Spaces of regular abstract martingales

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.