A Homological Lower Bound for Order Dimension of Lattices

Victor Reiner, Volkmar Welker

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

We prove that if a finite lattice L has order dimension at most d, then the homology of the order complex of its proper part Lo vanishes in dimensions d - 1 and higher. If L can be embedded as a join-sublattice in Nd, then Lo actually has the homotopy type of a simplicial complex with d vertices.

Original languageEnglish (US)
Pages (from-to)165-170
Number of pages6
JournalOrder
Volume16
Issue number2
DOIs
StatePublished - 1999

Bibliographical note

Funding Information:
The authors thank Irena Peeva for the initial conversations leading to this work and to [2], and the Mathematisches Forschungsinstitut Oberwolfach for their hospitality. They also thank Günter M. Ziegler for helpful conversations and edits. Vic Reiner was supported by a University of Minnesota McKnight-Land Grant Fellowship and a Sloan Fellowship, and Volkmar Welker was supported by Deutsche Forschungsgemeinschaft (DFG).

Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.

Keywords

  • Lattices
  • Order complex
  • Order dimension
  • Topology of posets

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