On the isomorphism problem for power semigroups

Denote by P(S) the semigroup obtained by equipping the family of all non-empty subsets of a (multiplicatively written) semigroup S with the operation of setwise multiplication induced by S itself. We call a subsemigroup P of P(S) downward complete if any element of S lies in at least one set X ∈ P and any non-empty subset of a set in P is again in P. We prove, for a commutative semigroup S, a characterization of the cancellative elements of a downward complete subsemigroup of P(S) in terms of the cancellative elements of S. Consequently, we show that, if H and K are cancellative semigroups and either of them is commutative, then every isomorphism from a downward complete subsemigroup of P(H) to a downward complete subsemigroup of P(K) restricts to an isomorphism from H to K. This solves an open case of a problem of Tamura and Shafer from the late 1960s and generalizes a recent result by Bienvenu and Geroldinger, where it is assumed, among other conditions, that H and K are numerical monoids.