Journal of the
Korean Mathematical Society JKMS

For any $m\times n$ nonbinary Boolean matrix $A$, its spanning column rank is the minimum number of the columns of $A$ that spans its column space. We have a characterization of spanning column rank-$1$ nonbinary Boolean matrices. We investigate the linear operators that preserve the spanning column ranks of matrices over the nonbinary Boolean algebra. That is, for a linear operator $T$ on $m\times n$ nonbinary Boolean matrices, it preserves all spanning column ranks if and only if there exist an invertible nonbinary Boolean matrix $P$ of order $m$ and a permutation matrix $Q$ of order $n$ such that $T(A)=PAQ$ for all $m\times n$ nonbinary Boolean matrix $A$. We also obtain other characterizations of the (spanning) column rank preserver.

Keywords: spanning column rank, constituent, linear operator, congruence operator

MSC numbers: Primary 15A03, 15A86, 15B34

Supported by: This work was supported under the framework of international cooperation program managed by the National Research Foundation of Korea (NRF-2018K1A3A1A 38057138).