Matrices over semirings
Let \(R\) be a commutative semiring with multiplicative identity \(1 \neq 0\) (for example, the nonnegative integers), and let \(M_{n}(R)\) denote the semiring of \(n \times n\) matrices over \(R\). Because the condition of invertibility in \(M_{n}(R)\) is rather restrictive in this situation, the author introduces the concept of ``semi-invertibility'': \(A \in M_{n}(R)\) is semi-invertible if there exist \(A_{1}, A_{2} \in M_{n}(R)\) such that \(I+ AA_{1} = AA_{2}\) and \(I+ A_{1}A = A_{2}A\). Some more or less straightforward generalizations of criteria for invertibility when \(R\) is a ring are made to give criteria for semi-invertibility in the semiring case.
scientific article; zbMATH DE number 1071950
scientific article