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The fundamental theorem of finite semidistributive lattices

Nathan Reading, David E Speyer, Hugh Thomas

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.

The fundamental theorem of finite semidistributive 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.

Paper Structure

This paper contains 24 sections, 93 theorems, 24 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The fundamental theorem
  3. Conventions about partially ordered sets
  4. Fundamentals of factorization systems
  5. From $\kappa$-lattices to factorization systems
  6. From factorization systems to $\kappa$-lattices
  7. Finite semidistributive lattices
  8. The infinite case
  9. Conditions on infinite complete lattices
  10. Proof of Theorem \ref{['infinite main']}
  11. Counterexamples
  12. Factorization systems for intervals
  13. Covers and canonical join representations
  14. Covers
  15. Canonical join representations
  16. ...and 9 more sections

Key Result

Theorem 1.1

A finite poset $L$ is a distributive lattice if and only if it is isomorphic to $\operatorname{Downsets}(P)$ for some finite poset $P$. In this case, $P$ is isomorphic to $\operatorname{Ji} L$. The map $x\mapsto\{ j \in \operatorname{Ji} L: j \leq x \}$ is an isomorphism from $L$ to $\operatorname{D

Figures (8)

  • Figure 1: A two-acyclic factorization system
  • Figure 2: The semidistributive lattice associated to Figure \ref{['2afs fig']}
  • Figure 3: Portions of the lattices from Examples \ref{['comp SD not cov sep']} and \ref{['comp SD not cov det']}
  • Figure 4: The factorization system and lattice $\operatorname{Pairs}({\mathord{\to}})$ in Example \ref{['EGKappaNotPairs']}
  • Figure 5: A factorization system for which $\operatorname{Pairs}({\mathord{\to}})$ is not semidistributive
  • ...and 3 more figures

Theorems & Definitions (174)

  • Theorem 1.1: FTFDL
  • Theorem 1.2: FTFSDL
  • Example 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 2.3
  • proof
  • ...and 164 more