The canonical join complex for biclosed sets |
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Authors: | Alexander Clifton Peter Dillery Alexander Garver |
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Institution: | 1.Department of Mathematics and Computer Science,Emory University,Atlanta,USA;2.Department of Mathematics,University of Michigan,Ann Arbor,USA;3.Laboratoire de Combinatoire et d’Informatique Mathématique,Université du Québec à Montréal,Montreal,Canada |
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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. |
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