Cubical convex ear decompositions (Q2380319)

Summary: We consider the problem of constructing a convex ear decomposition for a poset. The usual technique, introduced 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 finite 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 \(P_1\times P_2\), \(P_1\check\times P_2\), and \(P_1\hat\times P_2\) 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.