The Catenary Degree of Krull Monoids I

TL;DR

This paper studies the catenary degree of Krull monoids with finite class groups, providing a new characterization, bounds, and conditions for small catenary degrees, enhancing understanding of factorizations in algebraic structures.

Contribution

It introduces a new simple characterization of the catenary degree for Krull monoids with finite class groups under mild conditions.

Findings

New upper bound on the catenary degree

Characterization of when the catenary degree is at most 4

Clarification of the relationship between catenary degree and set of distances

Abstract

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree $\mathsf c (H)$ of $H$ is the smallest integer $N$ with the following property: for each $a \in H$ and each two factorizations $z, z'$ of $a$, there exist factorizations $z = z_0, ..., z_k = z'$ of $a$ such that, for each $i \in [1, k]$, $z_i$ arises from $z_{i-1}$ by replacing at most $N$ atoms from $z_{i-1}$ by at most $N$ new atoms. Under a very mild condition on the Davenport constant of $G$, we establish a new and simple characterization of the catenary degree. This characterization gives a new structural understanding of the catenary degree. In particular, it clarifies the relationship between $\mathsf c (H)$ and the set of distances of $H$ and opens the way towards obtaining more detailed results on the catenary degree. As first applications, we give a new upper bound on $\mathsf c(H)$ and characterize when $\mathsf c(H)\leq 4$.