\subseteq 
\bigoplus_{i \geq -1} {\mathcal F}_i {\otimes}{\mathcal F}_{n-i-1}.$ Also let $\varepsilon \colon {\hat{\mathcal F}}\rightarrow k$ be given by the delta function $\delta_{e}$. It is easy to see that ${\hat{\mathcal F}}$ is a coalgebra, with ${\hat \Delta}$ as the coproduct and $\varepsilon$ as the counit. The process of passing from ${{\mathcal F}}$ to ${\hat{\mathcal F}}$ just described is the standard way of adding a counit to a coalgebra. For matters of notational convenience, we let ${e}v = v {e}= 0.$ With this convention, it is still true that ${\hat \Delta}(u v) = {\hat \Delta}(u) v + u {\hat \Delta}(v)$ for $u,v \in {\hat{\mathcal F}}$. Hence the extended object ${\hat{\mathcal F}}$ is also an infinitesimal bialgebra.

$$\begin{array}{c | c c c}
      & {\hat \Delta}& \hat G & \hat H \\ \hline
{e}& {e}{\otimes}{e}& 1 & \\
1     & {e}{\otimes}1 + 1 {\otimes}{e}& {{\bf c}}& {{\bf c}}\\
{{\bf c}}& 2 (1 {\otimes}1) + {{\bf c}}{\otimes}{e}+ {e}{\otimes}{{\bf c}}& {{\bf c}}^2 + {{\bf d}}& {{\bf c}}^2 + 2 {{\bf d}}\\
{{\bf d}}& 1 {\otimes}{{\bf c}}+ {{\bf c}}{\otimes}1 + {{\bf d}}{\otimes}{e}+ {e}{\otimes}{{\bf d}}& {{\bf c}}{{\bf d}}+ {{\bf d}}{{\bf c}}& {{\bf d}}+ 2 {{\bf d}}{{\bf c}}\end{array}$$

Let $\hat G \colon {\hat{\mathcal F}}\rightarrow {\hat{\mathcal F}}$ be the linear 