A signed analog of the Birkhoff transform

We construct a family of posets, called signed Birkhoff posets, that may be viewed as signed analogs of distributive lattices. Our posets are generally not lattices, but they are shown to posses many combinatorial properties corresponding to well-known properties of distributive lattices. They have the additional virtue of being face posets of regular cell decompositions of spheres. We relate the zeta polynomial of a signed Birkhoff poset to Stembridge's enriched order polynomial and give a combinatorial description the cd-index of a signed Birkhoff poset in terms of peak sets of linear extensions of an associated labeled poset. Our description is closely related to a result of Billera, Ehrenborg, and Readdy's expressing the cd-index of an oriented matroid in terms of the flag f-vector of the underlying geometric lattice.