Björner and Wachs introduced CL-shellability as a technique for studying the topological structure of order complexes of partially ordered sets. They also introduced the notion of recursive atom ordering, and they proved that a finite bounded poset is CL-shellable if and only if it admits a recursive atom ordering.

In this paper, a generalization of the notion of recursive atom ordering is introduced. A finite bounded poset is proven to admit such a generalized recursive atom ordering if and only if it admits a traditional recursive atom ordering. This is also proven equivalent to admitting a CC-shelling (a type of shelling introduced by Kozlov) with a further property called self-consistency. Thus, CL-shellability is proven equivalent to self-consistent CC-shellability. As an application, the uncrossing posets, namely the face posets for stratified spaces of planar electrical networks, are proven to be dual CL-shellable.

Mathematics Subject Classifications: 05E45, 06A07

Keywords: Poset topology, lexicographic shellability, EC-shellability, recursive atom ordering, uncrossing order