The system of sets of lengths in Krull monoids under set addition

TL;DR

This paper characterizes when the system of sets of lengths in Krull monoids with prime divisors in each class is closed under set addition, based on the structure of the class group.

Contribution

It provides a characterization of the additively closed systems of sets of lengths in Krull monoids in terms of their class group structure.

Findings

The system of sets of lengths is additively closed if and only if the class group has a specific structure.

The paper links algebraic properties of the class group to combinatorial properties of factorization lengths.

It advances understanding of factorization theory in Krull monoids with prime divisors.

Abstract

Let $H$ be a Krull monoid with class group $G$ and suppose that each class contains a prime divisor. Then every element $a \in H$ has a factorization into irreducible elements, and the set $\mathsf L (a)$ of all possible factorization lengths is the set of lengths of $a$. We consider the system $\mathcal L (H) = \{ \mathsf L (a) \mid a \in H \}$ of all sets of lengths, and we characterize (in terms of the class group $G$) when $\mathcal L (H)$ is additively closed under set addition.