Generalizations of Laver tables

TL;DR

This paper extends the concept of Laver tables to more complex algebraic structures with multiple generators, operations, and generalized self-distributivity, aiming to model rank-into-rank embeddings.

Contribution

It introduces a broad generalization of Laver tables to algebras with multiple operations, higher arity, and partial self-distributivity, connecting to rank-into-rank embeddings.

Findings

Defined new classes of generalized Laver algebras.

Established structural properties analogous to classical Laver tables.

Linked algebraic structures to rank-into-rank embedding models.

Abstract

We shall generalize the notion of a Laver table to algebras which may have many generators, several fundamental operations, fundamental operations of arity higher than 2, and to algebras where only some of the operations are self-distributive or where the operations satisfy a generalized version of self-distributivity. These algebras mimic the algebras of rank-into-rank embeddings $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ in the sense that composition and the notion of a critical point make sense for these sorts of algebras.