Generalizations of Laver tables

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.