The Characterization and the Product of Quasi-Ehresmann transversals

Abstract

Wang (Filomat 29(5), 985-1005, 2015) introduced and investigated quasi-Ehresmann transversals of semi-abundant semigroups as the generalizations of
orthodox transversals of regular semigroups in the semi-abundant case. In this paper, we give two characterizations for a generalized quasi-Ehresmann transversal to be a quasi-Ehresmann transversal. These results further demonstrate that quasi-Ehresmann transversals are the ``real" generalizations of orthodox transversals in the semi-abundant case. Moreover, we obtain the main result that the product of any two quasi-ideal quasi-Ehresmann transversals of a semi-abundant semigroup $S$ which satisfy the regularity condition is a quasi-ideal quasi-Ehresmann transversal of $S$. Furthermore, all quasi-ideal quasi-Ehresmann transversals of $S$ form a rectangular band.