Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

Document Type

Article

Publication Date

1-1-2010

Abstract

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.

Keywords

Boolean matrix, Bijective linear mapping, Rank preserver

Publication Title

Linear Algebra And Its Applications

Volume

433

Issue

7

Publisher

Elsevier Science Inc

Publisher Location

360 PARK AVE SOUTH, NEW YORK, NY 10010-1710 USA

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