Chainable properties of semigroups of non negative matrices

The theorem by Protasov and Voynov on the combinatorial structure of semigroups of nonnegative matrices extends a well-known result- of Frobenius on the canonical form of an irreducible nonnegative matrix. We generalize the Protasov — Voynov theorem to not necessarily irreducible semigroups of matrices. For this purpose, an extensions of the concepts of imprimitivity index and canonical partition arc introduced which arc based on the chain properties of nonnegative matrices.