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arxiv: 2605.03913 · v1 · submitted 2026-05-05 · 🧮 math.CO

Lattice characterization of cyclic interval hypergraphic posets

F\'elix G\'elinas , Yirong Yang This is my paper

Pith reviewed 2026-05-07 15:14 UTC · model grok-4.3

classification 🧮 math.CO MSC 05C6506A07
keywords cyclic interval hypergraphshypergraphic posetshypergraphic polytopeslattice characterizationinterval hypergraphsposet latticescombinatorial polytopes
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The pith

Cyclic interval hypergraphs have hypergraphic posets that form lattices precisely when their edges satisfy explicit combinatorial conditions.

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

The paper determines exactly when the poset obtained by orienting the 1-skeleton of a hypergraphic polytope becomes a lattice, restricted to the case of cyclic interval hypergraphs. This class includes ordinary interval hypergraphs, whose lattice property was already characterized, and the complete cyclic interval hypergraph. The result matters because hypergraphic posets encompass familiar objects such as the weak order on permutations and the Tamari lattice, so the characterization draws a precise boundary between those that are lattices and those that are not. The conditions can be read directly from the hypergraph's edge set.

Core claim

For a cyclic interval hypergraph H the hypergraphic poset P_H, obtained by orienting the 1-skeleton of the hypergraphic polytope via a generic linear functional, is a lattice if and only if H satisfies a specific set of conditions on the overlaps and cyclic arrangements of its hyperedges. This supplies the complete characterization promised in the title and recovers both the earlier result for interval hypergraphs and the result for the complete cyclic interval hypergraph as special cases.

What carries the argument

The hypergraphic poset P_H, defined as the orientation of the 1-skeleton of the hypergraphic polytope Delta_H by a generic linear functional; it encodes the combinatorial order whose lattice property is characterized for the cyclic interval case.

If this is right

  • The known lattice criteria for interval hypergraphs appear as the special case when the cyclic structure collapses to a linear one.
  • The complete cyclic interval hypergraph yields a lattice, matching the prior result of Adenbaum et al.
  • Special instances recover the Tamari lattice and the weak order as lattices.
  • The lattice property of P_H can be decided by direct inspection of the cyclic intervals of H without constructing the full poset.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same overlap conditions may supply a template for lattice characterizations in larger families of hypergraphs.
  • The result links hypergraphic posets to other well-known combinatorial lattices that arise from polytopes.
  • Small cyclic interval hypergraphs can be enumerated and checked computationally to map the boundary of the lattice condition.

Load-bearing premise

The hypergraph must be cyclic interval and the orientation must be induced by a generic linear functional on the hypergraphic polytope.

What would settle it

A cyclic interval hypergraph that violates the stated edge-overlap conditions yet still produces a lattice under generic orientation, or one that meets the conditions yet fails to be a lattice.

Figures

Figures reproduced from arXiv: 2605.03913 by F\'elix G\'elinas, Yirong Yang.

Figure 1
Figure 1. Figure 1: The cyclic interval hypergraph H = {1256, 123, 23456} on the left and the orientation A given by the source sequence SA = (6, 2, 4) on the right. Note that this orientation is acyclic view at source ↗
Figure 2
Figure 2. Figure 2: The hypergraph H = {12, 34, 124, 134} on the left consist of a hugging quadruple without a fix. The hypergraph H = {12, 23, 34, 124, 134} on the right consist of a hugging quadruple and a fix. Definition 3.5. A hypergraph H is closed under intersection if for any hyperedges H, H′ ∈ H such that H ∩ H′ ̸= ∅, their intersection H ∩ H′ is a hyperedge in H view at source ↗
Figure 3
Figure 3. Figure 3: The weak order on 4 elements. Assuming n = 4, the colors represent parts of equivalent classes used in the proof of Proposition 4.5. Since PH is a lattice, there exists an orientation M such that A, B ≤ M and M ≤ C, D. By Theorem 2.5, SM is one of the following source sequences: (1, 1, n, n − 1, . . .), (1, 1, n, n, . . .), (1, 2, n, n − 1, . . .), (1, 2, n, n, . . .). However, these four orientations are … view at source ↗
Figure 4
Figure 4. Figure 4: The cyclic interval hypergraph H = {1236, 234, 1256, 34, 56}. On the left we can see the juxtaposition of the orientations A with sources represented with • and B with sources represented with • respectively given by the source sequence SA = (1, 3, 5, 3, 5) and SB = (2, 2, 2, 4, 6). On the right we can see the orientation given by the pseudo-join X of A and B given by the source sequence SX = (3, 4, 6, 4, … view at source ↗
Figure 5
Figure 5. Figure 5: The hypergraph H = {123, 345, 567, 178}. On the left, an counter￾clockwise minimal cycle S = (178, 123, 345, 567) given by the source sequence (1, 3, 5, 7). On the right, a clockwise minimal cycle S = (567, 345, 123, 178) given by the source sequence (3, 5, 7, 1). Case 1 (Counterclockwise). Suppose that for all Hi ∈ S we have that A(Hi) (resp. B(Hi)) and X(Hi) are both in H + i or H − i . Since X(Hi+1) ∈ H… view at source ↗
Figure 6
Figure 6. Figure 6: Four hypergraphs H with the juxtaposed acyclic orientations A and B with sources respectively represented with • and •. Those four hypergraphs include one instance of each case discussed in the proof of Lemma 5.8. Proposition 5.9. Let A and B be two acyclic orientations of H. If XAB is cyclic with a 2-cycle given by some H ∈ Hcyc and H′ ∈ Hreg, then there exists an interval D ⊆ [n] such that H|D has a hugg… view at source ↗
Figure 7
Figure 7. Figure 7: Example of hyperedges illustrating the information at the beginning of the proof of Lemma 5.7 We first show that there exists an interval D1 ⊆ [n] such that H|D1 has a hugging quadruple. Let D1 = [ℓ − 1, h˜ r]. • Let I 1 = H′ ∩ D1 . First, ℓ − 1, ℓ ∈ [X(H′ ), X(H)] ⊆ H′ . Secondly, h˜ r ∈/ H′ , other￾wise (H′ i , h′ i ) s i=1∥(H˜ i , h˜ i) r i=1 is an (H′ , H′ AB)-sequence. With H′ ∈ Hreg, we conclude that… view at source ↗
Figure 8
Figure 8. Figure 8: Example of hyperedges illustrating the information given by Claim 5.9.1 in the proof of Lemma 5.7. Let D2 = [ℓ − 1, min((L 1 ) +)] ⊂ D1 and consider H|D2 . We find a new hugging quadruple as follows. • Let I 2 = H′ ∩ D2 . As before, ℓ − 1, ℓ ∈[X(H′ ), X(H)] ⊆ H′ . Secondly, min((L 1 ) +) ∈/ H′ , otherwise h˜ 2 ∈ [X(H′ ), min((L 1 ) +)] ⊆ H′ , causing at least one of A and B to be cyclic. Therefore, togethe… view at source ↗
Figure 9
Figure 9. Figure 9: Examples of the first case discussed in the proof of Proposition 5.11. On the left, an instance of the first subcase of Case 1 and on the right, an instance of the second subcase of Case 1. For both scenarios, let H∩ = H∩|D ∈ H and observe that [X′ (H), H¯AB] ⊆ H∩ ⊆ H ∩ H¯ . The acyclicity of A and B implies that (H∩)AB ≥ (H¯ )AB. Since X′ > A, B, we obtain the following chain of inequalities view at source ↗
read the original abstract

Hypergraphic polytopes $\Delta_{\mathbb{H}}$ arise as Minkowski sums of simplices indexed by the hyperedges of a hypergraph $\mathbb{H}$. Orienting the $1$-skeleton of such a polytope by a certain generic linear functional gives rise to the hypergraphic poset $P_{\mathbb{H}}$. Hypergraphic posets include the weak order for the permutahedron and the Tamari lattice for the associahedron. This motivates the problem of determining when $P_{\mathbb{H}}$ is a lattice. In this paper, we give a complete lattice characterization for cyclic interval hypergraphs, extending the result of Bergeron and Pilaud for interval hypergraphs, and the result of Adenbaum et al. for the complete cyclic interval hypergraph.

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.

Referee Report

1 major / 2 minor

Summary. The paper gives a complete lattice characterization for hypergraphic posets P_H arising from cyclic interval hypergraphs H, obtained by orienting the 1-skeleton of the hypergraphic polytope Δ_H via a generic linear functional. This extends the Bergeron-Pilaud characterization for interval hypergraphs and the Adenbaum et al. result for the complete cyclic interval case by supplying necessary and sufficient conditions on the hypergraph structure.

Significance. If the if-and-only-if characterization holds, the result unifies several known lattices (weak order, Tamari lattice) under a single combinatorial condition on cyclic interval hypergraphs and supplies a precise criterion for when the oriented hypergraphic poset is a lattice. The argument proceeds by defining the relevant poset operations, verifying the lattice axioms from the cyclic interval property, and confirming the characterization is exhaustive within the stated class.

major comments (1)
  1. [§3] §3 (or the main theorem statement): the necessity direction relies on the cyclic interval property to ensure the join and meet operations are well-defined and closed; the manuscript should explicitly exhibit a counter-example hypergraph that is not cyclic interval but satisfies the stated combinatorial condition, to confirm the condition is not strictly stronger than the hypothesis.
minor comments (2)
  1. [Abstract] The abstract and introduction should clarify whether the generic linear functional is required only for the orientation or also for the lattice property itself; the current wording leaves open whether the characterization is independent of the choice of functional.
  2. [§2] Notation for the hypergraphic polytope Δ_H and the poset P_H is introduced without a forward reference to the precise definition of the covering relations; adding a short diagram or explicit covering-relation formula in §2 would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the significance, and recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (or the main theorem statement): the necessity direction relies on the cyclic interval property to ensure the join and meet operations are well-defined and closed; the manuscript should explicitly exhibit a counter-example hypergraph that is not cyclic interval but satisfies the stated combinatorial condition, to confirm the condition is not strictly stronger than the hypothesis.

    Authors: We agree that an explicit counterexample would help clarify that the cyclic interval hypothesis is essential and that our combinatorial condition is not strictly stronger. In the revised manuscript we will add, in §3 immediately after the statement of the main theorem, a concrete hypergraph H that satisfies the combinatorial condition but is not cyclic interval. For this H we will verify directly that the join operation on P_H fails to be closed (hence P_H is not a lattice). This example will illustrate that the cyclic interval property cannot be omitted from the necessity direction. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper defines hypergraphic posets from Minkowski sums of simplices and orients them via generic linear functionals, then verifies lattice axioms directly from the cyclic interval hypergraph property to obtain an if-and-only-if characterization. This extends Bergeron-Pilaud and Adenbaum et al. without reducing any central claim to a self-definition, fitted input renamed as prediction, or load-bearing self-citation chain. No ansatz is smuggled via citation, and no uniqueness theorem is imported from the authors' own prior work. The argument remains independent of the target result and is externally falsifiable via the stated hypergraph class.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The central claim rests on the standard definitions of hypergraphic polytopes and the generic linear functional orientation; no free parameters, invented entities, or ad-hoc axioms are visible.

axioms (2)
  • domain assumption Hypergraphic polytopes are Minkowski sums of simplices indexed by hyperedges of H.
    Invoked in the first sentence of the abstract as the source of the 1-skeleton that is oriented.
  • domain assumption A generic linear functional orients the 1-skeleton to produce the hypergraphic poset P_H.
    Stated as the construction that yields the poset whose lattice property is characterized.

pith-pipeline@v0.9.0 · 5420 in / 1253 out tokens · 39575 ms · 2026-05-07T15:14:55.197651+00:00 · methodology

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