Linear operators that preserve Boolean rank of Boolean matrices

LeRoy B. Beasley; Seok-Zun Song

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 2, page 435-440
  • ISSN: 0011-4642

Abstract

top
The Boolean rank of a nonzero m × n Boolean matrix A is the minimum number k such that there exist an m × k Boolean matrix B and a k × n Boolean matrix C such that A = B C . In the previous research L. B. Beasley and N. J. Pullman obtained that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks 1 and 2 . In this paper we extend this characterizations of linear operators that preserve the Boolean ranks of Boolean matrices. That is, we obtain that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks 1 and k for some 1 < k ≤ m .

How to cite

top

Beasley, LeRoy B., and Song, Seok-Zun. "Linear operators that preserve Boolean rank of Boolean matrices." Czechoslovak Mathematical Journal 63.2 (2013): 435-440. <http://eudml.org/doc/260698>.

@article{Beasley2013,
abstract = {The Boolean rank of a nonzero $m\times n$ Boolean matrix $A$ is the minimum number $k$ such that there exist an $m\times k$ Boolean matrix $B$ and a $k\times n$ Boolean matrix $C$ such that $A=BC$. In the previous research L. B. Beasley and N. J. Pullman obtained that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks $1$ and $2$. In this paper we extend this characterizations of linear operators that preserve the Boolean ranks of Boolean matrices. That is, we obtain that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks $1$ and $k$ for some $1<k\le m$.},
author = {Beasley, LeRoy B., Song, Seok-Zun},
journal = {Czechoslovak Mathematical Journal},
keywords = {Boolean matrix; Boolean rank; Boolean linear operator; Boolean matrix; Boolean rank; Boolean linear operator; preserver},
language = {eng},
number = {2},
pages = {435-440},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Linear operators that preserve Boolean rank of Boolean matrices},
url = {http://eudml.org/doc/260698},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Beasley, LeRoy B.
AU - Song, Seok-Zun
TI - Linear operators that preserve Boolean rank of Boolean matrices
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 2
SP - 435
EP - 440
AB - The Boolean rank of a nonzero $m\times n$ Boolean matrix $A$ is the minimum number $k$ such that there exist an $m\times k$ Boolean matrix $B$ and a $k\times n$ Boolean matrix $C$ such that $A=BC$. In the previous research L. B. Beasley and N. J. Pullman obtained that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks $1$ and $2$. In this paper we extend this characterizations of linear operators that preserve the Boolean ranks of Boolean matrices. That is, we obtain that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks $1$ and $k$ for some $1<k\le m$.
LA - eng
KW - Boolean matrix; Boolean rank; Boolean linear operator; Boolean matrix; Boolean rank; Boolean linear operator; preserver
UR - http://eudml.org/doc/260698
ER -

References

top
  1. Beasley, L. B., Li, C.-K., Pierce, S., 10.1080/03081089208818185, Linear Multilinear Algebra 33 (1992), 109-119. (1992) Zbl0767.15006MR1346786DOI10.1080/03081089208818185
  2. Beasley, L. B., Pullman, N. J., 10.1016/0024-3795(84)90158-7, Linear Algebra Appl. 59 (1984), 55-77. (1984) Zbl0536.20044MR0743045DOI10.1016/0024-3795(84)90158-7
  3. Kang, K.-T., Song, S.-Z., Heo, S.-H., Jun, Y.-B., Linear preserves of regular matrices over general Boolean algebras, Bull. Malays. Math. Sci. Soc. 34 (2011), 113-125. (2011) MR2783783
  4. Kim, K. H., Boolean Matrix Theory and Applications, Pure and Applied Mathematics 70 Marcel Dekker, New York (1982). (1982) Zbl0495.15003MR0655414
  5. Song, S.-Z., 10.1090/S0002-9939-1993-1184086-1, Proc. Am. Math. Soc. 119 (1993), 1085-1088. (1993) Zbl0802.15006MR1184086DOI10.1090/S0002-9939-1993-1184086-1

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.