$\cPA$-isomorphisms of inverse semigroups

TL;DR

This paper investigates the structure of inverse semigroups through their partial automorphism monoids, establishing conditions under which classes of inverse semigroups are uniquely determined by these monoids.

Contribution

It proves that certain classes of inverse semigroups are $ ext{PA}$-closed and that specific combinatorial inverse semigroups are $ ext{PA}$-determined, advancing understanding of their structural uniqueness.

Findings

The class of inverse semigroups with no maximal isolated subgroup as a direct product is $ ext{PA}$-closed.

The class of all combinatorial inverse semigroups is $ ext{PA}$-closed.

Certain combinatorial inverse semigroups are uniquely determined by their partial automorphism monoids.

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 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.