set of all group congruences on a semigroup S, the minimal element (for the partial order defined by inclusion of sets) need not be the smallest element. In the specific case in which S is an inverse semigroup σ is the smallest congruence on S such that S/σ is a group, that is, if τ is any other congruence on S with S/τ a group, then σ is contained in τ. The congruence σ is called the minimum group congruence on S. The minimum group congruence can be used to give a characterisation of E-unitary inverse semigroups (see below). A congruence ρ on an inverse semigroup S is called idempotent pure if E-unitary inverse semigroups One class of inverse semigroups that has been studied extensively over the years is the class of E-unitary inverse semigroups: an inverse semigroup S (with semilattice E of idempotents) is E-unitary if, for all e in E and all s in S, Equivalently, One further characterisation of an E-unitary inverse semigroup S is the following: if e is in E and e ≤ s, for some s in S, then s is in E. Theorem. Let S be an inverse semigroup with semilattice E of idempotents, and minimum group congruence σ. Then the following are equivalent: S is E-unitary; σ is idempotent pure; = σ, where is the compatibility relation on S, defined by are idempotent. McAlister's Covering Theorem. Every inverse semigroup S has a E-unitary cover; that is there exists an idempotent separating surjective homomorphism from some E-unitary semigroup T onto S. Central to the study of E-unitary inverse semigroups is the following construction. Let be a partially ordered set, with ordering ≤, and let be a subset of with the properties that is a lower semilattice, that is, every pair of elements A, B in has a greatest lower bound A B in (with respect to ≤); is an order ideal of , that is, for A, B in , if A is in and B ≤ A, then B is in . Now let G be a group that acts on (on the left), such that for all g in G and all A, B in , gA = gB if, and only if, A = B; for each g in G and each B in , there exists an A in such that gA = B; for all A, B in