Inverse semigroups determined by their partial automorphism monoids

TL;DR

This paper proves that certain inverse semigroups are uniquely determined by their partial automorphism monoids, with specific conditions ensuring their structural uniqueness up to isomorphism or duality.

Contribution

It establishes that tightly connected fundamental inverse semigroups without isolated nontrivial subgroups are uniquely determined by their partial automorphism monoids, up to isomorphism or duality.

Findings

Inverse semigroups are lattice determined modulo semilattices.

Partial automorphism monoids characterize inverse semigroups up to isomorphism or duality.

Conditions like tight connectivity and absence of isolated subgroups are crucial for these characterizations.

Abstract

The partial automorphism monoid of an inverse semigroup is an inverse monoid consisting of all isomorphisms between its inverse subsemigroups. We prove that a tightly connected fundamental inverse semigroup $S$ with no isolated nontrivial subgroups is lattice determined "modulo semilattices" and if $T$ is an inverse semigroup whose partial automorphism monoid is isomorphic to that of $S$, then either $S$ and $T$ are isomorphic or they are dually isomorphic chains relative to the natural partial order; a similar result holds if $T$ is any semigroup and the inverse monoids consisting of all isomorphisms between subsemigroups of $S$ and $T$, respectively, are isomorphic. Moreover, for these results to hold, the conditions that $S$ be tightly connected and have no isolated nontrivial subgroups are essential.