Kobayashi's problem for power semigroups of inverse semigroups

For a semigroup $S$ (written multiplicatively), let $\mathcal{P}(S)$ denote the semigroup of all non-empty subsets of $S$ under setwise multiplication. The finitary power semigroup ${\mathcal{P}}_{\text{fin}}(S)$ is the subsemigroup of $\mathcal{P}(S)$ consisting of all non-empty finite subsets of $S$. Kobayashi’s problem asks for which classes $\mathcal{O}$ of semigroups every isomorphism $\mathcal{P}(H)\to \mathcal{P}(K)$ with $H,K\in \mathcal{O}$ maps singletons bijectively onto singletons and hence induces an isomorphism $H\to K$. We give an affirmative answer for inverse semigroups, extending previous results for groups, semilattices, and, more generally, Clifford semigroups. We also establish the corresponding rigidity theorem for finitary power semigroups. To this end, we develop a new framework that provides a common approach to both settings.