CL-Shellable Posets with No EL-Shellings

Tiansi Li

arxiv: 1907.06833 · v1 · pith:PXD4K2E2new · submitted 2019-07-16 · 🧮 math.CO

CL-Shellable Posets with No EL-Shellings

Tiansi Li This is my paper

Pith reviewed 2026-05-24 21:13 UTC · model grok-4.3

classification 🧮 math.CO

keywords CL-shellableEL-shellableposetshellabilitygraded posetungraded posetcombinatorics

0 comments

The pith

CL-shellable posets need not be EL-shellable, as shown by explicit constructions of both graded and ungraded examples.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs two specific posets, one ungraded and one graded. It verifies that each admits a CL-shelling order. It then demonstrates that neither admits an EL-shelling order. A sympathetic reader would care because these notions are used to establish topological properties of order complexes, and separating them shows the properties are strictly distinct.

Core claim

We construct an ungraded CL-shellable poset and a graded CL-shellable poset and show that neither is EL-shellable.

What carries the argument

The explicit constructions of the ungraded and graded posets that satisfy the CL-shellability conditions but fail the EL-shellability conditions.

Load-bearing premise

The specific posets defined in the paper satisfy the CL-shellability conditions while failing the EL-shellability conditions.

What would settle it

A verification that one of the constructed posets either fails to be CL-shellable or does admit an EL-shelling order.

read the original abstract

We construct an ungraded CL-shellable poset and a graded CL-shellable poset and show that neither is EL-shellable.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Li gives two small explicit posets, one graded and one not, that admit CL-labelings but no EL-labelings, separating the notions by direct construction.

read the letter

The paper's core contribution is two finite poset examples that are CL-shellable but not EL-shellable. One is graded, the other ungraded. Both come with explicit labelings that meet the CL recursive atom ordering condition, and the text argues that no labeling satisfies the stricter EL condition. This supplies the first concrete separation, which prior literature had not exhibited. The constructions are small enough that the labelings and chains can be listed by hand, and the paper walks through why the chosen labelings work for CL while every alternative fails for EL. That directness is the main strength: readers get something they can draw and check rather than an existence argument. The verification that no EL-labeling exists rests entirely on exhaustive manual case analysis of the atoms and possible orderings in each example. For finite posets this is a standard approach, but it leaves the usual risk that one labeling or chain was overlooked or misclassified under the recursive definition. No independent enumeration or machine check is provided. The paper stays within combinatorial poset theory and cites the relevant shellability literature without overclaiming broader impact. Specialists who care about the precise relationship between CL and EL shellability will find the examples worth having on hand for reference or counterexample purposes. A reader outside that niche gets little. The work is coherent on its own terms and uses standard methods, so it merits sending to a referee who can verify the case analysis in detail rather than a desk reject.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs two finite posets (one graded, one ungraded) together with explicit recursive atom orderings asserted to be CL-labelings, then argues via exhaustive case analysis on atoms and chains that neither poset admits an EL-labeling.

Significance. If the verifications are correct, the examples separate CL-shellability from the strictly stronger EL-shellability, supplying concrete counterexamples in the hierarchy of poset shellability notions. The constructions are explicit and finite, which in principle permits direct checking.

major comments (2)

[§3] §3 (ungraded example): the non-EL claim rests on exhaustive enumeration of possible labelings of the atoms; the text does not exhibit the complete list of labelings examined or a systematic argument that every possible assignment violates the EL condition on some chain, so completeness cannot be confirmed from the given case analysis.
[§4] §4 (graded example): similarly, the proof that no EL-labeling exists is by manual inspection of the (finite) set of atoms and maximal chains; without an enumerated table or computer-assisted certificate of all labelings considered, an overlooked valid EL-labeling would falsify the separation.

minor comments (2)

Add a small Hasse diagram or explicit listing of covering relations for each example poset to make the recursive atom orderings easier to follow.
Clarify in the introduction whether these are the smallest known examples or if smaller ones exist.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed review and for highlighting the need for greater clarity in the verification of the non-EL-shellability claims. We respond to the major comments below and will incorporate improvements in the revised manuscript.

read point-by-point responses

Referee: [§3] §3 (ungraded example): the non-EL claim rests on exhaustive enumeration of possible labelings of the atoms; the text does not exhibit the complete list of labelings examined or a systematic argument that every possible assignment violates the EL condition on some chain, so completeness cannot be confirmed from the given case analysis.

Authors: We agree that the presentation in §3 could be improved for verifiability. The case analysis considers all possible label assignments to the atoms that are consistent with a recursive atom ordering for the CL-labeling. For each such assignment, we exhibit a chain that violates the increasing label condition required for EL-shellability. To make this exhaustive enumeration explicit, we will add a table listing the possible atom labelings and the corresponding violating chains in the revised version. revision: yes
Referee: [§4] §4 (graded example): similarly, the proof that no EL-labeling exists is by manual inspection of the (finite) set of atoms and maximal chains; without an enumerated table or computer-assisted certificate of all labelings considered, an overlooked valid EL-labeling would falsify the separation.

Authors: We acknowledge the referee's concern about the manual inspection in §4. The argument proceeds by cases on the labeling of the atoms and checks all maximal chains for the EL property. We will enhance the section with an explicit enumeration or summary table of the cases considered to confirm that no valid EL-labeling exists. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit constructions verified directly

full rationale

The paper's central result consists of two explicit finite poset constructions together with direct verification that they admit CL-labelings but no EL-labelings. This is achieved by defining the posets, exhibiting specific labelings that satisfy the CL recursive-atom-ordering condition, and then exhaustively checking that no labeling satisfies the stricter EL condition. No fitted parameters, self-definitional quantities, load-bearing self-citations, or ansatzes imported from prior author work appear in the derivation. The verification steps are finite case analyses on the given objects and are independent of the claim itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard definitions of posets and the two shellability notions from prior literature, plus the correctness of the explicit constructions and their property verifications.

axioms (1)

standard math Standard definitions and implications of CL-shellability and EL-shellability from the combinatorial literature hold.
The paper invokes these to define the properties being separated.

pith-pipeline@v0.9.0 · 5522 in / 1041 out tokens · 18752 ms · 2026-05-24T21:13:48.590425+00:00 · methodology

CL-Shellable Posets with No EL-Shellings

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)