Invertible incline matrices and Cramerʹs rule over inclines Original Research Article

Inclines are the additively idempotent semirings in which the products are less than or equal to factors. Thus inclines generalize Boolean algebra, fuzzy algebra and distributive lattice. And the Boolean matrices, the fuzzy matrices and the lattice matrices are the prototypical examples of the incline matrices (i.e., the matrices over inclines). In this paper, the complete description of the invertible incline matrices is given. Some necessary and sufficient conditions for an incline matrix to be invertible are studied, Cramerʹs rule over inclines is presented and the group of invertible incline matrices is investigated. The main results in the present paper generalize and develop the corresponding results in the literatures for the Boolean matrices, the fuzzy matrices and the lattice matrices.