Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank - Lim, Ming Huat; Tan, Sin Chee (Elsevier Science Inc, 2010)
Let M(m,n) (B) be the semimodule of all m x 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 buective linear mapping on B (m, n, k), then there exist permutation matrices P and Q such that T (A) = PAQ for all A is an element of B (m, n, k) or m = n and T (A) = PA(l)Q for all A is an element of B (m, n, k) This result follows from a more general theorem we prove concerning the structure of linear mappings on B (m, n, k) that preserve bot h 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 (C) 2010 Elsevier Inc All rights reserved.