Factorizations of matrices over semirings. (Q1414145)

A semiring \(R\) with identity satisfies all ring axioms but one: an additive inverse of an element in \(R\) is not required. All matrices below have entries in \(R\). The semiring rank of a matrix \(A\) is the smallest \(r\) such that \(A=BC\), where \(B\) is an \(n\times r\) matrix and \(C\) is an \(r\times n\) matrix. A matrix \(E\in M_n(R)\) is called an elementary matrix if \(E\) is permutationally equivalent to a direct sum of \(\begin{pmatrix} a_1 & a_2\\ 0 & a_3\end{pmatrix}\) and an \((n-2)\times (n-2)\) monomial matrix, where the \(a_i\)'s are nonzero elements of \(R\). A matrix \(A\) is called semiprime if for every factorization \(A=BC\) either \(B\) or \(C\) is a monomial matrix. The authors show: If \(R\) is a chain semiring and \(A\in M_n(R)\) a nonmonomial matrix with full semiring rank, then \(A=P_dP_{d-1}\cdots P_1\), where \(d\geq1\) and each \(P_i\) is an elementary matrix or a semiprime matrix in \(M_n(R)\).