Spanning column ranks of non-binary Boolean matrices and their preservers
J. Korean Math. Soc. 2019 Vol. 56, No. 2, 507-521
Published online 2019 Mar 01
Kyung-Tae Kang, Seok-Zun Song
Jeju National University; Jeju National University
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.
Keywords : spanning column rank, constituent, linear operator, congruence operator
MSC numbers : Primary 15A03, 15A86, 15B34
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