Consider a set S of countable non-negative matrices satisfying the property that for any two indices i, j, for some n ≧ 1 there are matrices M
2, · · ·, Mn
in S with (M
2 · · · Mn
>0. For non-negative vectors x set Tx = sup
Mx, where the supremum is taken separately in each coordinate. Assume that for each x with Tx finite in each coordinate there is a matrix in S which achieves the supremum simultaneously for all coordinates. With these two assumptions on S, the R-theory for a countable irreducible matrix is extended to the operator T. The results are used to consider the existence of stationary optimal policies for Markov decision processes with multiplicative rewards.