On algebraic properties of power monoids of numerical monoids

Let $S \subset \mathbb{N}_0$ be a numerical monoid and let $\mathcal P_{\mathrm{fin}} (S)$, resp $\mathcal P_{\mathrm{fin},0}(S)$, denote the power monoid, resp. the restricted power monoid, of $S$, that is the set of all finite nonempty subsets of $S$, resp. the set of all finite nonempty subsets of $S$ containing 0, with set addition as operation. The arithmetic of power monoids received some attention in recent literature. We complement these investigations by studying algebraic properties of power monoids, such as their prime spectrum. Moreover, we prove that almost all elements of $\mathcal P_{\mathrm{fin},0} (S)$ are irreducible (i.e., they are not proper sumsets), quantitatively improving a result of Shitov along the way.