids

A Puiseux monoid is a submonoid of $(\mathbb{Q},+)$ consisting of nonnegative rational numbers. Although the operation of addition is continuous with respect to the standard topology, the set of irreducibles of a Puiseux monoid is, in general, difficult to describe. In this paper, we use topological density to understand how much a Puiseux monoid, as well as its set of irreducibles, spread through $\mathbb{R}_{\ge 0}$. First, we separate Puiseux monoids according to their density in $\mathbb{R}_{\ge 0}$, and we characterize monoids in each of these classes in terms of generating sets and sets of irreducibles. Then we study the density of the difference group, the root closure, and the conductor semigroup of a Puiseux monoid. Finally, we prove that every Puiseux monoid generated by a strictly increasing sequence of rationals is nowhere dense in $\mathbb{R}_{\ge 0}$ and has empty conductor.