Linear preservers of extremes of matrix pairs over nonbinary Boolean algebra

The m×n Boolean matrix A is said to be of Boolean rank r if there exist m×r Boolean matrix B and r×m Boolean matrix C such that A=BC and r is the smallest positive integer that such a factorization exists. We characterize linear operators that preserve the sets of matrix ordered pairs which satisfy extremal properties with respect to Boolean rank inequalities of matrices over nonbinary Boolean algebras.