Left modularity and extremality for (some) infinite lattices

1. Problem Statement and Motivation

The paper addresses the generalization of two fundamental lattice-theoretic concepts—left modularity and extremality—from the well-studied realm of finite lattices to specific families of infinite complete lattices.

• Context: In finite lattice theory, particularly for semidistributive lattices, it is a known result (by Thomas, McAllister, and others) that left modularity and extremality are equivalent properties. These properties generalize the concept of distributivity.
• The Gap: While finite lattices of torsion classes for finite-dimensional algebras are well-understood, the lattice of torsion classes for general finite-dimensional algebras can be infinite (uncountable). Existing definitions of left modularity and extremality rely on finite parameters (e.g., the number of join-irreducibles equals the length of the longest chain), which are ill-defined or inapplicable in the infinite setting.
• Objective: The authors aim to define robust generalizations of these concepts for infinite lattices, prove their equivalence under specific structural conditions, and apply these results to the representation theory of finite-dimensional algebras, specifically characterizing when the lattice of torsion classes is left modular/extremal.

2. Methodology and Framework

The authors employ a dual approach combining lattice theory and representation theory.

A. Lattice-Theoretic Generalizations

The paper defines two distinct families of infinite lattices and generalizes the concepts within each:

1. Well-separated $\kappa$-lattices: A generalization of finite semidistributive lattices involving a bijection $\kappa$ between completely join-irreducible ($JI_c$) and completely meet-irreducible ($MI_c$) elements.
2. Weakly atomic completely semidistributive lattices: A broader class that includes the torsion class lattices of algebras.

Generalized Definitions:

• Left Modularity: An element $x$ is left modular if $(y \vee x) \wedge z = y \vee (x \wedge z)$ for all $y \leq z$. A lattice is left modular if it possesses an inclusion-maximal chain consisting entirely of left modular elements.
• Extremality (Infinite): The authors introduce a technical definition involving a bijection $\lambda: JI_c(L) \to MI_c(L)$ and a maximal chain $X$. For any step $x \lessdot y$ in the chain, there must be a unique $j \in JI_c(L)$ and $m \in MI_c(L)$ such that $j$ is the unique element below $y$ but not $x$, and $m$ is the unique element above $x$ but not $y$, with $\lambda(j)=m$.

B. Representation-Theoretic Application

The authors apply these abstract results to the lattice of torsion classes, denoted tors A, for a finite-dimensional algebra $A$.

• They utilize the concept of brick-splitting torsion pairs, where every brick (a module with a division ring endomorphism algebra) belongs either to the torsion class or its torsion-free complement.
• They introduce brick-directed algebras, defined as algebras where the "brick quiver" (a directed graph of bricks with non-zero morphisms) contains no oriented cycles.

3. Key Contributions and Results

I. Equivalence of Left Modularity and Extremality

The paper proves that for two specific families of infinite lattices, the generalized notions of left modularity and extremality are equivalent. This extends the finite result (Theorem 2.2) to the infinite case.

• Theorem 1.1:
  1. For a well-separated $\kappa$-lattice, $L$ is left modular $\iff$ $L$ is extremal.
  2. For a weakly atomic completely semidistributive lattice, $L$ is left modular $\iff$ $L$ is extremal.
  • Note: The authors emphasize that these two families are distinct in the infinite case (neither implies the other), though they coincide for finite lattices.

II. Structural Characterizations

For weakly atomic completely semidistributive lattices, the authors provide deep structural insights:

• Characterization of Elements (Theorem 1.3): An element $x$ is left modular if and only if for every $j \in JI_c(L)$, either $j \leq x$ or $x \leq \kappa(j)$ (but not both).
• Lattice Structure of Left Modular Elements (Theorem 1.3 & Prop 4.6): The set of left modular elements, $LM(L)$, forms a complete distributive sublattice of $L$. This generalizes the concept of the "spine" of a finite trim lattice.
• Combinatorial Realization (Theorem 1.2 & 1.4): There is a bijection between left modular elements and successor-closed sets of the labelling quiver $Q_L$ (a quiver on $JI_c(L)$). If $L$ is extremal, $Q_L$ is acyclic, and extremal chains correspond bijectively to linear extensions of the partial order on $Q_L$.

III. Application to Torsion Classes and Brick-Directed Algebras

The paper establishes a complete algebraic characterization for when the lattice of torsion classes is left modular/extremal.

• Theorem 1.5 & Corollary 1.6: For a finite-dimensional algebra $A$, the lattice tors A is left modular (equivalently, extremal) if and only if $A$ is brick-directed.
  • This is equivalent to the brick quiver $Q_b(A)$ having no oriented cycles.
  • It is also equivalent to the existence of a maximal chain in tors A where every element is a brick-splitting torsion class.
• Specific Examples (Corollaries 1.7 & 1.8):
  • For path algebras $kQ$, tors A is left modular iff $Q$ is a Dynkin quiver or the Kronecker quiver.
  • For Weyl groups of simply laced Dynkin diagrams, only the trivial elements ($\hat{0}, \hat{1}$) are left modular.
  • The authors construct infinite families of algebras yielding left modular/extremal lattices of arbitrary finite length $m > 4$ and arbitrary infinite cardinality.

4. Significance and Impact

1. Unification of Finite and Infinite Theory: The paper successfully bridges the gap between finite lattice theory (where extremality and left modularity are well-understood) and infinite lattice theory, providing a rigorous framework for the latter.
2. New Class of Algebras: By introducing brick-directed algebras, the authors generalize the classical notion of representation-directed algebras. This class includes tame and wild algebras (e.g., Kronecker algebra) that were previously not known to possess such strong lattice-theoretic properties.
3. Combinatorial Tools: The introduction of the labelling quiver and successor-closed sets provides a powerful combinatorial tool for analyzing infinite lattices, allowing for the construction of extremal chains via linear extensions of partial orders.
4. Connections to Open Problems: The results connect to the Strong Semibrick Conjecture and the study of infinite semibricks. The paper suggests that the existence of $2^{\aleph}$ maximal chains of left modular elements is linked to the existence of infinite semibricks, opening new avenues for research in representation theory.
5. Concrete Examples: The paper moves beyond abstract existence proofs by providing explicit constructions of algebras (via gluing techniques and gentle algebras) that generate left modular/extremal lattices of specific lengths and cardinalities, enriching the catalog of known examples in the field.

In summary, this work provides a foundational extension of lattice theory concepts to the infinite realm, driven by and deeply integrated with the representation theory of finite-dimensional algebras, offering new characterizations, structural theorems, and a wealth of concrete examples.