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arxiv: 2605.15319 · v1 · pith:QYGACBGOnew · submitted 2026-05-14 · 🧮 math.CO

Extended Abstract: Canonical join complex and cubical coordinates for all framing lattices

Jonah Berggren , Cl\'ement Chenevi\`ere This is my paper

Pith reviewed 2026-05-19 15:57 UTC · model grok-4.3

classification 🧮 math.CO
keywords framing latticescanonical join representationsjoin-irreducible elementsbrick cliquescubical coordinatesnoncrossing arc diagramsTamari latticeweak order
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The pith

Bricks and brick cliques give a uniform combinatorial model for canonical join representations in every framing lattice.

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

The paper defines bricks and brick cliques to represent join-irreducible elements and their canonical joins inside framing lattices, which include the Tamari lattice and weak orders as special cases. This model extends noncrossing arc diagrams and builds in a natural correspondence between join and meet representations together with duality under reflections of the underlying graph. A bijective proof supplies an explicit two-step reconstruction algorithm, while an auxiliary notion of cornered cliques produces cubical coordinates that generalize bracket vectors and support direct comparison of lattice elements.

Core claim

We define bricks and brick cliques as a combinatorial model for join-irreducible elements and canonical join representations in all framing lattices, generalizing noncrossing arc diagrams. Our model captures the natural bijection between join and meet canonical representations, as well as duality upon reflections of the framed graph. The proof is bijective, with an explicit reconstruction algorithm in two steps. Cornered cliques serve as an intermediate construction that yields cubical coordinates on a framing lattice.

What carries the argument

Bricks and brick cliques, which encode join-irreducible elements and canonical join representations while supporting a two-step bijective reconstruction.

If this is right

  • Every framing lattice element admits an explicit description by its set of bricks.
  • The two-step reconstruction algorithm computes the lattice element directly from any valid brick clique.
  • Reflection of the framed graph interchanges canonical join and meet representations in a uniform way.
  • Cubical coordinates obtained from cornered cliques give a direct comparison test between any two elements.
  • The construction applies without extra restrictions to framing lattices not previously treated by arc diagrams.

Where Pith is reading between the lines

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

  • Cubical coordinates may support new enumeration or sorting algorithms on these lattices.
  • The brick model could be tested for compatibility with other known lattice constructions that generalize the Tamari or weak orders.
  • If the reconstruction remains efficient, it might yield practical ways to compute meets and joins in large framing lattices.

Load-bearing premise

The newly defined bricks and brick cliques correctly capture the join-irreducible elements and canonical join representations for every framing lattice.

What would settle it

A single framing lattice in which some brick clique fails to reconstruct to the claimed lattice element via the two-step algorithm, or in which the claimed bijection between join and meet representations breaks.

Figures

Figures reproduced from arXiv: 2605.15319 by Cl\'ement Chenevi\`ere, Jonah Berggren.

Figure 1
Figure 1. Figure 1: Objects and bijections involved in this article. Left-right and up-down reflections [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The caracol graph of size 3. The corresponding framing lattice is isomorphic to [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The bricks on the left are incoherent, while those on the right are coherent. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: An example of the reconstruction algorithm. Top left is displayed the framed graph [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
read the original abstract

This document is an extended abstract for two articles in preparation. Recently, framing lattices were introduced to generalize many classical lattices such as the Tamari lattice and the weak order on the symmetric group. We define bricks and brick cliques as a combinatorial model for join-irreducible elements and canonical join representations in all framing lattices, generalizing noncrossing arc diagrams of (Reading, 2015) for the weak order on the symmetric group. Our model captures the natural bijection between join and meet canonical representations, as well as duality upon reflections of the framed graph. The proof is bijective, with an explicit reconstruction algorithm in two steps. A useful intermediate construction in our bijective proof is our new definition of cornered cliques. These enable us to define cubical coordinates on a framing lattice, generalizing bracket vectors for the Tamari lattice and providing an efficient comparison criterion.

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

2 major / 2 minor

Summary. This extended abstract for two articles in preparation introduces bricks and brick cliques as a combinatorial model for join-irreducible elements and canonical join representations in all framing lattices, generalizing noncrossing arc diagrams. It claims a bijective proof via an explicit two-step reconstruction algorithm that also captures the natural bijection between join and meet representations as well as duality under reflections of the framed graph. An intermediate construction of cornered cliques is used to define cubical coordinates on framing lattices, generalizing bracket vectors for the Tamari lattice and providing an efficient comparison criterion.

Significance. If the claimed bijection and uniform correctness hold for arbitrary framing lattices, the work would supply a new combinatorial language applicable beyond classical cases such as the Tamari lattice and weak order on the symmetric group, potentially enabling systematic study of join-irreducibles, duality, and coordinate systems across a broader family of posets.

major comments (2)
  1. [Abstract] Abstract: the central assertion that bricks and brick cliques correctly identify join-irreducible elements and canonical join representations for every framing lattice (including those outside previously studied families) is stated without any explicit definition of a brick, any verification that the construction recovers the lattice order, or any outline of the two-step reconstruction algorithm.
  2. [Abstract] Abstract: the uniformity claim—that the model works without additional restrictions on the framed graph—is load-bearing for the generalization beyond noncrossing arc diagrams, yet no supporting argument, lemma, or check for novel framings is supplied in the text.
minor comments (2)
  1. The citation to Reading (2015) for noncrossing arc diagrams should be expanded to a full bibliographic entry.
  2. The abstract refers to 'two articles in preparation'; it would be helpful to indicate whether the full proofs and definitions will appear in those articles or whether additional detail is expected in a revised version of this document.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our extended abstract and for highlighting points that clarify the scope of this short document. As this is an extended abstract for two full articles in preparation, space constraints limit the inclusion of complete definitions and proofs; we address each major comment below by indicating where the requested material appears in the forthcoming papers and what, if anything, can be added here.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central assertion that bricks and brick cliques correctly identify join-irreducible elements and canonical join representations for every framing lattice (including those outside previously studied families) is stated without any explicit definition of a brick, any verification that the construction recovers the lattice order, or any outline of the two-step reconstruction algorithm.

    Authors: We agree that the extended abstract does not contain the explicit definition of a brick or a detailed outline of the two-step reconstruction algorithm. These are supplied in the two articles in preparation: the first article defines bricks combinatorially as a generalization of noncrossing arcs and proves that brick cliques are in bijection with canonical join representations; the second article presents the explicit two-step reconstruction algorithm and verifies that it recovers the original lattice element, thereby confirming that the construction respects the lattice order. The extended abstract only summarizes the bijective strategy and key properties. We can add a one-paragraph high-level sketch of the reconstruction algorithm to a revised version of the extended abstract if the editor permits. revision: partial

  2. Referee: [Abstract] Abstract: the uniformity claim—that the model works without additional restrictions on the framed graph—is load-bearing for the generalization beyond noncrossing arc diagrams, yet no supporting argument, lemma, or check for novel framings is supplied in the text.

    Authors: The uniformity claim follows from the general definition of framing lattices on arbitrary framed graphs (as introduced in the foundational paper on framing lattices) and from the design of our bijective proof, which does not impose extra restrictions beyond those already present in the framed graph. The supporting lemmas establishing correctness for general framings, including explicit checks on framings outside the Tamari and weak-order cases, appear in the full articles. The extended abstract states the uniformity as a consequence of the bijective reconstruction and the reflection duality. We can insert a clarifying sentence in a revision stating that the proof applies to all framed graphs without further restrictions. revision: partial

Circularity Check

0 steps flagged

No circularity: new definitions with independent bijective proof

full rationale

The extended abstract introduces bricks and brick cliques via explicit new combinatorial definitions as a model for join-irreducibles and canonical joins in all framing lattices, generalizing Reading (2015) on specific cases. It asserts a bijective proof together with a two-step reconstruction algorithm and an intermediate cornered-clique construction for cubical coordinates. No equations, fitted parameters, or self-citations appear that would make any claimed result equivalent to its inputs by construction; the central claim rests on the (still-in-preparation) proof rather than on renaming or self-referential fitting. The derivation is therefore self-contained as a fresh combinatorial model.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 3 invented entities

The work relies on standard lattice-theoretic notions of join-irreducible elements and canonical join representations; the new objects are introduced without independent prior evidence.

axioms (1)
  • standard math Standard definitions and properties of join-irreducible elements and canonical join representations in lattices
    Invoked when stating that bricks model join-irreducible elements and brick cliques model canonical join representations.
invented entities (3)
  • bricks no independent evidence
    purpose: Combinatorial model for join-irreducible elements in framing lattices
    New object defined to generalize noncrossing arc diagrams.
  • brick cliques no independent evidence
    purpose: Combinatorial model for canonical join representations
    New object defined to capture canonical join representations uniformly.
  • cornered cliques no independent evidence
    purpose: Intermediate construction enabling cubical coordinates
    New auxiliary object introduced for the bijective proof and coordinate definition.

pith-pipeline@v0.9.0 · 5681 in / 1451 out tokens · 38899 ms · 2026-05-19T15:57:04.104525+00:00 · methodology

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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    A unifying framework for the ν-Tamari lattice and principal order ideals in Young’s lattice

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