Equivalences and Distinctions in Lexicographic Shellability of Posets

TL;DR

The paper investigates how different notions of lexicographic shellability relate, introducing RFAS and LCRFAS as new recursive tools. It proves TCL-shellability is equivalent to CC-shellability and shows RC-type frameworks can characterize CC-shellability via RFAS, with LC ensuring a compatible labeling exists. It provides small counterexamples of posets that are CC-shellable but not CL-shellable, and establishes a tight equivalence between TCL-shellability and CC-shellability while clarifying when RFAS suffices. The work also outlines open questions about the limits of RFAS/LCRFAS and their connections to EC/EL frameworks, guiding future research in the topology of posets.

Abstract

We present two perhaps surprisingly small posets, one graded and one non-graded, that are CC-shellable in the sense of Kozlov and TCL-shellable in the sense of Hersh, but not CL-shellable in the sense of Björner and Wachs. In the spirit of Björner and Wachs' recursive atom orderings (RAO) and Hersh and Stadnyk's generalized recursive atom orderings (GRAO), we also introduce a notion called recursive first atom sets (RFAS). An RFAS is a set of conditions on the atoms of each interval in a finite bounded poset $P$ that are necessary for CC-shellability of $P$ and sufficient for shellability of $P$. We also prove that under an extra condition, $P$ is CC-shellable if and only if it admits an RFAS, in the same way that RAOs provide a reformulation of CL-shellability.

Figures (9)

• Figure 1: Relationships between different notions of lexicographic shellability and where the relationships are proved in this paper and elsewhere in the literature.
• Figure 2: A poset $P$ with an EL-labeling (left), a CL-labeling that is not an EL-labeling (middle), and a CC-labeling that is neither an EL-labeling nor a CL-labeling (right), as described in Example \ref{['ex: labelings']}.
• Figure 3: A graded TCL-shellable poset that is not CL-shellable. A TCL-labeling of the poset is given by the bold labels. A dual EL-labeling of the poset is given by the labels in parentheses.
• Figure 4: An nongraded TCL-shellable poset that is not CL-shellable. A TCL-labeling of the poset is shown on the left. A dual EL-labeling of the poset is shown on the right.
• Figure 5: The order complex of $(x,\hat{1})$.

Theorems & Definitions (55)

• Definition 2.1: bw
• Definition 2.2: kozlov
• Definition 2.3: hersh
• Remark 2.4
• Remark 2.5
• Example 2.6
• Definition 2.7: HerStad
• Theorem 3.1
• proof
• Remark 3.2