A realization theorem for sets of distances

Abstract

Let $H$ be an atomic monoid. The set of distances $\Delta (H)$ of $H$ is the set of all $d\in \mathbb{N}$ with the following property: there are irreducible elements $u\_1,\dots ,u\_k,v\_1\dots ,v\_k+d$ such that $u\_1\cdot \dots \cdot u\_k=v\_1\cdot \dots \cdot v\_k+d$ but $u\_1\cdot \dots \cdot u\_k$ cannot be written as a product of $\ell$ irreducible elements for any $\ell \in \mathbb{N}$ with $k<\ell <k+d$. It is well-known (and easy to show) that, if $\Delta (H)$ is nonempty, then $\min \Delta (H)=\gcd \Delta (H)$. In this paper we show conversely that for every finite nonempty set $\Delta \subset \mathbb{N}$ with $\min \Delta =\gcd \Delta$ there is a finitely generated Krull monoid $H$ such that $\Delta (H)=\Delta$.

Revised version. To appear in Journal of Algebra