  We determine all functional closure properties of finite
$\mathbb{N}$-weighted automata, even all multivariate ones, and in particular
all multivariate polynomials. We also determine all univariate closure
properties in the promise setting, and all multivariate closure properties
under certain assumptions on the promise, in particular we determine all
multivariate closure properties where the output vector lies on a monotone
algebraic graph variety.
