On elasticities of locally finitely generated monoids

Let $H$ be a commutative and cancellative monoid. The elasticity $\rho (a)$ of a non-unit $a\in H$ is the supremum of $m\mathrm{/}n$ over all $m,n$ for which there are factorizations of the form $a=u_1 \cdot \ldots \cdot u_m=v_1 \cdot \ldots \cdot v_{n}$,whereall$u_i$and$v_j$areirreducibles\mathrm{.}Theelasticity$\rho (H)$of$H$isthesupremumoverall$\rho (a)$. 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.