Sets of minimal distances and characterizations of class groups of Krull monoids

TL;DR

This paper investigates the structure of minimal distances in the sets of lengths of factorizations within Krull monoids, characterizing when these distances form intervals and confirming a conjecture relating class groups to factorization systems.

Contribution

It characterizes when the set of minimal distances is an interval and confirms a conjecture linking the system of sets of lengths to the class group for specific cases.

Findings

The set of minimal distances can be characterized when it forms an interval.

The system of all sets of lengths depends solely on the class group.

The conjecture relating the system of lengths to the class group is confirmed for certain class groups.

Abstract

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor. Then every non-unit $a \in H$ can be written as a finite product of atoms, say $a=u_1 \cdot \ldots \cdot u_k$. The set $\mathsf L (a)$ of all possible factorization lengths $k$ is called the set of lengths of $a$. There is a constant $M \in \mathbb N$ such that all sets of lengths are almost arithmetical multiprogressions with bound $M$ and with difference $d \in \Delta^* (H)$, where $\Delta^* (H)$ denotes the set of minimal distances of $H$. We study the structure of $\Delta^* (H)$ and establish a characterization when $\Delta^*(H)$ is an interval. The system $\mathcal L (H) = \{ \mathsf L (a) \mid a \in H \}$ of all sets of lengths depends only on the class group $G$, and a standing conjecture states that conversely the system $\mathcal L (H)$ is characteristic for the class group. We confirm this conjecture (among others) if the class group is isomorphic to $C_n^r$ with $r,n \in \mathbb N$ and $\Delta^*(H)$ is not an interval.