\cPA-isomorphisms of inverse semigroups

A partial automorphism of a semigroup $S$ is any isomorphism between its subsemigroups, and the set all partial automorphisms of $S$ with respect to composition is the inverse monoid called the partial automorphism monoid of $S$. Two semigroups are said to be \cPA-isomorphic if their partial automorphism monoids are isomorphic. A class \K of semigroups is called \cPA-closed if it contains every semigroup \cPA-isomorphic to some semigroup from \K. Although the class of all inverse semigroups is not \cPA-closed, we prove that the class of inverse semigroups, in which no maximal isolated subgroup is a direct product of an involution-free periodic group and the two-element cyclic group, is \cPA-closed. It follows that the class of all combinatorial inverse semigroups (those with no nontrivial subgroups) is \cPA-closed. A semigroup is called \cPA-determined if it is isomorphic or anti-isomorphic to any semigroup that is \cPA-isomorphic to it. We show that combinatorial inverse semigroups which are either shortly connected [5] or quasi-archimedean [10] are \cPA-determined