Generating modular lattices of up to 30 elements

TL;DR

This paper introduces an efficient algorithm for generating and counting various types of finite lattices up to 30 elements, significantly improving speed and enabling new enumerations.

Contribution

The paper presents a novel algorithm that efficiently generates finite modular, semimodular, graded, and geometric lattices, surpassing previous methods in speed and scale.

Findings

Counted modular lattices up to 30 elements

Counted semimodular lattices up to 25 elements

Provided statistics on small lattice shapes

Abstract

An algorithm is presented for generating finite modular, semimodular, graded, and geometric lattices up to isomorphism. Isomorphic copies are avoided using a combination of the general-purpose graph-isomorphism tool nauty and some optimizations that handle simple cases directly. For modular and semimodular lattices, the algorithm prunes the search tree much earlier than the method of Jipsen and Lawless, leading to a speedup of several orders of magnitude. With this new algorithm modular lattices are counted up to 30 elements, semimodular lattices up to 25 elements, graded lattices up to 21 elements, and geometric lattices up to 34 elements. Some statistics are also provided on the typical shape of small lattices of these types.