On the arithmetic of power monoids

Given a monoid H (written multiplicatively), the family 𝒫fin,1(H) 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 𝒫fin,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 𝒫fin,1(H) to admit unique minimal factorizations.