  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.
