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arxiv: 2606.00273 · v1 · pith:2L7S6QNXnew · submitted 2026-05-29 · 🧮 math.CO · math.RT

(P,φ)-Tamari and higher torsion lattices of type A

Adrien Segovia This is my paper

Pith reviewed 2026-06-28 21:47 UTC · model grok-4.3

classification 🧮 math.CO math.RT
keywords (P,φ)-Tamari latticeshigher torsion classesjoin-semidistributive latticesleft modular latticeshigher Auslander algebrashigher Nakayama algebrasTamari latticeposet constructions
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The pith

A general (P,φ)-Tamari construction produces lattices that are join-semidistributive, join-extremal and left modular, with higher torsion class lattices of type A algebras as special cases.

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

The paper introduces a general construction that takes any poset P and any chain φ inside P and produces a new poset called the (P,φ)-Tamari lattice. It proves that every such lattice is join-semidistributive, join-extremal and left modular. Particular choices of P and φ recover exactly the lattices of higher torsion classes of the higher Auslander algebras of type A, so those lattices inherit the three properties. The lattices of higher torsion classes of the higher Nakayama algebras of type A are shown to be lattice quotients of the Auslander versions. When P equals φ and both are chains, the construction recovers the classical Tamari lattice.

Core claim

The (P,φ)-Tamari construction associates to each poset P and each chain φ in P a lattice that is join-semidistributive, join-extremal and left modular. For a specific choice of P and φ this lattice coincides with the lattice of higher torsion classes of the higher Auslander algebra of type A. The corresponding lattice for the higher Nakayama algebra of type A is a lattice quotient of the Auslander lattice. When P equals φ is a chain the construction recovers the Tamari lattice.

What carries the argument

The (P,φ)-Tamari construction that builds a lattice from an arbitrary poset P and a chain φ inside P.

If this is right

  • The higher torsion class lattices of higher Auslander algebras of type A are join-semidistributive, join-extremal and left modular.
  • The higher torsion class lattices of higher Nakayama algebras of type A are lattice quotients of the Auslander versions.
  • All (P,φ)-Tamari lattices, including the classical Tamari lattice, satisfy join-semidistributivity, join-extremality and left modularity.
  • Lattice quotients of (P,φ)-Tamari lattices inherit the join-semidistributive and left modular properties.

Where Pith is reading between the lines

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

  • Other families of algebras whose torsion class lattices admit a similar poset-and-chain description may inherit the same three lattice properties without separate proofs.
  • Left modularity supplies a canonical way to label covering relations that could be used to compute canonical join representations in these torsion lattices.
  • The quotient relation between Auslander and Nakayama versions suggests a systematic way to obtain further quotients by varying the chain φ inside a fixed P.

Load-bearing premise

The combinatorial descriptions of higher torsion classes given by August et al. match exactly the elements of the (P,φ)-Tamari poset for the particular P and φ chosen in the paper.

What would settle it

An explicit higher torsion class in a higher Auslander algebra of type A that cannot be matched to any element of the corresponding (P,φ)-Tamari lattice under the map defined in the paper.

Figures

Figures reproduced from arXiv: 2606.00273 by Adrien Segovia.

Figure 1
Figure 1. Figure 1: We represent five successive doublings of a lattice by lower pseudo￾intervals, which are represented with thick red edges. ∆(P) of a poset P is the simplicial complex of vertex set P whose faces are the chains of P. An EL-labelling of a lattice L is an edge-labelling such that in any interval, when reading the labels following the maximal chains from bottom to top, there is a unique maximal increasing chai… view at source ↗
Figure 4
Figure 4. Figure 4: On the left we have (C ϕ P , ≤prod) from [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: On the left is a poset P with a chain ϕ whose elements are circled. On the right is C ϕ P , with the numbering from Section 3.4. it suffices to prove that y < y ′ . By definition of y ′ , we have y ≤ y ′ . If y = y ′ , then ϕ(Kj) ≤ y, and since ϕ(Ki) ≤ ϕ(Kj) because K is non-decreasing, it would give ϕ(Ki) ≤ y, which is absurd. Thus y < y ′ . This finishes the proof of the proposition. □ Remark 3.33. The h… view at source ↗
Figure 7
Figure 7. Figure 7: On the top right is represented the tree φ(u) for the Tamari word u = 1014012. On the left are represented B3(u) and E3(u). On the bottom right is the tree Tu of the interval [B3(u), E3(u)]. u of length n to the binary tree T whose weight sequence is (0, u0, u1, . . . , un−1, n + 1). For an example, see the top right of [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: On the left this is the mirror image of the diagram of u = 1014012 with an extra initial 0 and extra final 8 and lines of slope 1 drawn in red starting at each 0. The segments with the red lines form the binary tree φ(u), and ψ(u) is obtained by identifying the red points that are on the same red lines. On the right we show the partition of Tam(C4, ϕ2) with the intervals I2(u) in red, and we represented th… view at source ↗
Figure 9
Figure 9. Figure 9: On the left we have Cn with ϕ whose elements are circled. On the right we have an illustration of S(i, j) in the proof of Proposition 4.13 where the elements (ϕ(i), j) ∈ C ϕ P for i ≤ j < n are circled. Proof. Let T ⊆ os d+1 n . We prove that T satisfies both (1) and (2 ′ ) if and only if it satisfies both (1) and (2). We suppose that T satisfies (1). Suppose that T satisfies (2). Let i < j < n. Suppose th… view at source ↗
Figure 10
Figure 10. Figure 10: Illustration of the proof of Proposition 4.15 Conversely, suppose that T satisfies (2 ′ ). Let i < j < n. Suppose that (x1, . . . , xd, i) ∈ T and (i + 1, . . . , i + 1, j) ∈ T. We need to prove that (x1, . . . , xd, j) ∈ T. For k ∈ {0, 1, . . . , i}, let Xk ∶= (max(x1, i − k), max(x2, i − k), . . . , max(xd, i − k), i), Lk ∶= (max(x2, i − k) + 1, max(x3, i − k) + 1, . . . , max(xd, i − k) + 1, i + 1, j),… view at source ↗
read the original abstract

The goal of this work is to study the combinatorics of the lattices of higher torsion classes of the higher Auslander and Nakayama algebras of type \textbf{A}. Combinatorial descriptions of these higher torsion classes were recently obtained by August \textit{et al.} (2025), and it was observed that the lattices that they form are not semidistributive. We study in some depth these lattices, proving in particular that the lattices of higher torsion classes of the higher Auslander algebras of type \textbf{A} are join-semidistributive, join-extremal and left modular. We also prove that the lattices of higher torsion classes of the higher Nakayama algebras of type \textbf{A} are lattice quotients of them. In order to prove these results, we define a general construction that produces a lattice, which we call $(P,\phi)$-Tamari, for any choice of poset $P$ and chain $\phi$ in $P$. We prove the lattice results for this general construction. When $P=\phi$ is a chain, we recover the Tamari lattice, whereas the lattices of higher torsion classes that we study are obtained for another very particular choice.

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 / 0 minor

Summary. The paper introduces the (P,φ)-Tamari lattice construction for an arbitrary poset P and chain φ in P. It proves that all such lattices are join-semidistributive, join-extremal, and left modular. The classical Tamari lattice arises when P=φ is a chain. For a specific choice of P and φ, the construction yields the lattices of higher torsion classes of higher Auslander algebras of type A (which are therefore join-semidistributive, join-extremal, and left modular); the corresponding Nakayama-algebra torsion lattices are shown to be lattice quotients of these. The proofs rely on the general construction together with the combinatorial descriptions of the torsion classes given by August et al. (2025).

Significance. If the specific (P,φ) identification is correct, the work supplies a uniform combinatorial explanation for the lattice-theoretic properties of these higher torsion lattices and extends the known good behavior of Tamari lattices to a broader family. The general construction itself may be of independent interest in poset combinatorics.

major comments (1)
  1. [Section defining the specific (P,φ) for Auslander/Nakayama algebras] The load-bearing step is the assertion that a particular choice of poset P and chain φ inside it exactly reproduces the poset of higher torsion classes described by August et al. (2025). The manuscript must supply an explicit order-preserving bijection (including verification that covering relations match) rather than relying on cardinality agreement or informal correspondence; without this, the transfer of the proved join-semidistributivity, join-extremality, and left-modularity properties does not go through.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need to make the identification between the specific (P, φ) construction and the higher torsion class posets fully rigorous. We address the major comment below and will incorporate the requested strengthening in the revision.

read point-by-point responses

  1. Referee: [Section defining the specific (P,φ) for Auslander/Nakayama algebras] The load-bearing step is the assertion that a particular choice of poset P and chain φ inside it exactly reproduces the poset of higher torsion classes described by August et al. (2025). The manuscript must supply an explicit order-preserving bijection (including verification that covering relations match) rather than relying on cardinality agreement or informal correspondence; without this, the transfer of the proved join-semidistributivity, join-extremality, and left-modularity properties does not go through.

    Authors: We agree that the current presentation relies on the combinatorial descriptions in August et al. (2025) together with the explicit definition of our particular (P, φ) without a fully spelled-out order-preserving bijection and covering-relation check. This is a substantive gap for the transfer of the lattice-theoretic properties. In the revised manuscript we will add a dedicated subsection that constructs an explicit bijection between the elements of the (P, φ)-Tamari lattice for our chosen P and φ and the higher torsion classes, verifies that it is order-preserving in both directions, and confirms that covering relations are preserved (using the covering relations already described combinatorially by August et al.). This will establish a lattice isomorphism and thereby justify the application of the general theorems. revision: yes

Circularity Check

0 steps flagged

No significant circularity; general (P,φ)-Tamari construction derives lattice properties independently of the torsion-class application

full rationale

The paper defines a new general (P,φ)-Tamari lattice for arbitrary poset P and chain φ, proves join-semidistributive, join-extremal and left-modular properties directly for this construction, and recovers the classical Tamari lattice when P=φ is a chain. The claim that higher torsion lattices of higher Auslander/Nakayama algebras arise as instances for a particular choice of P and φ is an identification step that relies on the external combinatorial descriptions of August et al. (2025). Because the lattice-property proofs are carried out at the general level and do not reduce to or presuppose the specific identification, no step is self-definitional, no fitted input is relabeled as a prediction, and no load-bearing premise collapses to a self-citation chain. The derivation is therefore self-contained against the external benchmark of the cited combinatorial descriptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper relies on the combinatorial descriptions of higher torsion classes from August et al. (2025) to identify the relevant (P,φ) instances; the (P,φ)-Tamari construction itself is defined in the paper.

axioms (1)
  • domain assumption The combinatorial descriptions of higher torsion classes of the higher Auslander and Nakayama algebras of type A obtained by August et al. (2025) are correct.
    The paper uses these descriptions to select the specific P and φ that produce the torsion lattices.
invented entities (1)
  • (P,φ)-Tamari lattice no independent evidence
    purpose: A general construction that produces a lattice from any poset P and chain φ in P, with the Tamari lattice recovered when P=φ is a chain.
    Defined in the paper as the central new object used to prove the lattice properties.

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