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

Let Mm,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 Mm,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)=PAtQ for all A∈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 both the weight of each matrix and rank one matrices of weight k2. Here the weight of a Boolean matrix is the number of its nonzero entries.