On three families of dense Puiseux monoids

A positive monoid is a submonoid of the nonnegative cone of a linearly ordered abelian group. The positive monoids of rank $1$ are called Puiseux monoids, and their atomicity, arithmetic of length, and factorization have been systematically investigated for about ten years. Each Puiseux monoid can be realized as an additive submonoid of the nonnegative cone of $\mathbb{Q}$. We say that a Puiseux monoid is dense if it is isomorphic to an additive submonoid of $\mathbb{Q}_{\ge 0}$ that is dense in $\mathbb{R}_{\ge 0}$ with respect to the Euclidean topology. Every non-dense Puiseux monoid is known to be a bounded factorization monoid. However, the atomic structure as well as the arithmetic and factorization properties of dense Puiseux monoids turn out to be quite interesting. In this paper, we study the atomic structure and some arithmetic and factorization aspects of three families of dense Puiseux monoids.