  In \cite{Troitsky:05,Korostenski:08}, the authors introduced and studied the
space $\mathcal M_r$ of regular martingales on a vector lattice and the space
$M_r$ of bounded regular martingales on a Banach lattice. In this note, we
study these two spaces from the vector lattice point of view. We show, in
particular, that these spaces need not be vector lattices. However, if the
underlying space is order complete then $\mathcal M_r$ is a vector lattice and
$M_r$ is a Banach lattice under the regular norm.
