arxiv: 2604.20947 · v1 · submitted 2026-04-22 · 🧮 math.RA

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Left modularity and extremality for (some) infinite lattices

Charles Paquette, Kaveh Mousavand, Osamu Iyama, Sota Asai

Pith reviewed 2026-05-09 22:25 UTC · model grok-4.3

classification 🧮 math.RA

keywords left modularityextremalityinfinite latticessemidistributive latticestorsion classesbrick-directed algebraskappa-latticesweakly atomic lattices

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The pith

For well-separated kappa-lattices and weakly atomic completely semidistributive lattices, left modularity and extremality are equivalent.

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

The paper shows that in two important families of infinite lattices—well-separated κ-lattices and weakly atomic completely semidistributive lattices—left modularity and extremality are equivalent properties. It also provides several characterizations of left modular elements within the semidistributive family and proves that these elements form a complete distributive sublattice. When applied to the lattice of torsion classes over a finite-dimensional algebra, this equivalence translates to the algebra being brick-directed. This generalizes known results from finite lattices to the infinite setting and yields many concrete examples.

Core claim

For both well-separated κ-lattices and weakly atomic completely semidistributive lattices, extremality and left modularity imply each other. Furthermore, for weakly atomic completely semidistributive lattices, left modular elements are characterized in several ways and form a complete distributive sublattice. The lattice of torsion classes of a finite-dimensional algebra A is left modular, equivalently extremal, if and only if A is brick-directed.

What carries the argument

The mutual implication between extremality and left modularity, which holds once the lattice belongs to one of the two families (well-separated κ-lattices or weakly atomic completely semidistributive lattices).

Load-bearing premise

The lattices under study must belong to either the well-separated κ-lattices or the weakly atomic completely semidistributive lattices for the equivalence to hold.

What would settle it

A concrete counterexample would be a lattice from one of these families that is left modular but fails to be extremal, or a finite-dimensional algebra that is not brick-directed yet whose torsion-class lattice is left modular.

Figures

Figures reproduced from arXiv: 2604.20947 by Charles Paquette, Kaveh Mousavand, Osamu Iyama, Sota Asai.

**Figure 1.** Figure 1: Lattice of torsion classes of path algebra of Q : 2 → 1. Vertices are specified by τ -rigid modules generating torsion classes. To put our more general notion of extremality in Definition 3.2 into perspective, we compare it with the classical notion of extremality, only given for finite lattices (see Section 2.1). In particular, the following lemma shows that our notion of extremality coincides with the cl… view at source ↗

read the original abstract

For some important families of complete infinite lattices, we study some generalizations of two fundamental notions which are mostly treated for finite lattices. Specifically, for well-separated $\kappa$-lattices, and also for weakly atomic completely semidistributive lattices, we generalize the notions of left modularity and extremality. These two families of lattices coincide if restricted to finite lattices, but are distinct when infinite lattices are also included. For both families, we prove that extremality and left modularity imply each other. Furthermore, for weakly atomic completely semidistributive lattices, we give several conceptual characterizations of left modular elements, and show that the set of left modular elements form a complete distributive sublattice. Our results, combined with some recent work on finite lattices, imply that the weakly atomic completely semidistributive lattices that are left modular (or extremal) generalize the semidistributive trim lattices; from finite to infinite lattices. We then apply our results to the lattice of torsion classes of finite dimensional algebras, which are known to fall in the intersection of the two families treated in our work. For an algebra $A$, we obtain that the lattice of torsion classes is left modular (equivalently, extremal) if and only if $A$ is brick-directed. This leads to an abundance of concrete examples and non-examples.

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.

Desk Editor's Note private letter to a colleague

The paper extends left modularity and extremality to two families of infinite lattices, proves they coincide there, and gives a criterion for when torsion-class lattices are left modular.

read the letter

The main thing to know is that this work generalizes left modularity and extremality from finite lattices to well-separated kappa-lattices and to weakly atomic completely semidistributive lattices, shows the two properties are equivalent inside each family, and then applies the equivalence to torsion-class lattices to conclude that they are left modular exactly when the algebra is brick-directed. It also gives characterizations of left modular elements in the semidistributive case and shows those elements form a complete distributive sublattice, extending the finite semidistributive trim lattices to the infinite setting. The torsion-class application supplies concrete examples and non-examples, which is useful for people working with module categories. The definitions are built around covering relations and the structural axioms of each family, so the proofs can move between the properties without leaning on finiteness. The paper is upfront that the two families coincide only in the finite case and that the equivalence is stated only inside them. That scope is clear and avoids overclaiming. No circularity or hidden assumptions show up in the abstract or the stress-test note. The central claims rest on the stated separation and atomicity conditions, which the authors use directly. This is aimed at lattice theorists and representation theorists who already know the finite versions and want to handle infinite posets or torsion classes. A reader in that overlap will see the direct extension and the new criterion without needing to reinvent the definitions. It deserves peer review. The theorems are new, the application is concrete, and the setup looks careful enough to be worth a referee's time.

Referee Report

0 major / 2 minor

Summary. The manuscript generalizes the notions of left modularity and extremality from finite lattices to two families of complete infinite lattices: well-separated κ-lattices and weakly atomic completely semidistributive lattices. It proves that the two properties are equivalent within each family. For weakly atomic completely semidistributive lattices it supplies several conceptual characterizations of left modular elements and proves that the set of such elements forms a complete distributive sublattice. The results are applied to the lattice of torsion classes of a finite-dimensional algebra A, yielding the equivalence that this lattice is left modular (equivalently extremal) if and only if A is brick-directed.

Significance. The work extends two classical notions to infinite lattices under explicitly delimited structural hypotheses, supplies characterizations that generalize the finite semidistributive trim case, and furnishes a concrete algebraic criterion (brick-directedness) for a natural class of lattices arising in representation theory. These contributions are of interest to both lattice theorists and algebraists working with torsion classes.

minor comments (2)

[Introduction] The introduction states that the two families coincide on finite lattices but are distinct for infinite ones; a single concrete infinite example illustrating the distinction would improve accessibility.
[Application section] The claim that torsion-class lattices lie in the intersection of the two families is asserted on the basis of prior results; a one-sentence reminder of the relevant theorem or reference would make the application self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the manuscript, recognition of its significance for both lattice theory and representation theory, and recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper introduces definitions of generalized left modularity and extremality explicitly in terms of covering relations and the structural axioms of well-separated κ-lattices and weakly atomic completely semidistributive lattices. It then proves mutual implication directly from those axioms within each family. The torsion-class application rests on the independent fact that such lattices lie in the intersection of the two families, followed by the already-proved equivalence. No equation or claim reduces by construction to a fitted parameter, a self-definition, or an unverified self-citation chain; all load-bearing steps are external lattice-theoretic properties or prior non-circular results on finite cases.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper works entirely within standard lattice theory. No free parameters are fitted to data. The only background assumptions are the definitions of the two lattice families and the usual axioms of complete lattices and semidistributivity.

axioms (2)

standard math The structures satisfy the axioms of a complete lattice (every subset has sup and inf).
Invoked throughout to define the families and the notions of left modularity and extremality.
domain assumption The lattices are either well-separated kappa-lattices or weakly atomic completely semidistributive.
These are the two families for which the equivalence is proved; they are stated as hypotheses rather than derived.

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Works this paper leans on

28 extracted references · 2 canonical work pages

[1]

Adachi, O

T. Adachi, O. Iyama, I. Reiten, -tilting theory, Compos. Math. 150 (2014), no. 3, pp 415-452

2014
[2]

S. Asai, O. Iyama, K. Mousavand, C. Paquette, Brick-splitting Torsion Pairs and Left Modularity, arXiv:2506.13602 (2025)

work page arXiv 2025
[3]

Assem, D

I. Assem, D. Simson, and A. Skowronski, Elements of the representation theory of associative algebras: Techniques of representation theory. Cambridge University Press, 2006

2006
[4]

Br\"ustle, G

T. Br\"ustle, G. Douville, K. Mousavand, H. Thomas, E. Yıldırım, On the combinatorics of gentle algebras. Canadian Journal of Mathematics, 72(6):1551–1580, 2020

2020
[5]

Barnard, A

E. Barnard, A. T. Carroll, S. Zhu, Minimal inclusions of torsion classes, Algebr. Comb. 2 (2019), no. 5, 879-901

2019
[6]

Blass, B

A. Blass, B. Sagan, Möbius functions of lattices, Adv. Math. 127 (1997), 94–123

1997
[7]

Demonet, O

L. Demonet, O. Iyama, G. Jasso, -tilting finite algebras and g -vectors, Int. Math. Res. Not. (2019), no. 3, 852–892

2019
[8]

Demonet, O

L. Demonet, O. Iyama, N. Reading, I. Reiten, H. Thomas, Lattice theory of torsion classes, Trans. Amer. Math. Soc. Ser. B 10 (2023)

2023
[9]

Dummit, R.M

D.S. Dummit, R.M. Foote, Abstract Algebra, 3rd Edition, John Wiley and Sons, Inc., Hoboken, NJ (2003)

2003
[10]

Davey, H

B. Davey, H. Priestley Introduction to Lattices and Order, 2nd Ed., Cambridge University Press, London (2002)

2002
[11]

Enomoto, FD Applet

H. Enomoto, FD Applet. Web applet for special biserial algebras, available at https://haruhisa-enomoto.github.io/fd-applet/
[12]

Ingalls and H

C. Ingalls and H. Thomas, Noncrossing partitions and representations of quivers. Compos. Math. 145 (2009), no. 6, 1533–1562

2009
[13]

Garver, T

A. Garver, T. McConville, Lattice Properties of Oriented Exchange Graphs and Torsion Classes, Algebr. Represent. Theory 22 (2019), no. 1, 43–78

2019
[14]

Geuenich, String Applet

J. Geuenich, String Applet. Web applet for special biserial algebras, available at https://www.math.unibielefeld.de/ jgeuenich/string-applet/
[15]

Grätzer, General Lattice Theory, 2nd edition

G. Grätzer, General Lattice Theory, 2nd edition. Birkäuser, Basel, Switzerland (2003)

2003
[16]

S. Liu, B. Sagan, left modular elements of lattices, J. Comb. Theory Ser. A. 91 (2000), 369–385

2000
[17]

Markowsky, Primes, irreducibles and extremal lattices, Order 9 (1992), no

G. Markowsky, Primes, irreducibles and extremal lattices, Order 9 (1992), no. 3, 265–290

1992
[18]

Mizuno, Classifying -tilting modules over preprojective algebras of Dynkin type, Math

Y. Mizuno, Classifying -tilting modules over preprojective algebras of Dynkin type, Math. Z. 277 (2014), no. 3-4, 665–690

2014
[19]

Mousavand, -tilting finiteness of biserial algebras, Algebr

K. Mousavand, -tilting finiteness of biserial algebras, Algebr. Represent. Theory 26 (2023), 2485–2522

2023
[20]

Mousavand, C

K. Mousavand, C. Paquette, Biserial algebras and generic bricks, Math. Z. 310, 64 (2025)

2025
[21]

Mousavand, C

K. Mousavand, C. Paquette, Hom-orthogonal modules and brick-Brauer-Thrall conjectures, J. Algebra, 686 (2026), 650–676

2026
[22]

Mühle, Extremality, left modularity and Semidistributivity, Algebra Universalis 84, 16 (2023)

H. Mühle, Extremality, left modularity and Semidistributivity, Algebra Universalis 84, 16 (2023)

2023
[23]

Reading, D

N. Reading, D. E. Speyer, H. Thomas The fundamental theorem of finite semidistributive lattices, Selecta Math. (N.S.) 27 (2021), no. 4, Paper No. 59, 53 pp

2021
[24]

Segovia, Extremality in semidistributive lattices, arXiv:2511.18540 (2025)

A. Segovia, Extremality in semidistributive lattices, arXiv:2511.18540 (2025)

work page arXiv 2025
[25]

R. P. Stanley, Modular elements of geometric lattices, Algebra Universalis 1 (1971), 214-217

1971
[26]

R. P. Stanley, Supersolvable Lattices, Algebra Universalis 2 (1972), 197-217

1972
[27]

Thomas, An analogue of distributivity for ungraded lattices, Order 23 (2006), no

H. Thomas, An analogue of distributivity for ungraded lattices, Order 23 (2006), no. 2, 249–269

2006
[28]

Thomas, N

H. Thomas, N. Williams, Rowmotion in slow motion, Proc. Lond. Math. Soc. 119 (2019), 1149–1178

2019