  Paterson showed how to construct an etale groupoid from an inverse semigroup
using ideas from functional analysis. This construction was later simplified by
Lenz. We show that Lenz's construction can itself be further simplified by
using filters: the topological groupoid associated with an inverse semigroup is
precisely a groupoid of filters. In addition, idempotent filters are closed
inverse subsemigroups and so determine transitive representations by means of
partial bijections. This connection between filters and representations by
partial bijections is exploited to show how linear representations of inverse
semigroups can be constructed from the groups occuring in the associated
topological groupoid.
