is that the extended map ${\hat G}$ is a coderivation on ${\hat{\mathcal F}}$. The connection with the Boolean lattice is provided by the identity ${\hat G}(\Psi(B_n)) = \Psi(B_{n+1})$. The map $\hat G$ is our main tool for answering the questions in Section \[ss:questions\].

In Section \[s:duality\], we dualise the maps ${\hat \Delta}$ and $\hat G$ to get respectively a product (denoted by $\centerdot$) on ${\hat{\mathcal F}}$ and a derivation $S$, with the property $S(v) = S(\beta(v))$. We write down explicit formulas for the product $\centerdot$ and the derivation $S$. By way of motivation, we provide quick applications to the Boolean lattice. The explicit product $\centerdot$ has also appeared independently in the work of Stenson and Reading; see in particular  and  . The preprints are available on their respective homepages. I thank Ehrenborg for pointing these references to me.

In Section \[s:ln\], we restate all earlier results in an alternate notation for [[[[**c**]{}]{}[[**d**]{}]{}]{}]{}-monomials that involves lists. This notation is quite natural and easy to work with. In Sections \[s:unimodal\], \[s:maxima\] and \[s:balance\], we use the tools developed in earlier sections to target two specific problems, namely those raised in items (3) and (4) in Section \[ss:questions\]. Throughout these sections, we work 