Matrices comparable by columns and proportional by columns over lattices

Matrices over a lattice $(L,\le)$ comparable by columns are studied. (A matrix is comparable by columns iff its columns form a linearly ordered set with the partial order induced from`$L$.) Some properties of the matrices are obtained. Solvability of matrix equations in this class of matrices is studied.
The set of matrices proportional by columns is the subset of the set of matrices comparable by columns. Some properties as well as solvability of matrix equations are also studied for suth matrices.