On the atomic structure of exponential Puiseux monoids and semirings

TL;DR

This paper investigates the atomic structure of exponential Puiseux monoids and semirings, characterizing their atomic properties and conditions for factorization properties to hold.

Contribution

It provides a complete characterization of atomic exponential Puiseux monoids and explores conditions under which these monoids satisfy various factorization properties.

Findings

Atomic exponential Puiseux monoids are characterized explicitly.

The finite factorization, bounded factorization, and ACCP coincide in this context.

Conditions for exponential Puiseux monoids to satisfy the ACCP are established.

Abstract

A Puiseux monoid is an additive submonoid of the nonnegative cone of the rational numbers. We say that a Puiseux monoid $M$ is exponential provided that there exist a positive rational $r$ and a set $S$ consisting of nonnegative integers, which contains $0$, such that $M$ is generated by the set $\{r^s \mid s \in S\}$. If $M$ is multiplicatively closed then we say that $M$ is an exponential Puiseux semiring. Here we study the atomic properties of exponential Puiseux monoids and semirings. First, we characterize atomic exponential Puiseux monoids, and we prove that the finite factorization property, the bounded factorization property, and the ACCP coincide in this context. Then we proceed to offer a necessary condition and a sufficient condition for an exponential Puiseux monoid to satisfy the ACCP. We conclude by describing the exponential Puiseux monoids that are semirings.