On the Internal Sum of Puiseux Monoids

TL;DR

This paper explores how the atomicity and factorization properties of Puiseux monoids, which are submonoids of the nonnegative rationals, behave under their internal finite sum within the group of rationals, highlighting differences based on rank.

Contribution

It provides a detailed analysis of the factorization properties of internal sums of Puiseux monoids, especially regarding atomicity and bounded factorization, with new insights into their behavior under such sums.

Findings

Atomicity can be preserved or lost under internal sums depending on the monoids involved.

The behavior of factorization properties differs when summing with finitely generated Puiseux monoids.

Counterexamples show primary results do not extend to higher-rank torsion-free abelian groups.

Abstract

In this paper, we investigate the internal (finite) sum of submonoids of rank-$1$ torsion-free abelian groups. These submonoids, when not groups, are isomorphic to nontrivial submonoids of the nonnegative cone of $\mathbb Q$, known as Puiseux monoids, and have been actively studied during the last few years. Here we study how the atomicity and arithmetic of Puiseux monoids behave under their internal (finite) sum inside the abelian group $\mathbb Q$. We study the factorization properties of such internal sums, giving priority to Cohn's notion of atomicity and the classical bounded and finite factorization properties introduced and studied in 1990 by Anderson, Anderson, and Zafrullah in the setting of integral domains, and then generalized by Halter-Koch to commutative monoids. We pay special attention to how each of the considered properties behaves under the internal sum of a Puiseux monoid with a finitely generated Puiseux monoid. Throughout the paper, we also discuss examples showing that our primary results do not hold for submonoids of torsion-free abelian groups with rank larger than $1$.