Spaces of regular abstract martingales

In [15,10], the authors introduced and studied the space Mr of regular martingales on a vector lattice and the space Mr 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 Mr is a vector lattice and Mr is a Banach lattice under the regular norm.