The canonical join complex

TL;DR

This paper explores the combinatorics of canonical join representations in finite lattices, characterizing when the associated complex is flag and linking it to the lattice's topology.

Contribution

It introduces the canonical join complex, characterizes lattices with flag complexes, and relates the complex's structure to the topology of the lattice.

Findings

Characterization of finite lattices with flag canonical join complexes

Definition and analysis of the canonical join complex

Connection between the complex's properties and lattice topology

Abstract

In this paper, we study the combinatorics of a certain minimal factorization of the elements in a finite lattice $L$ called the canonical join representation. The join $\bigvee A =w$ is the canonical join representation of $w$ if $A$ is the unique lowest subset of $L$ satisfying $\bigvee A=w$ (where "lowest" is made precise by comparing order ideals under containment). When each element in $L$ has a canonical join representation, we define the canonical join complex to be the abstract simplicial complex of subsets $A$ such that $\bigvee A$ is a canonical join representation. We characterize the class of finite lattices whose canonical join complex is flag, and show how the canonical join complex is related to the topology of $L$.