  We develop the theory of distributive inverse semigroups as the analogue of
distributive lattices without top element and prove that they are in a duality
with those etale groupoids having a spectral space of identities, where our
spectral spaces are not necessarily compact. We prove that Boolean inverse
semigroups can be characterized as those distributive inverse semigroups in
which every prime filter is an ultrafilter; we also provide a topological
characterization in terms of Hausdorffness. We extend the notion of the patch
topology to distributive inverse semigroups and prove that every distributive
inverse semigroup has a Booleanization. As applications of this result, we give
a new interpretation of Paterson's universal groupoid of an inverse semigroup
and by developing the theory of what we call tight coverages, we also provide a
conceptual foundation for Exel's tight groupoid.
