A Perron-Frobenius-type Theorem for Positive Matrix Semigroups

One consequence of the Perron–Frobenius Theorem on indecomposable positive matrices is that whenever an \(n\times n\) matrix A dominates a non-singular positive matrix, there is an integer k dividing n such that, after a permutation of basis, A is block-monomial with \(k\times k\) blocks. Furthermore, for suitably large exponents, the nonzero blocks of \(A^m\) are strictly positive. We present an extension of this result for indecomposable semigroups of positive matrices.