On a groupoid construction for actions of certain inverse semigroups

We consider a version of the notion of F-inverse semigroup (studied in the algebraic theory of inverse semigroups). We point out that an action of such an inverse semigroup on a locally compact space has associated a natural groupoid construction, very similar to the one of a transformation group. We discuss examples related to Toeplitz algebras on subsemigroups of discrete groups,to Cuntz-Krieger algebras, and to crossed-products by partial automorphisms in the sense of Exel.