A representation for \(F\)-regular semigroups.
A regular semigroup \(S\) is called \(F\)-regular if each class of the least group-congruence \(\sigma\) on \(S\) contains a greatest element with respect to the natural partial order of \(S\): \(a\leq b\) iff \(a=eb=bf\) for some idempotents \(e,f\in E_S\). Observing that such a semigroup is an \(E\)-unitary regular monoid, \textit{E. Giraldes}, \textit{P. Marques-Smith} and \textit{H. Mitsch} [J. Algebra 274, No. 2, 491-510 (2004; Zbl 1060.20051)] provided a construction of all \(F\)-regular semigroups by means of Szendrei-triples. They characterized them as certain subsemigroups of the semidirect product of a strictly combinatorial semigroup by a group. Thereby the problem remained that in general strictly combinatorial semigroups are not well understood. The paper under review shows that a representation of \(F\)-regular semigroups as subsemigroups of the semidirect product of a band \(B\) and a group \(G\) is possible. In particular, \(B\) belongs to the band variety generated by \(E_S\) and \(G=S/\sigma\). The embedding of \(S\) into the semidirect (wreath) product of a certain band by the group \(G\) is explicitly given, and an abstract description of the image of \(S\) is provided. This generalizes the result on the representation of \(F\)-inverse semigroups due to \textit{D. B. McAlister} [Trans. Am. Math. Soc. 196, 351-370 (1974; Zbl 0297.20072)].
scientific article; zbMATH DE number 2173798