Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank
✍ Scribed by Ming-Huat Lim; Sin-Chee Tan
- Publisher
- Elsevier Science
- Year
- 2010
- Tongue
- English
- Weight
- 180 KB
- Volume
- 433
- Category
- Article
- ISSN
- 0024-3795
No coin nor oath required. For personal study only.
✦ Synopsis
Let M m,n (B) be the semimodule of all m × n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2 k min (m, n). Let B (m, n, k) denote the subsemimodule of M m,n (B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B (m, n, k), then there exist permutation matrices P and Q such that T(A) = PAQ for all A ∈ B (m, n, k) or m = n and T (A)
This result follows from a more general theorem we prove concerning the structure of linear mappings on B (m, n, k) that preserve both the weight of each matrix and rank one matrices of weight k 2 . Here the weight of a Boolean matrix is the number of its non-zero entries.