On nilpotency of 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, we study the nilpotency of matrices over commutative antirings. We first provide some properties and characterizations of the nilpotent matrices in terms of principal permanental minors, main diagonals and permanental adjoint matrices. When a family of matrices are simultaneously considered, we establish some characterizations of the simultaneous nilpotence for a family of matrices.