This paper investigates the semigroup of incline Hall matrices. An incline Hall matrix means an incline matrix which is greater than or equal to an incline invertible matrix. It is proved that any incline Hall matrix has index. Some necessary and sufficient conditions for an incline Hall matrix to be regular in the semigroup of incline Hall matrices are given and Green's relations on the such semigroup are described. Also the sandwich semigroup of incline Hall matrices is considered. The main results in the present paper generalize some previous results on Boolean matrices, fuzzy matrices and lattice matrices.