E-unitary covers for inverse semigroups
D. B. McAlister, Norman R. Reilly · Pacific Journal of Mathematics · 1977
An inverse semigroup is called E -unitary if the equations ea = e = e2 together imply a 2 = 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 G S, there exists g<ΞG such that s ^ g.Then {(5, g)G5 x 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 5 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 5 and G and so can be described in terms of 5 and G alone.The remainder of this paper is concerned with giving such a description.