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We obtain analogues, in the setting of semigroups with zero, of McAlister's covering theorem 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 semigroup 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 £*-unitary inverse semigroups introduced by Szendrei but properly contains the class of strongly categorical £*-unitary semigroups recently considered by Gomes and Howie.1991 Mathematics subject classification (Amer. Math. Soc): primary 20M18.

In 1971, inspired by the work of Lazard and Govorov for modules over a ring, Stenström proved that the strongly flat right acts AS over a monoid S (that is, the acts that are directed colimits of finitely generated free acts) are those for which the functor AS ⊗ − (from the category of left S-acts into the category of sets) preserves pullbacks and equalizers. He also provided interpolation-type conditions (now referred to in the literature as Property (P) and Property (E)) characterizing strong flatness. Unlike the situation for modules over a ring, strong flatness is strictly stronger than (mono-) flatness (wherein the functor AS ⊗ − is required only to preserve monomorphisms). The study of flatness properties of partially ordered monoids acting on partially ordered sets was initiated by S. M. Fakhruddin in the 1980s, and has been continued recently in the paper "Indecomposable, projective, and flat S-posets" by Shi, Liu, Wang, and Bulman-Fleming, Comm. Algebra 33, 235-251 (2005). In that paper, a criterion for the equality of elements in a tensor product of S-posets is given and a version of Property (P) is presented that, as in the unordered case, implies flatness and is implied by projectivity. The present paper introduces a corresponding Property (E) and establishes an analogue of the Lazard-Govorov-Stenström theorem in the context of S-posets.