Generalized recursive atom ordering and equivalence to CL-shellability

TL;DR

This paper generalizes recursive atom ordering for posets, proving its equivalence to CL-shellability and CC-shellability with self-consistency, and applies it to electrical network face posets.

Contribution

It introduces a generalized recursive atom ordering, establishing its equivalence to CL-shellability and CC-shellability with self-consistency.

Findings

Generalized recursive atom ordering is equivalent to CL-shellability.

Posets of electrical networks are dual CL-shellable.

Self-consistent CC-shellability characterizes CL-shellability.

Abstract

Bj\"orner and Wachs introduced CL-shellability as a technique for studying the topological structure of order complexes of partially ordered sets (posets). 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.