On the arithmetic of power monoids

Given a monoid H (written multiplicatively), the family $P_{\mathrm{fin},1}\left(H\right)$ of all non-empty finite subsets of H containing the identity element 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 $P_{\mathrm{fin},1}\left(H\right)$ 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 $P_{\mathrm{fin},1}\left(H\right)$ to admit unique minimal factorizations.