On the enumeration of finite $L$-algebras

TL;DR

This paper uses constraint satisfaction methods to enumerate finite $L$-algebras, revealing their vast diversity and connections to combinatorial objects, and establishing bijections with Bell numbers and Young diagrams.

Contribution

It introduces a computational approach to classify finite $L$-algebras and uncovers their links to well-known combinatorial structures, providing new insights into their enumeration.

Findings

377 million isomorphism classes of size eight

Bell numbers count finite linear $L$-algebras

Finite regular $L$-algebras correspond to Young diagrams

Abstract

We use Constraint Satisfaction Methods to construct and enumerate finite $L$-algebras up to isomorphism. These objects were recently introduced by Rump and appear in Garside theory, algebraic logic, and the study of the combinatorial Yang-Baxter equation. There are 377322225 isomorphism classes of $L$-algebras of size eight. The database constructed suggest the existence of bijections between certain classes of $L$-algebras and well-known combinatorial objects. On the one hand, we prove that Bell numbers enumerate isomorphism classes of finite linear $L$-algebras. On the other hand, we also prove that finite regular $L$-algebras are in bijective correspondence with infinite-dimensional Young diagrams.