LINEAR PRESERVERS OF BOOLEAN RANK BETWEEN DIFFERENT MATRIX SPACES
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The Boolean rank of a nonzero m × n Boolean matrix A is the least integer k such that there are an m× k Boolean matrix B and a k × n Boolean matrix C with A = BC. We investigate the structure of linear transformations T : M m,n → M p,q which preserve Boolean rank. We also show that if a linear transformation preserves the set of Boolean rank 1 matrices and the set of Boolean rank k matrices for any k, 2 ≤ k ≤ min{m, n} (or if T strongly preserves the set of Boolean rank 1 matrices), then T preserves all Boolean ranks.
Abstract
The Boolean rank of a nonzero m × n Boolean matrix A is the least integer k such that there are an m× k Boolean matrix B and a k × n Boolean matrix C with A = BC. We investigate the structure of linear transformations T : M m,n → M p,q which preserve Boolean rank. We also show that if a linear transformation preserves the set of Boolean rank 1 matrices and the set of Boolean rank k matrices for any k, 2 ≤ k ≤ min{m, n} (or if T strongly preserves the set of Boolean rank 1 matrices), then T preserves all Boolean ranks.
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