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On the Homological Dimension of Lattices

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Abstract

Let L be a finite lattice and let \({\widehat{L}}=L-\{{\hat{0}},{\hat{1}}\}\). It is shown that if the order complex \(\Delta({\widehat{L}})\) satisfies \({\tilde{\rm H}}_{k-2}\bigl(\Delta({\widehat{L}})\bigr) \neq 0\) then |L| ≥ 2k. Equality |L| = 2k holds iff L is isomorphic to the Boolean lattice {0,1}k.

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Authors and Affiliations

  1. Department of Mathematics, Technion, Haifa, 32000, Israel

    Roy Meshulam

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  1. Roy Meshulam
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Correspondence to Roy Meshulam.

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Research supported by the Israel Science Foundation.

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Meshulam, R. On the Homological Dimension of Lattices. Order 25, 153–155 (2008). https://doi.org/10.1007/s11083-008-9086-4

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  • DOI: https://doi.org/10.1007/s11083-008-9086-4

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