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

A realization result for systems of sets of lengths