Congruences on abundant semigroups associated with {G}reen's \(*\)-relations
\textit{F. Pastijn} and \textit{M. Petrich} [Boll. Unione Mat. Ital., VII. Ser., B 1, 591--603 (1987; Zbl 0621.20036)] comprehensively studied the congruences on regular semigroups associated with Green's relations. The purpose of this paper is to generalize those results to ``good'' congruences on abundant semigroups. Recall that a semigroup \(S\) is \textit{abundant} if each class of the generalized Green relations \(\mathcal L^\ast\) and \(\mathcal R^\ast\) contains an idempotent; the good congruences are those that respect \(\mathcal L^\ast\) and \(\mathcal R^\ast\), a natural restriction in this context. Direct analogues are found for many of the results of Pastijn and Petrich [loc. cit.]. As an example, the least good congruence generated by \(\mathcal L^\ast\) is the least good congruence whose quotient is a right regular band. The cited work also considered congruences on regular semigroups generated by the restrictions of Green's relations to the idempotents. These results are also generalized, although in order to do so the authors have to impose a further condition on an abundant semigroup: that the regular elements form a subsemigroup.
scientific article; zbMATH DE number 6837918
scientific article