On the Approximation of Random Variables on a Finite Chain

We consider the transformations of independent random variables over a linearly ordered finite set (a chain) by the join and meet operations. We investigate the possibility of approximating an arbitrary probability distribution on a chain by means of a (possibly iterated) application of the join and meet operations to independent random variables with distributions from a given set. We establish some conditions under which the approximation is impossible and the conditions when it becomes possible.