Multiplicativity of permanents over matrix semirings

In this paper, we investigate the conditions for the multiplicativity of the permanent over a matrix semiring. We prove that if S is either a commutative antiring or a commutative semiring where the set $V(S)$ of all additively invertible elements coincides with the set of all nilpotents, then the permanent is multiplicative on the group of invertible matrices over S if and only if 1+2V(S)²=1. We then use this result to investigate the number of invertible matrices over S with a specified permanent.