  We use the theory of bialgebras to provide the algebraic background for state
space realization theorems for input-output maps of control systems. This
allows us to consider from a common viewpoint classical results about formal
state space realizations of nonlinear systems and more recent results involving
analysis related to families of trees. If $H$ is a bialgebra, we say that $p
\in H^*$ is differentially produced by the algebra $R$ with the augmentation
$\epsilon$ if there is right $H$-module algebra structure on $R$ and there
exists $f \in R$ satisfying $p(h) = \epsilon(f \cdot h)$. We characterize those
$p \in H^*$ which are differentially produced.
