map defined by ${\hat G}({e}) = 1$ and ${\hat G}(u) = G(u) + u {{\bf c}}$ for $u \in {{\mathcal F}}$. Note that the definition of $\hat G$ is arranged so that the equation $$\label{e:g}
{\hat G}(\Psi(B_n)) = \Psi(B_{n+1}) \quad \text{holds for} \quad n \geq 0.$$ This can be seen from Proposition \[p:g\]. It is clear that understanding $\hat G$ is crucial for our purposes. At least, that is the approach we take.

It is easy to see that for $u,v \in {{\mathcal F}}$, we have $\hat G(u v) = \hat G(u) v + u \hat G(v) 
        - u {{\bf c}}v$. We will use this identity later in the proof of Theorem \[t:coderivation\]. However, it does not hold in ${\hat{\mathcal F}}.$ Hence the terms $\hat G({e}v)$ and $\hat G(u {e})$ need to be handled with care.

Figure \[f:val\] illustrates the maps ${\hat \Delta}$ and $\hat G$. The map $\hat H$ is the analogue of the map $\hat G$ for the cubical lattice. This will be explained in Appendix \[s:cube\].

More definitions
----------------

Apart from ${\hat \Delta}$ and $\hat G$, we define a third map $\mu \colon {\hat{\mathcal F}}{\otimes}{\hat{\mathcal F}}\rightarrow {\hat{\mathcal F}}$ of degree $2$ by $\mu({e}{\otimes}{e}) = 2, \: \mu({e}{\otimes}v) 