Locally strongly primitive semigroups of nonnegative matrices

The class of locally strongly primitive semigroups of nonnegative matrices is introduced. It is shown that, by a certain permutation similarity, all the matrices of a semigroup of the class considered can be brought to block monomial form; moreover, any matrix product of sufficient length has positive nonzero blocks only. This shows that the following known property of an imprimitive nonnegative matrix in Frobenius form is inherited. If such a matrix is raised to a sufficiently high power, then all its nonzero blocks are positive. A combinatorial criterion of the locally strong primitivity of a semigroup of nonnegative matrices is found.