Actions of Inverse Semigroups and their Ample Groupoids

We study actions of inverse semigroups on arbitrary spaces, and relate properties of the action to topological properties of the associated groupoid of germs. Then, following the construction of Paterson in his 1999 book ``Inverse Semigroups, Groupoids and their Operator Algebras'', we investigate the intrinsic actions of inverse semigroups on their character spaces, and their groupoids of germs. We draw inspiration from a well-known result that characterizes sub-semigroups of the inverse semigroup of open bisections on an étale groupoid, which we call bisection wideness. Using Paterson's approach, we construct a number of ample groupoids associated to an inverse semigroup, these being the universal groupoid and the groupoid of ultragerms, and determine conditions under which an inverse semigroup S and a sub-semigroup W give rise to the same groupoid.