On \(q^*\)-bisimple IC semigroups of type-\(E\). (Q1014786)

Let \(S\) be a quasi-adequate semigroup, i.e., each \(\mathcal L^*\)-class and each \(\mathcal R^*\)-class of \(S\) contains at least one of the idempotents of \(S\) and its set of idempotents forms a semigroup \(E(S)\). \(S\) is called type-\(E\) if \(E(S)\) is a direct product of a rectangular band \(E\) and an \(\omega\)-chain. An IC (idempotent-connected) quasi-adequate \(^*\)-bisimple (i.e., there is precisely one \(\mathcal D^*\)-class and one regular \(\mathcal D\)-class) semigroup of type-\(E\) is called a \(q^*\)-bisimple IC semigroup of type-\(E\) mentioned in the title of the paper. Such semigroups are characterized by means of so called generalized Bruck-Reilly extensions.