Rank and perimeter preserver of rank-1 matrices over max algebra

For a rank-1 matrix ${\displaystyle A=a\otimes b^t}$ over max algebra, we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the linear operators which preserve the rank and perimeter of rank-1 matrices over max algebra. That is, a linear operator T preserves the rank and perimeter of rank-1 matrices if and only if it has the form T(A) = U ⊗ A ⊗ V, or ${\displaystyle T\left(A\right)=U\otimes A^t\otimes V}$ with some monomial matrices U and V.