  We study a non-commutative generalization of Stone duality that connects a
class of inverse semigroups, called Boolean inverse $\wedge$-semigroups, with a
class of topological groupoids, called Hausdorff Boolean groupoids. Much of the
paper is given over to showing that Boolean inverse $\wedge$-semigroups arise
as completions of inverse semigroups we call pre-Boolean. An inverse
$\wedge$-semigroup is pre-Boolean if and only if every tight filter is an
ultrafilter, where the definition of a tight filter is obtained by combining
work of both Exel and Lenz. A simple necessary condition for a semigroup to be
pre-Boolean is derived and a variety of examples of inverse semigroups are
shown to satisfy it. Thus the polycyclic inverse monoids, and certain Rees
matrix semigroups over the polycyclics, are pre-Boolean and it is proved that
the groups of units of their completions are precisely the Thompson-Higman
groups $G_{n,r}$. The inverse semigroups arising from suitable directed graphs
are also pre-Boolean and the topological groupoids arising from these graph
inverse semigroups under our non-commutative Stone duality are the groupoids
that arise from the Cuntz-Krieger $C^{\ast}$-algebras.
