On the incomparability of systems of sets of lengths

Abstract

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor. We consider the system $\mathcal{L}(H)$ of all sets of lengths of $H$ and study when $\mathcal{L}(H)$ contains or is contained in a system $\mathcal{L}(H^{\prime })$ of a Krull monoid $H^{\prime }$ with finite class group $G^{\prime }$, prime divisors in all classes and Davenport constant $\mathsf{D}(G^{\prime })=\mathsf{D}(G)$. Among others, we show that if $G$ is either cyclic of order $m\ge 7$ or an elementary $2$-group of rank $m-1\ge 6$, and $G^{\prime }$ is any group which is non-isomorphic to $G$ but with Davenport constant $\mathsf{D}(G^{\prime })=\mathsf{D}(G)$, then the systems $\mathcal{L}(H)$ and $\mathcal{L}(H^{\prime })$ are incomparable.

arXiv commentEuropean J. Combinatorics, to appear