Vol. 68, No. 1, 1977

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E-unitary covers for inverse semigroups

D. B. McAlister and Norman R. Reilly

Vol. 68 (1977), No. 1, 161–174

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.

Mathematical Subject Classification 2000
Primary: 20M15
Received: 24 May 1976
Published: 1 January 1977
D. B. McAlister
Norman R. Reilly