Cubical convex ear decompositions

TL;DR

This paper develops a framework for constructing convex ear decompositions of posets using CL- and EL-labelings, demonstrating their existence for specific lattices and closure properties under products, with implications for polytopal complexes.

Contribution

It axiomatizes conditions for convex ear decompositions via CL- and EL-ceds, constructs new labelings for specific lattices, and proves that these decompositions are preserved under certain poset products.

Findings

Constructed EL-labelings for d-divisible partition lattice and coset lattice.

Established convex ear decompositions formed by face lattices of hypercubes.

Proved that products of posets with convex ear decompositions also admit such 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.