Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

EXTREME SETS OF RANK INEQUALITIES OVER BOOLEAN MATRICES AND THEIR PRESERVERS

  • Song, Seok Zun (Department of Mathematics Jeju National University) ;
  • Kang, Mun-Hwan (Department of Mathematics Jeju National University) ;
  • Jun, Young Bae (Department of Mathematics Education (and RINS) Gyeongsang National University)
  • Received : 2011.03.16
  • Published : 2013.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

We consider the sets of matrix ordered pairs which satisfy extremal properties with respect to Boolean rank inequalities of matrices over nonbinary Boolean algebra. We characterize linear operators that preserve these sets of matrix ordered pairs as the form of $T(X)=PXP^T$ with some permutation matrix P.

Keywords

References

  1. L. B. Beasley and A. E. Guterman, Rank inequalities over semirings, J. Korean Math. Soc. 42 (2005), no. 2, 223-241. https://doi.org/10.4134/JKMS.2005.42.2.223
  2. L. B. Beasley, A. E. Guterman, Y. B. Jun, and S. Z. Song, Linear preservers of extremes of rank inequalities over semirings: row and column ranks, Linear Algebra Appl. 413 (2006), no. 2-3, 495-509. https://doi.org/10.1016/j.laa.2005.03.024
  3. L. B. Beasley, A. E. Guterman, and C. L. Neal, Linear preservers for Sylvester and Frobenius bounds on matrix rank, Rocky Mountain J. Math. 36 (2006), no. 1, 67-80. https://doi.org/10.1216/rmjm/1181069488
  4. L. B. Beasley, S. J. Kirkland, and B. L. Shader, Rank comparisons, Linear Algebra Appl. 221 (1995), 171-188. https://doi.org/10.1016/0024-3795(93)00242-R
  5. L. B. Beasley, S. G. Lee, and S. Z. Song, Linear operators that preserve pairs of matrices which satisfy extreme rank properties, Linear Algebra Appl. 350 (2002), 263-272. https://doi.org/10.1016/S0024-3795(02)00293-8
  6. L. B. Beasley and N. J. Pullman, Boolean-rank-preserving operators and Boolean rank-1-spaces, Linear Algebra Appl. 59 (1984), 55-77. https://doi.org/10.1016/0024-3795(84)90158-7
  7. S. Kirkland and N. J. Pullman, Linear operators preserving invariants of nonbinary matrices, Linear and Multilinear Algebra 33 (1992), 295-300. https://doi.org/10.1080/03081089308818200
  8. C. K. Li and S. Pierce, Linear preserver problems, Amer. Math. Monthly 108 (2001), no. 7, 591-605. https://doi.org/10.2307/2695268
  9. S. Pierce and others, A survey of linear preserver problems contents, Linear and Multilinear Algebra 33 (1992), no. 1-2, 1-119. https://doi.org/10.1080/03081089208818176
  10. S. Z. Song and K. T. Kang, Linear maps that preserve commuting pairs of matrices over general Boolean algebra, J. Korean Math. Soc. 43 (2006), no. 1, 77-86. https://doi.org/10.4134/JKMS.2006.43.1.077

Cited by

  1. EXTREME PRESERVERS OF TERM RANK INEQUALITIES OVER NONBINARY BOOLEAN SEMIRING vol.51, pp.1, 2014, https://doi.org/10.4134/JKMS.2014.51.1.113