J. Aust. Math. Soc.  81 (2006), 185-198
Inverse semigroups determined by their partial automorphism monoids

Simon M. Goberstein
  Department of Mathematics and Statistics
  California State University
  Chico, CA 95929
  USA
  SGoberstein@csuchico.edu


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.
Download the article in PDF format (size 121 Kb)
Australian Mathematical Publishing Association Inc. ©  Australian MS