On elasticities of locally finitely generated monoids

Abstract

Let $H$ be a commutative and cancellative monoid. The elasticity $ρ(a)$ 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 $ρ(H)$ of $H$ is the supremum over all $ρ(a)$. We establish a characterization, valid for finitely generated monoids, when every rational number $q$ with $1< q < ρ(H)$ 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.