<i>E</i>-unitary covers for inverse semigroups

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Abstract

An inverse semigroup is called E -unitary if the equations ea = e = e 2 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. 1.E~unitary covers.An inverse semigroup is called Eunitary if the equations ea = e = e 2 together imply a 2 = a.It was shown in [4] that every inverse semigroup S has an £-unitary cover in the sense that there is an E-unitary inverse semigroup P together with an idempotent separating homomorphism θ of P onto S. It was further shown in [5] that every E-unitary inverse semigroup is isomorphic to a P(G, X, &) where if is a down directed partially ordered set with 3/ an ideal and subsemilattice of #? and where G acts on d£ by order automorphisms in such a way that % = G°ίJ\ see [5] for details.The group G in P = P(G, #?, ( S/) is isomorphic to the maximum group homomorphic image P/σ of P where σ = {(a, b)E P x P: ea = eb for some e 2 = e E P).DEFINITION 1.1.Let 5 be an inverse semigroup and let G be a group.Then an E-unitary inverse semigroup P is an E-unitary cover of S through G if (i) P/σ -G (ii) there is an idempotent separating homomorphism θ of P onto 5. 161 162 D B McALISTER AND N R REILLYThus, if P = P(G,%, <&) is an E-unitary cover of 5 then P is an JB-unitary cover of 5 through G.As stated, the problem of finding £-unitary covers of an inverse semigroup S consists of finding homomorphisms onto 5.The main result of this section shows that this problem can be replaced by an embedding problem.DEFINITION 1.2. [2] Let S = S 1 be an inverse semigroup, with group of units G. Then S is a factorizable inverse semigroup if and only if, for each a E 5 there exists g E G such that a ^ g.Chen and Hsieh showed in [1] that every inverse semigroup S can be embedded in a factorizable inverse semigroup.Indeed, let θ be .ahomomorphism of 5 into the symmetric inverse semigroup $ x on a set Xfor some permutation γ of Y} is a factorizable inverse semigroup which contains Sθ.PROPOSITION 1.3.Let F be a factorizable inverse semigroup with group of units G and let θ be a one-to-one homomorphism of an inverse semigroup S into F. Suppose that for each g E G there exists s E S such that sθ g g.Then