Constructing unlabelled lattices

TL;DR

This paper introduces an improved algorithm for constructing all unlabelled lattices up to a certain size, significantly increasing efficiency and enabling enumeration of lattices with 20 elements.

Contribution

The authors develop a stabiliser chain-based algorithm that grows lattices layer-by-layer, achieving 10 to 100 times faster construction of unlabelled lattices.

Findings

Speed increased by one to two orders of magnitude.

Successfully computed unlabelled lattices on 20 elements.

Enhanced algorithmic approach for lattice enumeration.

Abstract

We present an improved orderly algorithm for constructing all unlabelled lattices up to a given size, that is, an algorithm that constructs the minimal element of each isomorphism class relative to some total order. Our algorithm employs a stabiliser chain approach for cutting branches of the search space that cannot contain a minimal lattice; to make this work, we grow lattices by adding a new layer at a time, as opposed to adding one new element at a time, and we use a total order that is compatible with this modified strategy. The gain in speed is between one and two orders of magnitude. As an application, we compute the number of unlabelled lattices on 20 elements.