The Furstenberg property in Puiseux monoids

TL;DR

This paper explores the Furstenberg property in Puiseux monoids, analyzing how non-invertible elements relate to atoms and connecting these properties with existing atomic results in these additive submonoids of rationals.

Contribution

It investigates the Furstenberg property within Puiseux monoids, introducing weaker related properties and linking them to known atomic structures in these monoids.

Findings

Characterization of Furstenberg property in Puiseux monoids

Identification of properties weaker than Furstenberg in this context

Connections established between these properties and atomic results

Abstract

Let $M$ be a commutative monoid. The monoid $M$ is called atomic if every non-invertible element of $M$ factors into atoms (i.e., irreducible elements), while $M$ is called a Furstenberg monoid if every non-invertible element of $M$ is divisible by an atom. Additive submonoids of $\mathbb{Q}$ consisting of nonnegative rationals are called Puiseux monoids, and their atomic structure has been actively studied during the past few years. The primary purpose of this paper is to investigate the property of being Furstenberg in the context of Puiseux monoids. In this direction, we consider some properties weaker than being Furstenberg, and then we connect these properties with some atomic results which have been already established for Puiseux monoids.