On Factorizations of Upper Triangular Nonnegative Matrices of Order Three

Abstract

LetT3(N0)denote the semigroup of3×3upper 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 subsemigroupSof the matrices inT3(N0)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 casen=3. And then, we consider some factorization cases of matrixAinSwithρ(A)=1and give formulas for the minimum factorization length of some special matrices inS.