We study some properties of the { \bf cd}-index of the Boolean lattice. They are extremely similar to the properties of the { \ab}-index, or equivalently, the flag $h$-vector of the Boolean lattice and hence may be viewed as their { \bf cd}-analogues. We define a different algebra structure on the polynomial algebra $k < \cv, \dv>$ and give a derivation on this algebra. It is of significance for the Boolean lattice and forms our main tool. Using similar methods, we also prove some results for the { \bf cd}-index of the cubical lattice. We show that the Dehn-Sommerville relations for the flag $f$-vector of an Eulerian poset are equivalent to certain simple identities that exist in our algebra.
Categories: math.CO