\(\mathcal{P}\mathcal{A}{\text{-isomorphisms}}\) of inverse semigroups

Abstract.

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 an inverse monoid called the partial automorphism monoid of S. Two semigroups are said to be \(\mathcal{P}\mathcal{A}{\text{-isomorphic}}\) if their partial automorphism monoids are isomorphic. A class \(\mathbb{K}\) of semigroups is called \(\mathcal{P}\mathcal{A}{\text{-closed}}\) if it contains every semigroup \(\mathcal{P}\mathcal{A}{\text{-isomorphic}}\) to some semigroup from \(\mathbb{K}.\) Although the class of all inverse semigroups is not \(\mathcal{P}\mathcal{A}{\text{-closed}},\) 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 \(\mathcal{P}\mathcal{A}{\text{-closed}}.\) It follows that the class of all combinatorial inverse semigroups (those with no nontrivial subgroups) is \(\mathcal{P}\mathcal{A}{\text{-closed}}.\) A semigroup is called \(\mathcal{P}\mathcal{A}{\text{-determined}}\) if it is isomorphic or antiisomorphic to any semigroup that is \(\mathcal{P}\mathcal{A}{\text{-isomorphic}}\) to it. We show that combinatorial inverse semigroups which are either shortly connected [5] or quasi-archimedean [10] are \(\mathcal{P}\mathcal{A}{\text{-determined}}.\)