Approximating length-based invariants in atomic Puiseux monoids

TL;DR

This paper explores the relationship between numerical monoids and Puiseux monoids, showing how factorization invariants in the latter can be approximated through a limiting process based on the former, extending known results.

Contribution

It introduces a method to approximate invariants of Puiseux monoids using numerical monoids, bridging the two classes of monoids and extending existing results.

Findings

Factorization invariants of Puiseux monoids relate to those of numerical monoids via a limiting process.

The technique recovers various known results about Puiseux monoids.

Puiseux monoids can be studied through their approximation by numerical monoids.

Abstract

A numerical monoid is a cofinite additive submonoid of the nonnegative integers, while a Puiseux monoid is an additive submonoid of the nonnegative cone of the rational numbers. Using that a Puiseux monoid is an increasing union of copies of numerical monoids, we prove that some of the factorization invariants of these two classes of monoids are related through a limiting process. This allows us to extend results from numerical to Puiseux monoids. We illustrate the versatility of this technique by recovering various known results about Puiseux monoids.