On the system of sets of lengths and the elasticity of submonoids of a finite-rank free commutative monoid

Abstract

Let $H$ be an atomic monoid. For $x\in H$, let $\mathsf{L}(x)$ denote the set of all possible lengths of factorizations of $x$ into irreducibles. The system of sets of lengths of $H$ is the set $\mathcal{L}(H)=\{\mathsf{L}(x)\mid x\in H\}$. On the other hand, the elasticity of $x$, denoted by $\rho (x)$, is the quotient $\sup \mathsf{L}(x)/\inf \mathsf{L}(x)$ and the elasticity of $H$ is the supremum of the set $\{\rho (x)\mid x\in H\}$. The system of sets of lengths and the elasticity of $H$ both measure how far is $H$ from being half-factorial, i.e., $|\mathsf{L}(x)|=1$ for each $x\in H$. Let $\mathcal{C}$ denote the collection comprising all submonoids of finite-rank free commutative monoids, and let ${\mathcal{C}}_d=\{H\in \mathcal{C}\mid \text{rank}(H)=d\}$. In this paper, we study the system of sets of lengths and the elasticity of monoids in $\mathcal{C}$. First, we construct for each $d\ge 2$ a monoid in ${\mathcal{C}}_d$ having extremal system of sets of lengths. It has been proved before that the system of sets of lengths does not characterize (up to isomorphism) monoids in ${\mathcal{C}}_1$. Here we use our construction to extend this result to ${\mathcal{C}}_d$ for any $d\ge 2$. On the other hand, it has been recently conjectured that the elasticity of any monoid in $\mathcal{C}$ is either rational or infinite. We conclude this paper by proving that this is indeed the case for monoids in ${\mathcal{C}}_2$ and for any monoid in $\mathcal{C}$ whose corresponding convex cone is polyhedral.

19 pages

References in corpus (3)

• Atomicity and boundedness of monotone Puiseux monoids
• On the Atomic Structure of Puiseux Monoids
• Sets of Arithmetical Invariants in Transfer Krull Monoids

Cited by in corpus (1)

• On strongly primary monoids, with a focus on Puiseux monoids