Séminaire Lotharingien de Combinatoire, 80B.2 (2018), 12 pp.

Alexander Clifton, Peter Dillery, and Alexander Garver

The Canonical Join Complex for Biclosed Sets

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.

Received: November 14, 2017. Accepted: February 17, 2018. Final version: April 1, 2018.

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