  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.
