# Cubical convex ear decompositions

Russ Woodroofe

Arxiv ID: 0709.2793•Last updated: 7/8/2020

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.

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