On the \(E\)-unitary covers for a Bruck semigroup (Q1318962)

Let \(S = B(T,\alpha)\) be a Bruck semigroup over an inverse semigroup \(T\), let \(T'\) be an \(E\)-unitary cover of \(T\) with an idempotent separating homomorphism \(\theta\) from \(T'\) onto \(T\), and let \(\alpha'\) be an idempotent pure homomorphism from \(T'\) into its group of units such that there exists an element \(z\) which is a unit of \(T\) with \(\alpha' \theta \varepsilon_ z=\theta \alpha\), where \(a \varepsilon_ z = z^{-1} az\) for all \(a \in T\). Define \(\eta : B(T', \alpha') \to B(T,\alpha)\) by \(\eta : (m, a, n) \mapsto (m, z^{-1}_ m \cdot a\theta \cdot z_ n, n)\). Then \(\eta\) is an idempotent separating homomorphism, and \(B(T',\alpha')\) is an \(E\)-unitary cover of \(B(T,\alpha)\).