# The Elasticity of Puiseux Monoids

## TL;DR

This paper investigates the factorization properties of Puiseux monoids, characterizing their elasticity, classifying those with accepted elasticity, and exploring the topology of elasticities, including special cases like bifurcus monoids.

## Contribution

It provides a comprehensive analysis of the elasticity of Puiseux monoids, including formulas, classifications, and topological descriptions, advancing the understanding of their factorization behavior.

## Key findings

- Finite elasticity characterized and formulas derived
- Classification of Puiseux monoids with accepted elasticity
- Topology of elasticities and bounded factorization monoids described

## Abstract

Let $M$ be an atomic monoid and let $x$ be a non-unit element of $M$. The elasticity of $x$, denoted by $\rho(x)$, is the ratio of its largest factorization length to its shortest factorization length, and it measures how far is $x$ from having a unique factorization. The elasticity $\rho(M)$ of $M$ is the supremum of the elasticities of all non-unit elements of $M$. The monoid $M$ has accepted elasticity if $\rho(M) = \rho(m)$ for some $m \in M$. In this paper, we study the elasticity of Puiseux monoids (i.e., additive submonoids of $\mathbb{Q}_{\ge 0}$). First, we characterize the Puiseux monoids $M$ having finite elasticity and find a formula for $\rho(M)$. Then we classify the Puiseux monoids having accepted elasticity in terms of their sets of atoms. When $M$ is a primary Puiseux monoid, we describe the topology of the set of elasticities of $M$, including a characterization of when $M$ is a bounded factorization monoid. Lastly, we give an example of a Puiseux monoid that is bifurcus (that is, every nonzero element has a factorization of length at most $2$).