  The flag vector contains all the face incidence data of a polytope, and in
the poset setting, the chain enumerative data. It is a classical result due to
Bayer and Klapper that for face lattices of polytopes, and more generally,
Eulerian graded posets, the flag vector can be written as a cd-index, a
non-commutative polynomial which removes all the linear redundancies among the
flag vector entries. This result holds for regular CW complexes.
  We relax the regularity condition to show the cd-index exists for Whitney
stratified manifolds by extending the notion of a graded poset to that of a
quasi-graded poset. This is a poset endowed with an order-preserving rank
function and a weighted zeta function. This allows us to generalize the
classical notion of Eulerianness, and obtain a cd-index in the quasi-graded
poset arena. We also extend the semi-suspension operation to that of embedding
a complex in the boundary of a higher dimensional ball and study the simplicial
shelling components.
