Abstract: Inclines are additively idempotent semi-rings in which the products are less than or equal to either factor. Some properties of the invertible incline matrices are discussed, and the ranks of the invertible incline matrices are proved to be full. The row ranks, the column ranks and the Schein ranks of incline matrices are studied and important properties of the Schein ranks of incline matrices are given.

Key words: rank of incline matrix , invertible incline matrix, incline matrix, incline