On elasticities of locally finitely generated monoids

TL;DR

This paper characterizes when rational numbers between 1 and the elasticity of finitely generated monoids can be realized as element elasticities, extending results to locally finitely generated monoids and certain algebraic domains.

Contribution

It provides a characterization for the realization of rational elasticities in finitely generated and locally finitely generated monoids, including applications to algebraic domains.

Findings

Characterization for finitely generated monoids regarding elasticity realization

Extension of results to locally finitely generated monoids

Application to Krull and weakly Krull domains

Abstract

Let $H$ be a commutative and cancellative monoid. The elasticity $\rho(a)$ of a non-unit $a \in H$ is the supremum of $m/n$ over all $m, n$ for which there are factorizations of the form $a=u_1 \cdot \ldots \cdot u_m=v_1 \cdot \ldots \cdot v_{n}$, where all $u_i$ and $v_j$ are irreducibles. The elasticity $\rho (H)$ of $H$ is the supremum over all $\rho (a)$. We establish a characterization, valid for finitely generated monoids, when every rational number $q$ with $1< q < \rho (H)$ can be realized as the elasticity of some element $a \in H$. Furthermore, we derive results of a similar flavor for locally finitely generated monoids (they include all Krull domains and orders in Dedekind domains satisfying certain algebraic finiteness conditions) and for weakly Krull domains.