with the list notation. Wherever convenient, we also state our results in terms of the original notation of monomials.

There are four appendices. In Appendix \[s:connection\], we give some connection between the [**ab**]{} and the [**cd**]{}-index and explain why one expects results about the [[[[**a**]{}]{}[[**b**]{}]{}]{}]{}-index to carry over to the [**cd**]{}-index. In Appendix \[s:recur\], we give a recursion for computing the ${{{\bf c}}{{\bf d}}}$-index of an Eulerian poset in terms of certain polynomial sequences. These may be of independent interest. Appendix \[s:cube\] deals with the cubical lattice, which is the face lattice of the cube. Just as the simplex is the simplest polytope, the cube is the simplest zonotope and is an object of interest in its own right. Usually, techniques that work for the Boolean lattice also work for the cubical lattice with minor modifications; see  . Following this general principle, we establish similar results for the [**cd**]{}-index of the cubical lattice. In Appendix \[s:fun\], we show that ${\hat{\mathcal F}}$ is a free algebra with the $\centerdot$ product (Theorem \[t:free\]). We also show that the Dehn-Sommerville relations satisfied by the flag $f$-vector of an Eulerian poset are equivalent to certain simple identities that hold in ${\hat{\mathcal F}}$.

The polynomial 