On the enumeration of finite 𝐿-algebras

We use Constraint Satisfaction Methods to construct and enumerate finite L 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 377,322,225 isomorphism classes of L L -algebras of size eight. The database constructed suggests the existence of bijections between certain classes of L L -algebras and well-known combinatorial objects. We prove that Bell numbers enumerate isomorphism classes of finite linear L L -algebras. We also prove that finite regular L L -algebras are in bijective correspondence with infinite-dimensional Young diagrams.