\sum_{{\hat{0}}< x < {\hat{1}}} \Psi([{\hat{0}},x]) {\otimes}\Psi([x,{\hat{1}}])$$ holds for any Eulerian poset $P$; see .

We will use this later to derive a basic result about the Boolean lattice; see Lemma \[l:easy\]. We will also return to it briefly in Appendix \[s:fun\], when we discuss the Dehn-Sommerville equations.

\[p:g\] There is a well-defined linear map $G \colon {{\mathcal F}}\rightarrow {{\mathcal F}}$ given by the initial conditions $$G(1) = 0, \: \: G({{\bf c}}) = {{\bf d}}, \: \: G({{\bf d}}) = {{\bf c}}{{\bf d}}$$ and the rule $G(u v) = G(u) v + u G(v)$, such that $$\Psi(B_{n+1}) = \Psi(B_n) {{\bf c}}+ G(\Psi(B_n)).$$

The importance of the map $G$ is that it gives an inductive way of computing $\Psi(B_n)$.

An extension of the basic setup {#ss:ext}
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Consider ${\hat{\mathcal F}}= k  {e}\oplus {{\mathcal F}}$, where ${e}$ is a formal symbol with degree $-1$. We write ${\hat{\mathcal F}}= \oplus_{n \geq -1} {\mathcal F}_n$, where ${{\mathcal F}}_{-1} = k  {e}$. Define a coproduct ${\hat \Delta}\colon {\hat{\mathcal F}}\rightarrow {\hat{\mathcal F}}{\otimes}{\hat{\mathcal F}}$ by $${\hat \Delta}({e}) = {e}{\otimes}{e}\quad \text{and} \quad
{\hat \Delta}(u) = {\Delta}(u) + {e}{\otimes}u + u {\otimes}{e},$$ for $u \in {{\mathcal F}}$. Observe that ${\hat \Delta}$ has degree $-1$, that is, ${\hat \Delta}({\mathcal F}_{n}) 