Inverse semigroups with zero: Covers and their structure

We obtain analogues, in the setting of semigroups with zero, of McAlister's covering theorem and the structure theorems of McAlister, O'Carroll, and M argolis and Pin. The covers come from a class C of semigroups defined by mo difying one of the many characterisations of E-unitary inverse semigroups, namely, that an inverse semigroup is E-unitary if and only if it is an inve rse image of an idempotent-pure homomorphism onto a group. The class C is p roperly contained in the class of all E*-unitary inverse semigroups introdu ced by Szendrei but properly contains the class of strongly categorical Ef- unitary semigroups recently considered by Comes and Howie.