Cubical convex ear decompositions

Abstract

We consider the problem of constructing a convex ear decomposition for a poset. The usual technique, first used by Nyman and Swartz, starts with a CL-labeling and uses this to shell the `ears' of the decomposition. We axiomatize the necessary conditions for this technique as a "CL-ced" or "EL-ced". We find an EL-ced of the d-divisible partition lattice, and a closely related convex ear decomposition of the coset lattice of a relatively complemented group. Along the way, we construct new EL-labelings of both lattices. The convex ear decompositions so constructed are formed by face lattices of hypercubes. We then proceed to show that if two posets P_1 and P_2 have convex ear decompositions (CL-ceds), then their products P_1 \times P_2, P_1 \lrtimes P_2, and P_1 \urtimes P_2 also have convex ear decompositions (CL-ceds). An interesting special case is: if P_1 and P_2 have polytopal order complexes, then so do their products.