a particular basis for ${{\hat{\mathcal F}}}$ and are trying to study the [**cd**]{}-index in this basis.

Dualise the maps ${\hat \Delta}, \hat G$ and $\mu$ defined in the previous section to get the corresponding dual maps ${\hat \Delta}^*, \hat G^*$ and $\mu^*$. The map ${\hat \Delta}^*$ is the convolution product on ${\hat{\mathcal F}}^*$, which was first introduced by Kalai . The algebra $({\hat{\mathcal F}}^*,{\hat \Delta}^*)$ can be identified with the algebra $A_{\varepsilon}$ studied by Billera and Liu .

Now using the identification of ${\hat{\mathcal F}}^*$ with ${\hat{\mathcal F}}$ explained above, we transfer these maps back to ${\hat{\mathcal F}}$ and obtain three maps, $$\centerdot \colon {\hat{\mathcal F}}{\otimes}{\hat{\mathcal F}}\rightarrow {\hat{\mathcal F}}, \
S \colon {\hat{\mathcal F}}\rightarrow {\hat{\mathcal F}}\ \text{and} \
\mu^* \colon {\hat{\mathcal F}}\rightarrow {\hat{\mathcal F}}{\otimes}{\hat{\mathcal F}}.$$ We note that these maps have degrees $1$, $-1$ and $-2$ respectively. In other words, $$\centerdot \colon {{\mathcal F}}_i {\otimes}{{\mathcal F}}_j \rightarrow {{\mathcal F}}_{i+j+1}, \
S \colon {{\mathcal F}}_{i+1} \rightarrow {{\mathcal F}}_i \ \text{and} \
\mu^* \colon {{\mathcal F}}_n \rightarrow \bigoplus\limits_{i \geq -1} {{\mathcal F}}_i {\otimes}{{\mathcal F}}_{n-i-2}.$$ The definitions of these maps can be made very explicit, see Section \[ss:explicit\]. But before doing that, we will present the dual versions of the results of the 