# The fundamental theorem of finite semidistributive lattices

## TL;DR

This paper establishes a fundamental theorem characterizing finite semidistributive lattices via a set with additional structure, linking lattice theory with combinatorics and geometry.

## Contribution

It introduces a new fundamental theorem for finite semidistributive lattices, connecting them to structured sets and geometric interpretations.

## Key findings

- Characterization of finite semidistributive lattices through admissible subsets
- Clarification of lattice constructions like canonical join representations
- Extension of results to infinite lattices

## Abstract

We prove a Fundamental Theorem of Finite Semidistributive Lattices (FTFSDL), modelled on Birkhoff's Fundamental Theorem of Finite Distributive Lattices. Our FTFSDL is of the form "A poset L is a finite semidistributive lattice if and only if there exists a set Sha with some additional structure, such that L is isomorphic to the admissible subsets of Sha ordered by inclusion; in this case, Sha and its additional structure are uniquely determined by L." The additional structure on Sha is a combinatorial abstraction of the notion of torsion pairs from representation theory and has geometric meaning in the case of posets of regions of hyperplane arrangements. We show how the FTFSDL clarifies many constructions in lattice theory, such as canonical join representations and passing to quotients, and how the semidistributive property interacts with other major classes of lattices. Many of our results also apply to infinite lattices.