On three families of dense Puiseux monoids
Scott. T. Chapman, Felix Gotti, Marly Gotti, Harold Polo
TL;DR
The paper advances the understanding of dense Puiseux monoids by dissecting three representative families: k-prime reciprocal monoids, p-adic monoids and their internal sums, and multiplicatively closed Puiseux monoids arising from rational cyclic semirings. It develops canonical sum decompositions to analyze divisibility and factorization, establishes ACCP and various factorization properties, and characterizes atomicity, antimatter, and valuation behavior within each family. The results highlight how density near 0 interacts with atomic structure, yielding both constructive examples of dense atom sets and precise criteria for when density enforces or obstructs factorization properties. Together, these sections illuminate the intricate landscape of dense Puiseux monoids and provide tools for further exploration of their arithmetic and structural diversity, with implications for broader factorization theory in commutative algebra.
Abstract
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.