previous section and derive some immediate consequences from them. This would give some motivation for considering these dual maps.

By general principles of duality, ${\hat{\mathcal F}}$ is an associative algebra with unit ${e}$, with respect to the $\centerdot$ product. The dual versions of Theorem \[t:coderivation\] and Lemma \[l:coderivation\_identity\] are as follows.

\[t:derivation\] The map $S$ is a derivation on the algebra ${\hat{\mathcal F}}$, that is, $$S \circ \centerdot = \centerdot \circ ({\operatorname{id}}{\otimes}S + S {\otimes}{\operatorname{id}}).$$ This may be more familiarly expressed as $S(u \centerdot v) = S(u) \centerdot v + u \centerdot S(v)$ for $u,v \in {\hat{\mathcal F}}$. Also, $S(1) = {e}$ and $S({e}) = 0$.

\[l:derivation\_identity\] Let the maps $S$, $\centerdot$ and $\mu^*$ be as defined before. Then $2 S = \centerdot \circ \mu^*$.

We know from equation  that $\hat G$ satisfies the important property ${\hat G}(\Psi(B_n)) = \Psi(B_{n+1})$. We now state the dual version of this property.

\[l:reduction\] Let $v$ be any [**cd**]{}-monomial of non-negative degree. Then $\beta(S(v)) = \beta(v)$.

To illustrate how duality works, we give a proof of this lemma. Let $v$ be a monomial of degree $n$, with $n \geq 0$. Then, $$\begin{array}{r c c c l}
\beta(S(v)) & = & \langle \delta_{S(v)},\Psi(B_{n}) \rangle                          
     & 