The set of distances in seminormal weakly Krull monoids

TL;DR

This paper investigates the set of distances in seminormal weakly Krull monoids, proving that under certain conditions, this set forms an interval, which has implications for the factorization properties of these algebraic structures.

Contribution

It establishes that the set of distances in specific seminormal weakly Krull monoids is always an interval, extending understanding of their factorization behavior.

Findings

The set of distances is an interval for certain seminormal weakly Krull monoids.

Includes seminormal orders in holomorphy rings of global fields as special cases.

Provides new insights into the factorization structure of these 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 $u_1, \ldots, u_k, v_1, \ldots, v_{k+d}$ 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 $l$ 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.