Algebra universalis

, 79:84 | Cite as

The canonical join complex for biclosed sets

  • Alexander Clifton
  • Peter Dillery
  • Alexander GarverEmail author


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.


Canonical join representation Shard intersection order Biclosed set Weak order 

Mathematics Subject Classification

06B99 05E45 



This project started at the 2016 combinatorics REU at the School of Mathematics, University of Minnesota, Twin Cities, and was supported by NSF RTG grant DMS-1148634. The authors would like to thank Craig Corsi, Thomas McConville, Henri Mühle, and Vic Reiner for their valuable advice and for useful conversations. The authors also thank an anonymous referee for carefully reading and commenting on this manuscript. Alexander Garver also received support from NSERC grant RGPIN/05999-2014 and the Canada Research Chairs program during various parts of this project.


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Alexander Clifton
    • 1
  • Peter Dillery
    • 2
  • Alexander Garver
    • 3
    Email author
  1. 1.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA
  3. 3.Laboratoire de Combinatoire et d’Informatique MathématiqueUniversité du Québec à MontréalMontrealCanada

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