The bands satisfying the strong isomorphism property

The power semigroup, or global, of a semigroup S is the set P ( S ) of all nonempty subsets of S equipped with the naturally defined multiplication. A class K of semigroups is globally determined if any two semigroups of K with isomorphic globals are themselves isomorphic. A class K of semigroups is said to satisfy the strong isomorphism property if for each pair S , T ∈ K and each isomorphism ψ from P ( S ) onto P ( T ), ψ ( S ¯ ) = T ¯ , where S ¯ = { { s } ∣ s ∈ S } ⊆ P ( S ) , T ¯ = { { t } ∣ t ∈ T } ⊆ P ( T ) , and hence S ≅ T . In this paper we investigate classes of bands satisfying the strong isomorphism property. We provide a description of the largest subclass K ¯ of K satisfying the strong isomorphism property for a globally determined class K of semigroups, and give some characterizations of the members of B ¯ for class B of all bands.