Quasi-abelian and quasi-solvable regular semigroups

In this note quasi-abelian semigroups are studied and it is proved that they form an e-variety of orthodox semigroups. More, quasi-abelian regular Bruck-Reilly monoids are characterized as extensions of monoids which are (reverse) semidirect products of a group and a semilattice. At last, quasi-abelian congruences are studied and a definition of quasi-solvability is given which generalizes the notion of solvability to the class of regular semigroups.