Linear preservers of Boolean rank between different matrix spaces

摘要

The Boolean rank of a nonzero $mimes n$ Boolean matrix $A$ is the least integer $k$ such that there are an $mimes k$ Boolean matrix $B$ and a $kimes n$ Boolean matrix $C$ with $A=BC$. We investigate the structure of linear transformations $mnpq$ 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 $k$, $2le kle min{m,n}$ (or if $T$ strongly preserves the set of Boolean rank $1$ matrices), then $T$ preserves all Boolean ranks.