Minimal relations and catenary degrees in Krull monoids

TL;DR

This paper investigates the sets of catenary degrees and minimal relations in Krull monoids, showing that any finite subset of integers ≥ 2 can be realized as the catenary degrees set, and exploring their relationship under certain conditions.

Contribution

It demonstrates that every finite nonempty subset of integers ≥ 2 can be realized as the set of catenary degrees in Krull monoids with finite class groups, and analyzes the structure of minimal relations.

Findings

Any finite subset of integers ≥ 2 can be realized as catenary degrees.

Under certain conditions, the set of catenary degrees is contained in the set of minimal relations.

The set of minimal relations can contain a long interval and coincide with it under specific conditions.

Abstract

Let $H$ be a Krull monoid with class group $G$. Then $H$ is factorial if and only if $G$ is trivial. Sets of lengths and sets of catenary degrees are well studied invariants describing the arithmetic of $H$ in the non-factorial case. In this note we focus on the set $Ca (H)$ of catenary degrees of $H$ and on the set $\mathcal R (H)$ of distances in minimal relations. We show that every finite nonempty subset of $\mathbb N_{\ge 2}$ can be realized as the set of catenary degrees of a Krull monoid with finite class group. This answers Problem 4.1 of {arXiv:1506.07587}. Suppose in addition that every class of $G$ contains a prime divisor. Then $Ca (H)\subset \mathcal R (H)$ and $\mathcal R (H)$ contains a long interval. Under a reasonable condition on the Davenport constant of $G$, $\mathcal R (H)$ coincides with this interval and the maximum equals the catenary degree of $H$.