On the arithmetic of power monoids

Given a monoid $H$ (written multiplicatively), the family ${\mathcal{P}}_{\mathrm{f}\mathrm{i}\mathrm{n},1}(H)$ of all non-empty finite subsets of $H$ containing the identity element $1_H$ is itself a monoid, called the reduced finitary power monoid of $H$, under the operation of setwise multiplication induced by $H$. We investigate the arithmetic of ${\mathcal{P}}_{\mathrm{f}\mathrm{i}\mathrm{n},1}(H)$ from the perspective of minimal factorizations into irreducibles, paying particular attention to the potential presence of non-trivial idempotents. Among other results, we provide necessary and sufficient conditions on $H$ for ${\mathcal{P}}_{\mathrm{f}\mathrm{i}\mathrm{n},1}(H)$ to admit unique minimal factorizations. Our results generalize and shed new light on recent developments on the topic.