| Title:
|
Linear operators that preserve graphical properties of matrices: isolation numbers (English) |
| Author:
|
Beasley, LeRoy B. |
| Author:
|
Song, Seok-Zun |
| Author:
|
Jun, Young Bae |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
3 |
| Year:
|
2014 |
| Pages:
|
819-826 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $A$ be a Boolean $\{0,1\}$ matrix. The isolation number of $A$ is the maximum number of ones in $A$ such that no two are in any row or any column (that is they are independent), and no two are in a $2\times 2$ submatrix of all ones. The isolation number of $A$ is a lower bound on the Boolean rank of $A$. A linear operator on the set of $m\times n$ Boolean matrices is a mapping which is additive and maps the zero matrix, $O$, to itself. A mapping strongly preserves a set, $S$, if it maps the set $S$ into the set $S$ and the complement of the set $S$ into the complement of the set $S$. We investigate linear operators that preserve the isolation number of Boolean matrices. Specifically, we show that $T$ is a Boolean linear operator that strongly preserves isolation number $k$ for any $1\leq k\leq \min \{m,n\}$ if and only if there are fixed permutation matrices $P$ and $Q$ such that for $X\in {\mathcal M}_{m,n}(\mathbb B)$ $T(X)=PXQ$ or, $m=n$ and $T(X)=PX^tQ$ where $X^t$ is the transpose of $X$. (English) |
| Keyword:
|
Boolean matrix |
| Keyword:
|
Boolean rank |
| Keyword:
|
Boolean linear operator |
| Keyword:
|
isolation number |
| MSC:
|
15A04 |
| MSC:
|
15A86 |
| MSC:
|
15B34 |
| idZBL:
|
Zbl 06391527 |
| idMR:
|
MR3298562 |
| DOI:
|
10.1007/s10587-014-0134-5 |
| . |
| Date available:
|
2014-12-19T16:15:00Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144060 |
| . |
| Reference:
|
[1] Beasley, L. B.: Isolation number versus Boolean rank.Linear Algebra Appl. 436 (2012), 3469-3474. Zbl 1245.15003, MR 2900729 |
| Reference:
|
[2] Beasley, L. B., Li, C.-K., Pierce, S.: Miscellaneous preserver problems. A survey of linear preserver problems.Linear Multilinear Algebra 33 (1992), 109-119. MR 1346786 |
| Reference:
|
[3] Beasley, L. B., Pullman, N. J.: Linear operators that strongly preserve upper ideals of matrices.F. Hoffman et al. Proceedings of the Twenty-Third Southeastern International Conference on Combinatorics, Graph Theory, and Computing, Florida Atlantic University, Boca Raton, USA Congr. Numerantium. 88 Utilitas Mathematica Publishing, Winnipeg (1992), 229-243. Zbl 0835.15008, MR 1208933 |
| Reference:
|
[4] 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:
|
[5] Caen, D. de, Gregory, D. A., Pullman, N. J.: The Boolean rank of zero-one matrices.C. C. Cadogan Proceedings of the Third Caribbean Conference on Combinatorics and Computing, Bridgetown, 1981 Univ. West Indies, Cave Hill Campus, Barbados (1981), 169-173. Zbl 0496.20052, MR 0657185 |
| Reference:
|
[6] Kang, K.-T., Song, S.-Z., Beasley, L. B.: Linear preservers of term ranks of matrices over semirings.Linear Algebra Appl. 436 (2012), 1850-1862. Zbl 1236.15058, MR 2889962 |
| Reference:
|
[7] Kim, K. H.: Boolean Matrix Theory and Applications.Monographs and Textbooks in Pure and Applied Mathematics 70 Marcel Dekker, New York (1982). Zbl 0495.15003, MR 0655414 |
| Reference:
|
[8] 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 |
| . |
