A realization theorem for sets of distances

TL;DR

This paper proves a realization theorem showing that any finite nonempty set of distances with a specific gcd property can be realized as the set of distances in some finitely generated Krull monoid.

Contribution

It establishes a converse to a known property by constructing Krull monoids for arbitrary such sets of distances.

Findings

Any finite nonempty set of natural numbers with min equal to gcd can be realized as a set of distances.

Provides a method to construct finitely generated Krull monoids with prescribed sets of distances.

Extends the understanding of the structure of sets of distances in atomic monoids.

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 $u\_1, \ldots, u\_k, v\_1 \ldots, v\_{k+d}$ such that $u\_1 \cdot \ldots \cdot u\_k=v\_1 \cdot \ldots \cdot v\_{k+d}$ but $u\_1 \cdot \ldots \cdot u\_k$ cannot be written as a product of $\ell$ irreducible elements for any $\ell \in \mathbb{N}$ with $k\lt \ell \lt k+d$. 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 $\Delta (H)=\Delta$.