  Martingale-like sequences in vector lattice and Banach lattice frameworks are
defined in the same way as martingales are defined in [Positivity 9 (2005),
437--456]. In these frameworks, a collection of bounded $X$-martingales is
shown to be a Banach space under the supremum norm, and under some conditions
it is also a Banach lattice with coordinate-wise order. Moreover, a necessary
and sufficient condition is presented for the collection of
$\mathcal{E}$-martingales to be a vector lattice with coordinate-wise order. It
is also shown that the collection of bounded $\mathcal{E}$-martingales is a
normed lattice but not necessarily a Banach space under the supremum norm.
