A realization result for systems of sets of lengths

TL;DR

This paper constructs a Dedekind domain with a prescribed family of finite sets of lengths, demonstrating a realization result for systems of sets of lengths in algebraic structures.

Contribution

It proves that any family of finite sets of lengths satisfying certain properties can be realized as the system of sets of lengths of a Dedekind domain.

Findings

Existence of Dedekind domain matching given set family

Characterization of systems of sets of lengths

Construction method for such domains

Abstract

Let $\mathcal L^*$ be a family of finite subsets of $\mathbb N_0$ having the following properties. (a). $\{0\}, \{1\} \in \mathcal L^*$ and all other sets of $\mathcal L^*$ lie in $\mathbb N_{\ge 2}$. (b). If $L_1, L_2 \in \mathcal L^*$, then the sumset $L_1 + L_2 \in \mathcal L^*$. We show that there is a Dedekind domain $D$ whose system of sets of lengths equals $\mathcal L^*$.