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 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 $H$.

References in corpus (2)

• The Realization Problem for Delta Sets of Numerical Semigroups
• On the set of catenary degrees of finitely generated cancellative commutative monoids

Cited by in corpus (1)

• Realizable sets of catenary degrees of numerical monoids