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A Non-Multiplicable Upho Poset Constructed from the Petersen Graph

T0 review · 1 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read A finitary upho poset built from walks on the Petersen graph admits no compatible monoid structure, disproving the conjecture that all finitary upho posets are multiplicable.

desk verdict This gives a concrete counterexample to the Fu-Peng-Zhang conjecture via the Petersen graph, but the step that turns multiplicability into a regular subgroup of Aut(G) is the one that needs the closest look. read the letter →

arxiv 2606.17549 v3 pith:L4FFMC3I submitted 2026-06-16 math.CO

Ryunosuke Matsuoka This is my paper · ORCID
classification math.CO
keywords uphoposetmultiplicablePetersengraphCayleyfinitaryvertex-transitivemonoidstructureautomorphismgroup
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper constructs, for any vertex-transitive graph G, a finitary upho poset P(G,v0) whose elements are finite walks starting at a fixed vertex v0 and whose order is by extension. When G is the Petersen graph, multiplicability of this poset would require the automorphism group of G to contain a regular subgroup. That would make G a Cayley graph, contradicting the known fact that the Petersen graph is not Cayley. The same construction applied to the line graph of the Petersen graph does produce a multiplicable poset, showing that non-Cayleyness alone does not decide multiplicability.

What carries the argument

The poset P(G,v0) whose elements are finite walks on G that begin at a distinguished vertex v0, ordered by one walk being an initial segment of another.

What would settle it

An explicit monoid multiplication on the walks of the Petersen-graph poset whose left-divisibility order recovers the poset order would falsify the non-multiplicability claim.

Watch

Extended reading notes

Core claim

For every vertex-transitive graph G the poset P(G,v0) of finite walks beginning at a fixed vertex v0 is always finitary and upho. Multiplicability of P(G,v0) for the Petersen graph would force Aut(G) to contain a regular subgroup and hence force G itself to be Cayley, which it is not. Therefore P(G,v0) supplies a non-multiplicable finitary upho poset. The analogous poset for the line graph of the Petersen graph is multiplicable.

Load-bearing premise

Multiplicability of P(G,v0) necessarily forces the automorphism group of G to contain a regular subgroup.

Editorial extensions

If this is right

  • Not every finitary upho poset admits a left-cancellative invertible-free monoid whose left-divisibility order matches the poset order.
  • Multiplicability of the constructed poset is independent of whether the underlying graph is Cayley.
  • There exist both multiplicable and non-multiplicable finitary upho posets arising from the same construction on non-Cayley graphs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same walk-based construction applied to other known non-Cayley vertex-transitive graphs is likely to produce further non-multiplicable examples.
  • The presence or absence of a regular subgroup in Aut(G) may be the precise graph-theoretic property that decides multiplicability of P(G,v0).
  • One could search for a different monoid structure on the Petersen poset that is not induced by a regular subgroup action, though the paper's implication shows no such structure can exist.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 2 minor

Summary. The manuscript constructs, for any vertex-transitive graph G and fixed vertex v0, a finitary upper homogeneous (upho) poset P(G,v0) whose elements are walks starting at v0, ordered by extension. It proves that multiplicability of this poset (existence of a compatible left-cancellative monoid whose left-divisibility order recovers the poset) would force Aut(G) to contain a regular subgroup, hence G would be a Cayley graph. Specializing to the Petersen graph, which is known to be non-Cayley, yields a counterexample to the Fu–Peng–Zhang conjecture that every finitary upho poset is multiplicable. The paper also exhibits a multiplicable poset arising from the line graph of the Petersen graph.

Significance. If the central implication holds, the result supplies the first explicit counterexample to the conjecture and supplies a general construction that ties the algebraic structure of upho posets to the existence of regular automorphism subgroups. The contrast between the Petersen graph and its line graph clarifies that non-Cayleyness of the underlying graph is not by itself decisive for non-multiplicability.

major comments (1)
  1. [proof of the main theorem (the implication from multiplicability to regular subgroup)] The load-bearing step is the claim that any left-cancellative monoid structure on P(G,v0) whose left-divisibility order coincides with the given poset order necessarily induces a regular subgroup of Aut(G) via the natural action on walks. The abstract states this implication, but the derivation—how monoid multiplication corresponds to walk concatenation while preserving the vertex-transitive action—must be spelled out with explicit maps and verification that the resulting action is free and transitive; without this, the contradiction with the known non-Cayley property of the Petersen graph does not go through.
minor comments (2)
  1. [Section 2 (construction)] The notation P(G,v0) is introduced without an explicit recursive definition of the order relation on walks; a displayed equation or short paragraph clarifying the covering relations would improve readability.
  2. [final section] The line-graph example is asserted to be multiplicable, but the explicit monoid operation is not exhibited; adding a brief description or reference to the construction used would make the contrast with the Petersen case self-contained.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, accurate summary of the results, and constructive major comment. We address the point below and will incorporate the requested clarifications in a revised version.

read point-by-point responses

  1. Referee: [proof of the main theorem (the implication from multiplicability to regular subgroup)] The load-bearing step is the claim that any left-cancellative monoid structure on P(G,v0) whose left-divisibility order coincides with the given poset order necessarily induces a regular subgroup of Aut(G) via the natural action on walks. The abstract states this implication, but the derivation—how monoid multiplication corresponds to walk concatenation while preserving the vertex-transitive action—must be spelled out with explicit maps and verification that the resulting action is free and transitive; without this, the contradiction with the known non-Cayley property of the Petersen graph does not go through.

    Authors: We agree that the current exposition of the implication (multiplicability implies existence of a regular subgroup of Aut(G)) would benefit from additional explicit detail. In the revised manuscript we will expand the proof of the main theorem by introducing explicit maps: the monoid multiplication on P(G,v0) is defined via concatenation of walks when the terminal vertex of the first matches the initial vertex of the second; the induced action of the monoid on the vertex set of G is obtained by sending each walk to the vertex it reaches from v0. We will add a lemma verifying that left-cancellativity together with the upho property implies the action is free, while vertex-transitivity of G together with the finitary upho structure ensures transitivity. The resulting image is therefore a regular subgroup, yielding the desired contradiction for the Petersen graph. These additions will be placed in the section containing the main theorem and will not alter any statements or results. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation is self-contained against external facts.

full rationale

The paper defines the poset construction P(G,v0) from walks on any vertex-transitive graph G independently of the multiplicability property. It then derives (within the paper) that multiplicability would imply a regular subgroup of Aut(G), hence that G is Cayley. This is contradicted by the externally known non-Cayley status of the Petersen graph. No equations reduce a prediction to a fitted input by construction, no self-citations are load-bearing for the central claim, and no ansatz or renaming is smuggled in. The argument stands on the independent construction plus a standard external theorem, so the derivation chain does not collapse to its own inputs.

Assumptions & free parameters 0 free parameters · 2 assumptions · 1 invented entities

The claim rests on a newly introduced construction of posets from graph walks together with standard facts from graph theory and order theory; no numerical parameters are fitted.

assumptions (2)
  • domain assumption The walk-based construction on any vertex-transitive graph yields a finitary upho poset
    Invoked for the general construction P(G,v0) before specializing to the Petersen graph
  • standard math Standard facts about automorphism groups, regular subgroups, and Cayley graphs
    Used to derive the contradiction from the assumption of multiplicability
invented entities (1)
  • P(G,v0) the finitary upho poset constructed from walks
    purpose: To serve as a concrete counterexample to the conjecture
    Newly defined object whose multiplicability properties are analyzed

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Cite this review

Pith. "Pith review of A Non-Multiplicable Upho Poset Constructed from the Petersen Graph." pith.science (2026). https://pith.science/paper/L4FFMC3I

@misc{pith2026260617549,
  author       = {Pith},
  title        = {Pith review of: A Non-Multiplicable Upho Poset Constructed from the Petersen Graph},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L4FFMC3I}},
  note         = {Machine review of arXiv:2606.17549}
}
abstract

An upper homogeneous (upho) poset is a poset whose every principal filter is isomorphic to the whole poset. Fu--Peng--Zhang conjectured that every finitary upho poset admits a compatible left-cancellative, invertible-free monoid structure whose left-divisibility order coincides with the given order. We disprove this conjecture. For every vertex-transitive graph $G$, we construct a finitary upho poset $P(G,v_0)$ from walks starting at a fixed vertex $v_0$. Applying this construction to the Petersen graph, we show that multiplicability of $P(G,v_0)$ would force the automorphism group of $G$ to contain a regular subgroup. This would imply that $G$ is a Cayley graph, contradicting the fact that the Petersen graph is not Cayley. Hence $P(G,v_0)$ is a non-multiplicable finitary upho poset. We also show that the analogous poset associated with the line graph of the Petersen graph is multiplicable, demonstrating that non-Cayleyness of the underlying graph alone does not determine multiplicability.

Figures

Figures reproduced from arXiv: 2606.17549 by the authors.

Figure 1
Figure 1. The Petersen graph 2.4 Cayley Graphs Definition 2.11. Let H be a finite group and S ⊂ H be a generating set of H such that e /∈ S, S = S −1 . The graph Cay(H, S) = (V, E) is defined as V := H, E := {{h, hs} | h ∈ H, s ∈ S}. Then Cay(H, S) (or graphs isomorphic to it) is called the Cayley graph of the group H with respect to the generating set S. Since S is a generating set, Cay(H, S) is connected. It is also vertex-… view at source ↗
Figure 2
Figure 2. The Hasse diagram of P(K3, v0) Remark 3.3. We can easily verify that for any graph G and (v, n),(w, m) ∈ P(G, v0), the covering relation ⋖ is characterized by (v, n) ⋖ (w, m) ⇔ m = n + 1 and v ∼ w. Proposition 3.4. If G is a vertex-transitive graph, then P is a finite-type N-graded upho poset. Proof. Take any a = (u, n) ∈ P. Since G is vertex-transitive, there exists σ ∈ Aut(G) satisfying σ(v0) = u. Then, the map ψa… view at source ↗

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Reference graph

Works this paper leans on

13 extracted references · 1 canonical work pages

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