Séminaire Lotharingien de Combinatoire, 95B.56 (2026), 12 pp.
Sota Asai, Osamu Iyama, Kaveh Mousavand and Charles Paquette
Left modular and Extremal Lattices Through Brick-Directed Algebras
Abstract.
We study some properties of the semidistributive lattices that arise as the lattice of torsion classes of finite dimensional algebras. First, we give a new realization of the left modular elements in terms of brick-splitting torsion classes. Restricted to finite lattices, this yields a full characterization of those lattices of torsion classes that are left modular (equivalently, extremal), thus trim. Beyond the setting of finite lattices, we also treat completely semidistributive weakly atomic lattices and generalize the notions of left modularity and extremality. Finally, we give an explicit construction to create arbitrarily large left modular and extremal semidistributive lattices.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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