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.

References in corpus (4)

• Systems of sets of lengths: Transfer Krull monoids versus weakly Krull monoids
• On the set of elasticities in numerical monoids
• Sets of Arithmetical Invariants in Transfer Krull Monoids
• On products of k atoms II

Cited by in corpus (6)

• Factorization under Local Finiteness Conditions
• Lattices over Bass rings and graph agglomerations
• On the finiteness of certain factorization invariants
• Definite orders with locally free cancellation
• Atomicity and Factorization of Puiseux Monoids
• Factorization in Finitely-Presented Monoids