On Factorizations of Upper Triangular Nonnegative Matrices of Order Three

Let T3 (N-0) denote the semigroup of 3 x 3 upper triangular matrices with nonnegative integral-valued entries. In this paper, we investigate factorizations of upper triangular nonnegative matrices of order three. Firstly, we characterize the atoms of the subsemigroup S of the matrices in T-3 (N-0) with nonzero determinant and give some formulas. As a consequence, problems 4a and 4c presented by Baeth et al. (2011) are each half-answered for the case n = 3. And then, we consider some factorization cases of matrix A in S with rho (A) = 1 and give formulas for the minimum factorization length of some special matrices in S.