On elasticities of locally finitely generated monoids

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 $a=u_1\cdot \dots \cdot u_m=v_1\cdot \dots \cdot v_n$, 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.