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  • P-ISSN1226-0657
  • E-ISSN2287-6081
  • KCI

ISSN : 1226-0657

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Vol.21 No.2

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CHARACTERIZATIONS OF BOOLEAN RANK PRESERVERS OVER BOOLEAN MATRICES

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics / Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, (P)1226-0657; (E)2287-6081
2014, v.21 no.2, pp.121-128
https://doi.org/10.7468/jksmeb.2014.21.2.121
Beasley, Leroy B.
Kang, Kyung-Tae
Song, Seok-Zun
Beasley,, L. B. , Kang,, K. , & Song,, S. (2014). CHARACTERIZATIONS OF BOOLEAN RANK PRESERVERS OVER BOOLEAN MATRICES. Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, 21(2), 121-128, https://doi.org/10.7468/jksmeb.2014.21.2.121

Abstract

The Boolean rank of a nonzero m <TEX>$m{\times}n$</TEX> Boolean matrix A is the least integer k such that there are an <TEX>$m{\times}k$</TEX> Boolean matrix B and a <TEX>$k{\times}n$</TEX> Boolean matrix C with A = BC. In 1984, Beasley and Pullman showed that a linear operator preserves the Boolean rank of any Boolean matrix if and only if it preserves Boolean ranks 1 and 2. In this paper, we extend this characterization of linear operators that preserve the Boolean ranks of Boolean matrices. We show that a linear operator preserves all Boolean ranks if and only if it preserves two consecutive Boolean ranks if and only if it strongly preserves a Boolean rank k with <TEX>$1{\leq}k{\leq}min\{m,n\}$</TEX>.

keywords
Boolean rank, linear operator, (strongly) preserve

Reference

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D. deCaen, D. Gregory & N. Pullman. (1981). The Boolean rank of zero one matrices (169-173). Proceedings of the Third Caribbean Conference on Combinatorics and Computing.

2.

L.B. Beasley & N.J. Pullman. (1984). Boolean rank preserving operators and Boolean rank-1 spaces. Linear Algebra and Its Applications, 59, 55-77. 10.1016/0024-3795(84)90158-7.

3.

Song, Seok-Zun;Beasley, Leroy B.;. (2013). LINEAR TRANSFORMATIONS THAT PRESERVE TERM RANK BETWEEN DIFFERENT MATRIX SPACES. Journal of the Korean Mathematical Society, 50(1), 127-136. 10.4134/JKMS.2013.50.1.127.

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L.B. Beasley & S.Z. Song. (2013). Linear operators that preserve Boolean rank of Boolean matrices. Czech. Math. J., 63, 435-440. 10.1007/s10587-013-0027-z.

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K.H. Kim. Boolean Matrix Theory and Applications. Pure Appl. Math. Vol. 70.

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S.J. Pierce et.al.. (1992). A Survey of Linear Preserver Problems. Linear and Multilinear Algebra, 33, 1-119. 10.1080/03081089208818176.

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics

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