  First we give a counterexample showing that recent results on separate order
continuity of Arens extensions of multilinear operators cannot be improved to
get separate order continuity on the product of the whole of the biduals. Then
we establish conditions on the operators and/or on the underlying Riesz
spaces/Banach lattices so that the extensions are order continuous on the
product of the whole biduals. We also prove that all Arens extensions of any
regular multilinear operator are order continuous in at least one variable and
we study when Arens extensions of regular homogeneous polynomials on a Banach
lattice $E$ are order continuous on $E^{\sim\sim}$.
