On the set of elasticities in numerical monoids

TL;DR

This paper investigates the relationship between elasticities and length sets in numerical monoids, showing that elasticities can fully determine length sets in arithmetical cases and describing their structure in general.

Contribution

It proves that elasticities determine length sets in arithmetical numerical monoids and characterizes elasticities in general numerical monoids.

Findings

Elasticities fully determine length sets in arithmetical numerical monoids.

Elasticities form specific monotone sequences with a common limit in general monoids.

The set of elasticities can be reconstructed from length sets in arithmetical cases.

Abstract

In an atomic, cancellative, commutative monoid $S$, the elasticity of an element provides a coarse measure of its non-unique factorizations by comparing the largest and smallest values in its set of factorization lengths (called its length set). In this paper, we show that the set of length sets $\mathcal L(S)$ for any arithmetical numerical monoid $S$ can be completely recovered from its set of elasticities $R(S)$; therefore, $R(S)$ is as strong a factorization invariant as $\mathcal L(S)$ in this setting. For general numerical monoids, we describe the set of elasticities as a specific collection of monotone increasing sequences with a common limit point of $\max R(S)$.