A matrix problem over a discrete valuation ring

A flat matrix problem of mixed type (over a discrete valuation ring and its skew field of fractions) is considered which naturally arises in connection with several problems in the theory of integer-valued representations and in ring theory. For this problem, a criterion for module boundedness is proved, which is stated in terms of a pair of partially ordered sets $\bigl(\mathscr P(A),\mathscr P(B)\bigr)$ associated with the pair of transforming algebras $(A,B)$ defining the problem. The corresponding statement coincides in effect with the formulation of Kleiner's well-known finite-type criterion for representations of pairs of partially ordered sets over a field. The proof is based on a reduction (which uses the techniques of differentiation) to representations of semimaximal rings (tiled orders) and partially ordered sets.