Up-martingales in vector lattices

We provide a construction in a general vector lattice similar to martingales, however not using both integrability and norms of elements. The idea is to replace the mean value with the infimum value in the conditional expectation of a positive element. On the one hand, the construction is quite general. But on the other hand, the corresponding operator playing role of a conditional expectation is both nonlinear and non-order continuous in the most natural examples of filtrations, that makes investigation to be non-trivial. Another difference from the standard martingales is that we consider arbitrary nets for indices of filtrations instead of just subsets of the real line. We also give an application of standard martingales over a general net of indices that cannot be obtained using countable martingales or martingales over the indices from a segment of the real line. Mathematics Subject Classification: Primary 46A35; secondary 46B15; 46A40; 46B42