Let \(\tau\colon S\to T\) be a morphism between inverse semigroups. An expansion is then a functor such that the morphism \(F(\tau)\colon F(S)\to F(T)\) satisfies \(F(\tau)\eta_T=\eta_S\tau\), where \(\eta_T,\eta_S\) are natural morphisms from \(F(T)\) and \(F(S)\) onto \(T\) and \(S\), respectively.
In this paper two new expansions are introduced. The first generalizes an expansion of Margolis and Meakin for the group case and is the freest idempotent-pure expansion of an inverse semigroup. This expansion is used to show, for example, the decidability of certain word problems in the Malcev product of varieties of inverse semigroups.
The second generalizes the Birget-Rhodes prefix expansion with an application outlined to partial actions of inverse semigroups. A new class of idempotent-pure homomorphisms is introduced called \(F\)-morphisms as they play the same role in the theory of idempotent-pure homomorphisms that \(F\)-inverse monoids play in the theory of \(E\)-unitary inverse semigroups.