  The concept of an adequate transversal of an abundant semigroup was
introduced by El-Qallali in [8] whilst in [7], he and Fountain initiated the
study of quasi-adequate semigroups as natural generalisations of orthodox
semigroups. In this work we provide a structure theorem for adequate
transversals of certain types of quasi-adequate semigroup and from this deduce
Saito's classic result on the structure of inverse transversals of orthodox
semigroups. We also apply it to left ample adequate transversals of left
adequate semigroups and provide a structure for these based on semidirect
products of adequate semigroups by left regular bands.
