Covers for regular semigroups (Q1345745)

D. B. McAlister proved that any inverse semigroup is an idempotent separating homomorphic image of an \(E\)-unitary inverse semigroup. This important theorem was extended by K. Takizawa (1979) and by M. Szendrei (1980) to orthodox semigroups. \textit{J. Almeida, J. Pin} and \textit{P. Weil} finally proved a general covering theorem for semigroups, whose idempotents form a subsemigroup [Math. Proc. Camb. Philos. Soc. 111, No. 2, 241-253 (1992; Zbl 0751.20042)]. The paper under review indicates how these results can be transferred to the class of all regular semigroups. This is done by replacing the set \(E_S\) of all idempotents in the regular semigroup \(S\) by the self- conjugate core \(C(S)\) defined as the least subsemigroup of \(S\) containing \(E_S\) and \(xC(S) x'\) for all \(x \in C(S)\), \(x' \in V(x)\). The covering theorem then states that every regular semigroup is a \(C(S)\)-separating homomorphic image of a \(C(S)\)-unitary regular semigroup (a proof of this and related results will be published by the first author in J. Pure Appl. Algebra). In particular, it is shown that any finite regular semigroup \(S\) has a finite \(C(S)\)-unitary regular cover and also a so- called finite type II-unitary cover.