| Title:
|
Linear operators that preserve Boolean rank of Boolean matrices (English) |
| Author:
|
Beasley, LeRoy B. |
| Author:
|
Song, Seok-Zun |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
63 |
| Issue:
|
2 |
| Year:
|
2013 |
| Pages:
|
435-440 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $1<k\leq m$. (English) |
| Keyword:
|
Boolean matrix |
| Keyword:
|
Boolean rank |
| Keyword:
|
Boolean linear operator |
| MSC:
|
15A04 |
| MSC:
|
15A86 |
| MSC:
|
15B34 |
| idZBL:
|
Zbl 06236421 |
| idMR:
|
MR3073968 |
| DOI:
|
10.1007/s10587-013-0027-z |
| . |
| Date available:
|
2013-07-18T14:58:10Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143322 |
| . |
| Reference:
|
[1] Beasley, L. B., Li, C.-K., Pierce, S.: Miscellaneous preserver problems.Linear Multilinear Algebra 33 (1992), 109-119. Zbl 0767.15006, MR 1346786, 10.1080/03081089208818185 |
| Reference:
|
[2] Beasley, L. B., Pullman, N. J.: Boolean-rank-preserving operators and Boolean-rank-1 spaces.Linear Algebra Appl. 59 (1984), 55-77. Zbl 0536.20044, MR 0743045, 10.1016/0024-3795(84)90158-7 |
| Reference:
|
[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. MR 2783783 |
| Reference:
|
[4] Kim, K. H.: Boolean Matrix Theory and Applications.Pure and Applied Mathematics 70 Marcel Dekker, New York (1982). Zbl 0495.15003, MR 0655414 |
| Reference:
|
[5] Song, S.-Z.: Linear operators that preserve column rank of Boolean matrices.Proc. Am. Math. Soc. 119 (1993), 1085-1088. Zbl 0802.15006, MR 1184086, 10.1090/S0002-9939-1993-1184086-1 |
| . |
