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 $\rho \left(x\right)$, is the ratio of its largest factorization length to its shortest factorization length, and it measures how far $x$ is from having all its factorizations of the same length. The elasticity $\rho \left(M\right)$ of $M$ is the supremum of the elasticities of all non-unit elements of $M$. In this paper, we study the elasticity of Puiseux monoids (i.e., additive submonoids of ${\mathbb{Q}}_{\ge 0}$). We characterize, in terms of the atoms, which Puiseux monoids $M$ have finite elasticity, giving a formula for $\rho \left(M\right)$ in this case. We also classify when $\rho \left(M\right)$ is achieved by an element of $M$. When $M$ is a primary Puiseux monoid (that is, a Puiseux monoid whose atoms have prime denominator), 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 $2$).