We obtain analogues, in the setting of semigroups with zero, of McAlister's convering theoren and the structure theorems of McAlister, O'Carroll, and Margolis and Pin. The covers come from a class C of semigroups defined by modifying one of the many characterisations of E-unitary inverse semigroups, namely, that an inverse semigroups is E-unitary if and only if it is an inverse image of an idempotent-pure homomorphism onto a group. The class C is properly contained in the class of all E*-unitary inverse semigroups introduced by Szendrei but properly contains the class of strongly categorical E*-unitary semigroups recently considered by Gomes and Howie.
