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

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http://hdl.handle.net/10138/304957

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Kohonen , J 2019 , ' Exponential lower bounds of lattice counts by vertical sum and 2-sum ' , Algebra Universalis , vol. 80 , no. 1 , 13 . https://doi.org/10.1007/s00012-019-0586-4

Title: Exponential lower bounds of lattice counts by vertical sum and 2-sum
Author: Kohonen, Jukka
Contributor: University of Helsinki, Department of Computer Science
Date: 2019-03
Language: eng
Number of pages: 11
Belongs to series: Algebra Universalis
ISSN: 0002-5240
URI: http://hdl.handle.net/10138/304957
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 2.2726n 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 2.1562n and 2.6797n for n large enough.
Subject: Modular lattices
Semimodular lattices
Vertical sum
Vertical 2-sum
Counting
111 Mathematics
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