Expansions of inverse semigroups

We construct the freest idempotent-pure expansion of an inverse semigroup, generalizing an expansion of Margolis and Meakin for the group case. We also generalize the Birget-Rhodes prefix expansion to inverse semigroups with an application to partial actions of inverse semigroups. In the process of generalizing the latter expansion, we are led to a new class of idempotent-pure homomorphisms which we term F -morphisms. These play the same role in the theory of idempotent-pure homomorphisms that F -inverse monoids play in the theory of E -unitary inverse semigroups.