Orderamarts: A class of asymptotic martingales

We extend the notion of real-valued asymptotic martingales to the Banach lattice valued case. Unlike the other extensions, the notion of “orderamart” preserves the lattice property of real amarts. We show also, a Riesz decomposition, a weak and strong convergence theorem, a probabilistic characterization of A-L spaces from which we can prove that a Banach lattice with the shur property and a quasi-interior point in the dual is an l¹(Γ).