Systems of sets of lengths of Puiseux monoids

TL;DR

This paper investigates the structure of sets of lengths in non-finitely generated Puiseux monoids, showing they do not uniquely characterize these monoids and connecting specific sets of lengths to Goldbach's conjecture.

Contribution

It demonstrates that systems of sets of lengths do not characterize non-finitely generated Puiseux monoids and extends known results from numerical monoids to this broader setting.

Findings

Existence of a BF-monoid with a full system of sets of lengths

Systems of sets of lengths do not characterize non-finitely generated Puiseux monoids

Connection between sets of lengths in a specific Puiseux monoid and Goldbach's conjecture

Abstract

In this paper we study the system of sets of lengths of non-finitely generated atomic Puiseux monoids (a Puiseux monoid is an additive submonoid of $\mathbb{Q}_{\ge 0}$). We begin by presenting a BF-monoid $M$ with full system of sets of lengths, which means that for each subset $S$ of $\mathbb{Z}_{\ge 2}$ there exists an element $x \in M$ whose set of lengths $\mathsf{L}(x)$ is $S$. It is well known that systems of sets of lengths do not characterize numerical monoids. Here, we prove that systems of sets of lengths do not characterize non-finitely generated atomic Puiseux monoids. In a recent paper, Geroldinger and Schmid found the intersection of systems of sets of lengths of numerical monoids. Motivated by this, we extend their result to the setting of atomic Puiseux monoids. Finally, we relate the sets of lengths of the Puiseux monoid $P = \langle 1/p \mid p \ \text{is prime} \rangle$ with the Goldbach's conjecture; in particular, we show that $\mathsf{L}(2)$ is precisely the set of Goldbach's numbers.