Exponential lower bounds of lattice counts by vertical sum and 2-sum
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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.
KeywordsModular lattices Semimodular lattices Vertical sum Vertical 2-sum Counting
Mathematics Subject Classification05C30 06C05 06C10
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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