Multiplicativity of Generalized Permanents over Semirings

We consider the matrix equation d H ( A B ) = d H ( A ) ⋅ d H ( B ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203719916/6ef71bf2-a4fa-438b-9c98-58b1246952d2/content/unequ9_121_1.tif"/> where d_H is a generalized permanent function corresponding to a subgroup H of S_n and the matrices A,B have entries in a semiring, Over antinegative cancellation semirings, we find results similar to the those for fields. For chain semirings we show the semigroup of those matrices A for which d_H(A) ≠ 0 is maximal with respect to the above equation over the 2-element Boolean semiring. For arbitrary chain semirings we show the “constant dominant diagonal” semigroup is always maximal with respect to the equation. It is further shown that the semigroup of matrices with at least one zero row (column) is always maximal.