  We use the rational tableaux introduced by Stembridge to give a bideterminant
basis for a normal reductive monoid and for its variety of noninvertible
elements. We also obtain a bideterminant basis for the full coordinate ring of
the general linear group and for all its truncations with respect to saturated
sets. Finally, we deduce an alternative proof of the double centraliser theorem
for the rational Schur algebra and the walled Brauer algebra over an arbitrary
infinite base field which was first obtained by Dipper, Doty and Stoll.
