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Matrices over semirings. (English) Zbl 0884.15010
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.

15B33Matrices over special rings (quaternions, finite fields, etc.)
Full Text: DOI
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