Completely simple congruences on \(E\)-inversive semigroups (Q2803549)

A semigroup \(S\) is \(E\)-inversive if for any \(a \in S\) there exists \(x \in S\) such that \(ax \in E(S)\), the set of idempotents of \(S\); equivalently if for each \(a\), there is a `weak inverse' \(x\), such that \(x = xax\). The \(E\)-inversive semigroups for which each weak inverse of \(a\) is actually an inverse are precisely the completely simple semigroups. The completely simple congruences on an \(E\)-inversive semigroup form a sublattice of the lattice of all congruences. It is shown that the `trace' relation, which identities congruences having the same restriction to the set of idempotents, is a complete congruence and that every congruence is determined by its trace and its kernel (the union of congruence classes of idempotents). Various ancillary results are also established. The reviewer notes that, according to the author, any class of such a congruence must contain a regular element and so the congruence is `strongly regular', in the terminology of \textit{X. Fan} et al. [J. Aust. Math. Soc. 100, No. 2, 199--215 (2016; Zbl 1344.20073)]. Thus there is overlap between the two papers.