  The complex algebra of an inverse semigroup with finitely many idempotents in
each $\mathcal D$-class is stably finite by a result of Munn. This can be
proved fairly easily using $C^*$-algebras for inverse semigroups satisfying
this condition that have a Hausdorff universal groupoid, or more generally for
direct limits of inverse semigroups satisfying this condition and having
Hausdorff universal groupoids. It is not difficult to see that a finitely
presented inverse semigroup with a non-Hausdorff universal groupoid cannot be a
direct limit of inverse semigroups with Hausdorff universal groupoids. We
construct here countably many non-isomorphic finitely presented inverse
semigroups with finitely many idempotents in each $\mathcal D$-class and
non-Hausdorff universal groupoids. At this time there is not a clear
$C^*$-algebraic technique to prove these inverse semigroups have stably finite
complex algebras.
