  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.
