Left modularity and extremality for (some) infinite lattices

TL;DR

This paper extends the concepts of left modularity and extremality from finite to certain infinite lattices, establishing their equivalence and exploring their properties, with applications to algebraic structures.

Contribution

It generalizes fundamental lattice notions to infinite cases, characterizes left modular elements, and applies results to the lattice of torsion classes in algebra.

Findings

Extremality and left modularity are equivalent in the studied infinite lattices.

Left modular elements form a complete distributive sublattice.

Lattice of torsion classes is left modular if and only if the algebra is brick-directed.

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.