The canonical join complex for biclosed sets

TL;DR

This paper classifies the faces of the canonical join complex for biclosed sets in a tree-supported lattice, revealing structural properties and proving the shard intersection order forms a lattice.

Contribution

It provides a combinatorial classification of the canonical join complex for biclosed sets and establishes the lattice structure of the shard intersection order.

Findings

Classification of faces of the canonical join complex

Description of elements in the shard intersection order

Proof that the shard intersection order is a lattice

Abstract

The canonical join complex of a semidistributive lattice is a simplicial complex whose faces are canonical join representations of elements of the semidistributive lattice. We give a combinatorial classification of the faces of the canonical join complex of the lattice of biclosed sets of segments supported by a tree, as introduced by the third author and McConville. We also use our classification to describe the elements of the shard intersection order of the lattice of biclosed sets. As a consequence, we prove that this shard intersection order is a lattice.