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Algebra universalis

, 80:13 | Cite as

Exponential lower bounds of lattice counts by vertical sum and 2-sum

  • Jukka KohonenEmail author
Open Access
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Abstract

We consider the problem of finding lower bounds on the number of unlabeled n-element lattices in some lattice family. We show that if the family is closed under vertical sum, exponential lower bounds can be obtained from vertical sums of small lattices whose numbers are known. We demonstrate this approach by establishing that the number of modular lattices is at least \(2.2726^n\) for n large enough. We also present an analogous method for finding lower bounds on the number of vertically indecomposable lattices in some family. For this purpose we define a new kind of sum, the vertical 2-sum, which combines lattices at two common elements. As an application we prove that the numbers of vertically indecomposable modular and semimodular lattices are at least \(2.1562^n\) and \(2.6797^n\) for n large enough.

Keywords

Modular lattices Semimodular lattices Vertical sum Vertical 2-sum Counting 

Mathematics Subject Classification

05C30 06C05 06C10 

Notes

Acknowledgements

Open access funding provided by University of Helsinki including Helsinki University Central Hospital. The author thanks the anonymous referees for invaluable comments that led to improvements in both the results and the exposition.

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Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of HelsinkiHelsinkiFinland

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