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

TL;DR

This paper develops methods to establish exponential lower bounds on the number of unlabeled lattices in certain families, using vertical sum and a new vertical 2-sum operation, with specific bounds for modular and semimodular lattices.

Contribution

It introduces a novel approach using vertical sum and vertical 2-sum to derive exponential lower bounds on lattice counts, including for vertically indecomposable lattices.

Findings

Number of modular lattices ≥ 2.2726^n for large n

Number of vertically indecomposable modular lattices ≥ 2.1562^n

Number of vertically indecomposable semimodular lattices ≥ 2.6797^n

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.