On global isomorphisms and a closure property of semigroups

Let S be a semigroup (written multiplicatively). Endowed with the operation of setwise multiplication induced by S on its parts, the non-empty subsets of S form themselves a semigroup, denoted by P(S). Accordingly, we say that a semigroup H is globally isomorphic to a semigroup K if P(H) is isomorphic to P(K); and that a class C of semigroups is globally closed if a semigroup in C can only be globally isomorphic to an isomorphic copy of a semigroup in the same class. We show that the classes of groups, torsion-free monoids, and numerical monoids are each globally closed. The first result extends a 1967 theorem of Shafer, while the last relies non-trivially on the second and on a classical theorem of Kneser from additive number theory.