Cartesian lattice counting by the vertical 2-sum

TL;DR

This paper introduces a method for counting and classifying certain lattices using vertical 2-sums, providing recurrence relations and asymptotic growth rates for various classes of lattices.

Contribution

It presents a new counting technique for nonisomorphic lattices via vertical 2-sums, including recurrence relations and asymptotic bounds for modular, distributive, and semimodular lattices.

Findings

Counted nonisomorphic modular and distributive lattices up to 35 and 60 elements.

Established asymptotic lower bounds for the number of these lattices.

Showed that semimodular lattices grow faster than any exponential in size.

Abstract

A vertical 2-sum of a two-coatom lattice $L$ and a two-atom lattice $U$ is obtained by removing the top of $L$ and the bottom of $U$, and identifying the coatoms of $L$ with the atoms of $U$. This operation creates one or two nonisomorphic lattices depending on the symmetry case. Here the symmetry cases are analyzed, and a recurrence relation is presented that expresses the number of such vertical 2-sums in some family of interest, up to isomorphism. Nonisomorphic, vertically indecomposable modular and distributive lattices are counted and classified up to 35 and 60 elements respectively. Asymptotically their numbers are shown to be at least $\Omega(2.3122^n)$ and $\Omega(1.7250^n)$, where $n$ is the number of elements. The number of semimodular lattices is shown to grow faster than any exponential in $n$.