The set of distances in seminormal weakly Krull monoids

Abstract

The set of distances of a monoid or of a domain 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 $l$ irreducible elements for any $l$ with $k<l<k+d$. We show that the set of distances is an interval for certain seminormal weakly Krull monoids which include seminormal orders in holomorphy rings of global fields.