IJPAM: Volume 106, No. 1 (2016)

INVERTIBLE MATRICES IN CERTAIN COMMUTATIVE
SUBSEMIRINGS OF FULL MATRIX SEMIRINGS

N. Sirasuntorn$^1$, R.I. Sararnrakskul$^2$
$^{1,2}$Department of Mathematics
Faculty of Science
Srinakharinwirot University
114 Sukhumvit 23, Wattana District, Bangkok 10110, THAILAND


Abstract. Then every field is a semifield. For a semifield $S$, we let $D_{n}(S)$ denote the set of all $A \in M_{n}(S)$ of the form

$\begin{bmatrix}
x_{1} & 0 & \cdots & 0 & y_{1} \\
0 & x_{2} & \cdots & y_{...
...2} & \cdots & x_{2} & 0 \\
y_{1} & 0 & \cdots & 0 & x_{1} \\
\end{bmatrix}$
where $M_{n}(S)$ is the full $n \times n$ matrix semiring over $S$. Then $D_{n}(S)$ is a maximal commutative subsemiring of the semiring $M_{n}(S)$. If $S$ is a field, it is known that $A \in D_{n}(S)$ is invertible if and only if $\det A \neq 0$. In this paper, invertible matrices in $D_{n}(S)$ where $S$ is a semifield which is not a field are characterized. It is shown that if $S$ is a semifield which is not a field, then $A \in D_{n}(S)$ is an invertible matrix over $S$ if and only if $(x_{i} = 0$ if and only if $ y_{i} \neq 0)$.

Received: Octobere 21, 2015

AMS Subject Classification: 16Y60, 20M17, 15A09

Key Words and Phrases: semiring, semifield, full matrix semiring, invertible matrix

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DOI: 10.12732/ijpam.v106i1.14 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2016
Volume: 106
Issue: 1
Pages: 191 - 197


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