\(E\)-unitary inversive covers for \(E\)-inversive semigroups whose idempotents form a subsemigroup (Q1326524)

In 1992, \textit{J. Almeida, J. E. Pin} and \textit{P. Weil} [Math. Proc. Camb. Philos. Soc. 111, 241-253 (1992; Zbl 0751.20042)] proved that every semigroup, in which the idempotents form a subsemigroup, has an \(E\)- unitary cover (i.e., there is a semigroup \(T\) satisfying: \(ae, e \in E_ T \Rightarrow a \in E_ T\), and an idempotent-separating homomorphism from \(T\) onto \(S\)). In particular, if \(S\) is \(E\)-inversive (i.e., for every \(a \in S\) there is some \(x \in S\) such that \(ax \in E_ S\)) and if the idempotents of \(S\) commute, then its \(E\)-unitary cover \(T\) can be chosen with the same properties (this result was proved already by \textit{J. Fountain} [Bull. Lond. Math. Soc. 22, 353-358 (1990; Zbl 0724.20040)]. In the paper under review, using the same method of weak categories the author shows that every \(E\)-inversive semigroup \(S\), whose set of idempotents belongs to a variety \(B\) of bands, has an \(E\)-unitary, \(E\)- inversive cover \(T\) such that \(E_ T \in B\) (see the proof of Theorem 2.1(2) in Almeida-Pin-Weil, [loc. cit.]). The result on the existence of a finite \(E\)-unitary cover for any finite semigroup, whose set of idempotents belongs to a pseudo-variety of (finite) bands, is the contents of Corollary 4.2 in Almeida-Pin-Weil [loc. cit.].