Minimal relations and catenary degrees in Krull monoids

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 F. Halter-Koch [Problem 4.1]{N-P-T-W16a}. Suppose, in addition, that every class of $G$ contains a prime divisor. Then, Ca $(H)\subset \mathcal R (H)$ and $\mathcal R (H)$ contain 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 $H$.