{e}^*)$. $$\begin{array}{r c l}
\beta(u \centerdot v) 
    & = &
           \beta(S(u \centerdot v)) \\
    & = &
           \beta(S(u) \centerdot v) + \beta(u \centerdot S(v)) \\
    & = &
           \beta(S(u) \centerdot v^*) + \beta(u \centerdot S(v^*)) \\
    & = &
           \beta(S(u \centerdot v^*)) \\
    & = &
           \beta(u \centerdot v^*).
\end{array}$$ We made use of the induction hypothesis in the third step.

The second statement follows from the first by the chain of inequalities shown below. $$\beta(u \centerdot v) = \beta((u \centerdot v)^*) = 
        \beta(v^* \centerdot u^*) = \beta(v \centerdot u).$$

An explicit description of the maps $\centerdot, S$ and $\mu^*$ {#ss:explicit}
---------------------------------------------------------------

We obtained the maps $\centerdot, S$ and $\mu^*$ by taking duals of certain other maps. In this section, we go through the duality grind to give explicit formulas for these maps. By way of justification, we give some applications in Section \[ss:ssa\].

\[l:bullet\] The $\centerdot$ product on ${\hat{\mathcal F}}$ is determined by the initial conditions $$1 \centerdot 1 = 2 {\bf c}, \: \:
1 \centerdot {\bf c} = {\bf c} \centerdot 1 = {\bf d} + 2 {\bf c}^{2}, \: \:
1 \centerdot {\bf d} = 2 {\bf c} {\bf d}, \: \:
{\bf d} \centerdot 1 