When is a Puiseux monoid atomic?

TL;DR

This paper surveys the atomicity of Puiseux monoids, providing characterizations for various algebraic properties and constructing classes satisfying certain factorization conditions, highlighting the complexity of their structure.

Contribution

It offers a comprehensive overview of conditions under which Puiseux monoids are atomic and related properties, including new constructions for specific factorization properties.

Findings

Characterizations of finitely generated, factorial, and half-factorial Puiseux monoids.

Construction of infinite classes satisfying ACCP, bounded, and finite factorization properties.

Abstract

A Puiseux monoid is an additive submonoid of the nonnegative rational numbers. If $M$ is a Puiseux monoid, then the question of whether each non-invertible element of $M$ can be written as a sum of irreducible elements (that is, $M$ is atomic) is surprisingly difficult. Although various techniques have been developed over the past few years to identify subclasses of Puiseux monoids that are atomic, no general characterization of such monoids is known. Here we survey some of the most relevant aspects related to the atomicity of Puiseux monoids. We provide characterizations of when $M$ is finitely generated, factorial, half-factorial, other-half-factorial, Pr\"ufer, seminormal, root-closed, and completely integrally closed. In addition to the atomicity, characterizations are also not known for when $M$ satisfies the ACCP, the bounded factorization property, or the finite factorization property. In each of these cases, we construct an infinite class of Puiseux monoids satisfying the corresponding property.