The minimum group congruence on certain eventually regular semigroups (Q1312271)

A semigroup \(S\) is called eventually regular if each element \(x \in S\) has a power which is regular. An eventually regular semigroup \(S\) is eventually orthodox if the set of idempotents \(E(S)\) of \(S\) forms a subsemigroup. Eventually conventional semigroups are a generalization of eventually orthodox semigroups having a slightly more complicated definition. The paper under review describes the minimum group congruence \(\sigma\) on an eventually conventional semigroup (and thus on an eventually orthodox semigroup) by \(\sigma = \{(a,b) \in S\times S\mid (\exists e \in E(S)) eae = ebe\}\) and thus generalizes known results for conventional semigroups. It should be noted that there is a relatively explicit description of the minimum group congruence on any \(E\)-inversive semigroup (which class is much more general than the class of eventually regular semigroups) [\textit{H. Mitsch}, J. Aust. Math. Soc., Ser. A 48, 66- 78 (1990; Zbl 0691.20050)]. It is, however, not clear whether the given description follows easily from the latter.