An inverse semigroup is
called E-unitary if the equations ea = e = e2 together imply a2= a. In a previous
paper, the first author showed that every inverse semigroup has an E. unitary cover.
That is, if S is an inverse semigroup, there is an E. unitary Inverse semigroup P
and an idempotent separating homomorphism of P onto S. The purpose of
this paper is to consider the problem of constructing E. unitary covers for
S.
Let S be an inverse semigroup and let F be an inverse semigroup, with group of
units G, containing S as an inverse subsemigroup and suppose that, for each s ∈ S,
there exists g ∈ G such that s ≦ g. Then {(s,g) ∈ S × G : s ≦ g} is an E-unitary
cover of S. The main result of §1 shows that every E-unitary cover of S can be
obtained in this way. It follows from this that the problem of finding E-unitary covers
for S can be reduced to an embedding problem. A further corollary to this result is
the fact that, if P is an E-unitary cover of S and P has maximal group homomorphic
image G, then P is a subdirect product of S and G and so can be described in terms
of S and G alone. The remainder of this paper is concerned with giving such a
description.
