Spanning column ranks of non-binary Boolean matrices and their preservers
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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.
Abstract
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.
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