On invertible matrices over antirings

An antiring is a semiring which is zerosumfree (i.e., a + b = 0 implies a = b = 0 for any a, b in this semiring). In this paper, the complete description of the invertible matrices over a commutative antiring is given and some necessary and sufficient conditions for a matrix over a commutative antiring to be invertible are obtained. Also, Cramer’s rule over commutative antirings is presented. The main results in this paper generalize and develop the corresponding results for the Boolean matrices, the fuzzy matrices, the lattice matrices and the incline matrices.