# A note on lattices with many sublattices

## TL;DR

This paper characterizes the number of sublattices in n-element lattices, showing they are limited to specific exponential forms and describing the lattices that attain these counts.

## Contribution

It establishes exact bounds on the number of subuniverses in n-element lattices and classifies those achieving specific counts.

## Key findings

- Number of subuniverses is 2^n, 13*2^{n-4}, 23*2^{n-5}, or less.
- Characterization of n-element lattices with exactly 2^n, 13*2^{n-4}, or 23*2^{n-5} subuniverses.

## Abstract

For every natural number $n\geq 5$, we prove that the number of subuniverses of an $n$-element lattice is $2^n$, $13\cdot 2^{n-4}$, $23\cdot 2^{n-5}$, or less than $23\cdot 2^{n-5}$. By a subuniverse, we mean a sublattice or the emptyset. Also, we describe the $n$-element lattices with exactly $2^n$, $13\cdot 2^{n-4}$, or $23\cdot 2^{n-5}$ subuniverses.