A realization theorem for sets of distances

Alfred Geroldinger (1) , Wolfgang Schmid (2)

1 IM - Institut für Mathematik [Graz]
2 LAGA - Laboratoire Analyse, Géométrie et Applications

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 $\Delta (H)=\Delta$.

Domaines: Algèbre commutative [math.AC]

Alfred Geroldinger, Wolfgang Schmid. A realization theorem for sets of distances. Journal of Algebra, 2017, 481, pp.188-198. ⟨10.1016/j.jalgebra.2017.03.003⟩. ⟨hal-01355253v2⟩