The Elasticity of Puiseux Monoids

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 $2$).

arXiv comment14 pages, 3 figures