A Homological Lower Bound for Order Dimension of Lattices

Victor Reiner; Volkmar Welker

doi:10.1023/A:1006325727816

A Homological Lower Bound for Order Dimension of Lattices

Victor Reiner, Volkmar Welker

Mathematics

Research output: Contribution to journal › Article › peer-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 L^o vanishes in dimensions d - 1 and higher. If L can be embedded as a join-sublattice in N^d, then L^o actually has the homotopy type of a simplicial complex with d vertices.

Original language	English (US)
Pages (from-to)	165-170
Number of pages	6
Journal	Order
Volume	16
Issue number	2
DOIs	https://doi.org/10.1023/A:1006325727816
State	Published - 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).

Keywords

Lattices
Order complex
Order dimension
Topology of posets

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title = "A Homological Lower Bound for Order Dimension of Lattices",

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.",

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author = "Victor Reiner and Volkmar Welker",

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doi = "10.1023/A:1006325727816",

language = "English (US)",

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AU - Reiner, Victor

AU - Welker, Volkmar

N1 - 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).

PY - 1999

Y1 - 1999

N2 - 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.

AB - 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.

KW - Lattices

KW - Order complex

KW - Order dimension

KW - Topology of posets

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JO - Order

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ER -

A Homological Lower Bound for Order Dimension of Lattices

Abstract

Bibliographical note

Keywords

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