[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/JKMS.j180241

SPANNING COLUMN RANKS OF NON-BINARY BOOLEAN MATRICES AND THEIR PRESERVERS

Kang, Kyung-Tae (Department of Mathematics Jeju National University)
Song, Seok-Zun (Department of Mathematics Jeju National University)

Publication Information

Journal of the Korean Mathematical Society / v.56, no.2, 2019 , pp. 507-521 More about this Journal

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;

Citations & Related Records

Reference

1	S.-Z. Song and S.-G. Lee, Column ranks and their preservers of general Boolean matrices, J. Korean Math. Soc. 32 (1995), no. 3, 531-540.
2	J. H. M. Wedderburn, Boolean linear associative algebra, Ann. of Math. (2) 35 (1934), no. 1, 185-194. DOI
3	L. B. Beasley and N. J. Pullman, Boolean-rank-preserving operators and Boolean-rank-1 spaces, Linear Algebra Appl. 59 (1984), 55-77. DOI
4	L. B. Beasley and N. J. Pullman, Semiring rank versus column rank, Linear Algebra Appl. 101 (1988), 33-48. DOI
5	L. B. Beasley, S. Z. Song, K. T. Kang, and B. K. Sarma, Column ranks and their pre- servers over nonnegative real matrices, Linear Algebra Appl. 399 (2005), 3-16. DOI
6	K. H. Kim, Boolean Matrix Theory and Applications, Monographs and Textbooks in Pure and Applied Mathematics, 70, Marcel Dekker, Inc., New York, 1982.
7	S. Kirkland and N. J. Pullman, Linear operators preserving invariants of nonbinary Boolean matrices, Linear and Multilinear Algebra 33 (1993), no. 3-4, 295-300. DOI
8	S.-Z. Song, Linear operators that preserve column rank of Boolean matrices, Proc. Amer. Math. Soc. 119 (1993), no. 4, 1085-1088. DOI
9	S.-Z. Song and S.-G. Hwang, Spanning column ranks and their preservers of nonnegative matrices, Linear Algebra Appl. 254 (1997), 485-495. DOI