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

Finitely presented inverse semigroups with finitely many idempotents in each $\mathcal D$-class and non-Hausdorff universal groupoids