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

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 for $a$ 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.