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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Erné, M., Heitzig, J., Reinhold, J.: On the number of distributive lattices. Electron. J. Combin. 9, Article #R24 (2002)
Gebhardt, V., Tawn, S.: Constructing unlabelled lattices. J. Algebra (2018). https://doi.org/10.1016/j.jalgebra.2018.10.017
Heitzig, J., Reinhold, J.: Counting finite lattices. Algebra Univers. 48, 43–53 (2002)
Jipsen, P., Lawless, N.: Generating all finite modular lattices of a given size. Algebra Univers. 74, 253–264 (2015)
Kleitman, D.J., Winston, K.J.: The asymptotic number of lattices. Ann. Discr. Math. 6, 243–249 (1980)
Klotz, W., Lucht, L.: Endliche Verbände. J. Reine Angew. Math 247, 58–68 (1971)
Kohonen, J.: Generating modular lattices of up to 30 elements. Order (2018). https://doi.org/10.1007/s11083-018-9475-2
Kohonen, J.: Lists of finite lattices (modular, semimodular, graded and geometric). https://b2share.eudat.eu/records/dbb096da4e364b5e9e37b982431f41de. Accessed 18 Feb 2019
Slomson, A., Allenby, R.B.J.T.: How to Count: An Introduction to Combinatorics, 2nd edn. Chapman and Hall/CRC, Boca Raton (2010)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This 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.
About this article
Cite this article
Kohonen, J. Exponential lower bounds of lattice counts by vertical sum and 2-sum. Algebra Univers. 80, 13 (2019). https://doi.org/10.1007/s00012-019-0586-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00012-019-0586-4