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

TL;DR

This paper investigates the sets of lengths and elasticity of submonoids of finite-rank free commutative monoids, constructing extremal examples and proving that elasticity is always rational or infinite in certain cases.

Contribution

It constructs monoids with extremal sets of lengths in all ranks and proves the rationality or infiniteness of elasticity for specific classes of these monoids.

Findings

Constructed monoids with extremal sets of lengths for each rank d ≥ 2.

Extended previous results to higher ranks regarding the characterization of monoids by their sets of lengths.

Proved that the elasticity of certain monoids is always rational or infinite.

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.