Multiplicativity of permanents over matrix semirings

Abstract

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.