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$(P,φ)$-Tamari lattices

Adrien Segovia

TL;DR

This work introduces the $(P,\phi)$-Tamari lattices as a broad combinatorial framework extending the classical Tamari lattice to arbitrary posets and chains. By encoding torclosed subsets of a structured component $\mathcal{C}_P^{\phi}$, the authors establish that these lattices are join-semidistributive and, under suitable conditions, semidistributive, while also proving left modularity via edge-labellings. They connect these lattices to higher torsion classes in type $A$ Auslander and Nakayama algebras, showing these representation-theoretic lattices admit the same structural properties and can be analyzed through the same combinatorial lens. The paper develops tools for left modularity, extremality, and congruence normality using EL-labellings and doubling theory, and provides explicit main examples including chain cases and the $d$-torsion class lattices, highlighting both the reach and limitations (e.g., semidistributivity conditions) of the construction. Overall, the results provide a unifying combinatorial framework for a class of lattices arising in representation theory and offer new methods for studying their congruence lattices and extremal properties.

Abstract

Given any poset $P$ and chain $φ$ in $P$, we define the $(P,φ)$-Tamari lattice. We study in depth these lattices and prove in particular that they are join-semidistributive, join-congruence uniform and left modular. We prove that the lattices of higher torsion classes of the higher Auslander and Nakayama algebras of type $\mathbb{A}$ are examples of $(P,φ)$-Tamari lattices and thus they inherit their properties. We also give general results related to left modular, extremal and congruence normal lattices.

$(P,φ)$-Tamari lattices

TL;DR

This work introduces the -Tamari lattices as a broad combinatorial framework extending the classical Tamari lattice to arbitrary posets and chains. By encoding torclosed subsets of a structured component , the authors establish that these lattices are join-semidistributive and, under suitable conditions, semidistributive, while also proving left modularity via edge-labellings. They connect these lattices to higher torsion classes in type Auslander and Nakayama algebras, showing these representation-theoretic lattices admit the same structural properties and can be analyzed through the same combinatorial lens. The paper develops tools for left modularity, extremality, and congruence normality using EL-labellings and doubling theory, and provides explicit main examples including chain cases and the -torsion class lattices, highlighting both the reach and limitations (e.g., semidistributivity conditions) of the construction. Overall, the results provide a unifying combinatorial framework for a class of lattices arising in representation theory and offer new methods for studying their congruence lattices and extremal properties.

Abstract

Given any poset and chain in , we define the -Tamari lattice. We study in depth these lattices and prove in particular that they are join-semidistributive, join-congruence uniform and left modular. We prove that the lattices of higher torsion classes of the higher Auslander and Nakayama algebras of type are examples of -Tamari lattices and thus they inherit their properties. We also give general results related to left modular, extremal and congruence normal lattices.

Paper Structure

This paper contains 10 sections, 34 theorems, 3 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background on posets and lattices
  3. New results on lattices
  4. The $(P,\phi)$-Tamari lattices
  5. Definition and first properties of $(P,\phi)$-Tamari
  6. Edge-labellings and congruences
  7. Main examples
  8. The case of the chains
  9. $d$-torsion classes of the higher Auslander algebras of type $\mathbb{A}$
  10. $d$-torsion classes of the higher Nakayama algebras of type $\mathbb{A}$

Key Result

Lemma 2.3

A lattice $L$ is SD if and only if $L$ is JSD and $|\mathrm{JIrr}(L)|=|\mathrm{MIrr}(L)|$ .

Figures (3)

  • Figure 1:
  • Figure 2: We represent $3$ successive doublings. The left modular elements are the blue dots. The thick red edges form the convex subsets $C$ that we double and we circled the elements of $H(C)$.
  • Figure 3:

Theorems & Definitions (54)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: freese1995free
  • Lemma 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7: freese1995free
  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • ...and 44 more