We study a non-commutative generalization of Stone duality that connects a class of inverse semigroups, called Boolean inverse ∧-semigroups, with a class of topological groupoids, called Hausdorff Boolean groupoids. Much of the paper is given over to showing that Boolean inverse ∧-semigroups arise as completions of inverse ∧-semigroups we call pre-Boolean. An inverse ∧-semigroup is pre-Boolean if and only if every tight filter is an ultrafilter, where tight filters are defined by combining ideas 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 Gn, 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*-algebras. An elementary application of our theory shows that the finite, fundamental Boolean inverse ∧-semigroups are just the finite direct products of finite symmetric inverse monoids. Finally, we explain how tight filters are related to prime filters setting the scene for future work.
