conditions $$\Delta(1) = 0, \: \: \Delta({\bf c}) = 2  (1 {\otimes}1), 
\: \: \Delta({\bf d}) = 1 {\otimes}{\bf c} + {\bf c} {\otimes}1$$ and the rule $\Delta(u  v) = \Delta(u)  v + u  \Delta(v)$ for $u,v \in {{\mathcal F}}.$

Under the coproduct ${\Delta}$, the vector space ${{\mathcal F}}$ is a coassociative coalgebra, but without a counit map. Further, the rule says that ${\Delta}$ is a derivation on ${{\mathcal F}}$ into the $({{\mathcal F}},{{\mathcal F}})$-bimodule $\mathcal F {\otimes}\mathcal F$. This makes ${{\mathcal F}}$ an infinitesimal bialgebra, also called a Newtonian coalgebra. This notion was first defined by Joni and Rota . For more recent work, see the papers of Aguiar . However, we will not use any facts about infinitesimal bialgebras.

The motivation for the definition of ${\Delta}$ is as follows. Consider the map $$\Psi : \{ \text{Eulerian posets} \} \rightarrow k \langle {{\bf c}}, {{\bf d}}\rangle,$$ which assigns to an Eulerian poset $P$ its ${{{\bf c}}{{\bf d}}}$-index $\Psi(P)$. The vector space spanned by all Eulerian posets is a coalgebra with the coproduct given by $$\Delta(P) = \sum_{{\hat{0}}< x < {\hat{1}}} [{\hat{0}},x] {\otimes}[x,{\hat{1}}].$$ And the map $\Psi$ is a morphism of coalgebras. In other words, the identity $$\label{e:coalg_morph}
\Delta(\Psi(P)) = 