In this paper we generalize Vere-JonesR-theory from irreducible to reducible nonnegative matrices of countably infinite dimension. We discuss excessive functions and eigenvectors associated with the convergence parameterR. Furthermore we show how generalized eigenvectors associated withRcan be constructed. Under special conditions related to strong ergodicity in Markov chains, these generalized eigenvectors can be chosen nonnegative. The results are extended to sets of nonnegative matrices which often appear in optimization problems (e.g. Markov decision chains, Leontief substitution systems).