A free two-generated left distributive algebra of elementary embeddings

TL;DR

This paper constructs a two-generated free left distributive algebra of elementary embeddings under a large cardinal assumption slightly weaker than I3, extending previous results that involved only a single generator.

Contribution

It demonstrates the existence of a two-generated free left distributive algebra of rank-to-rank embeddings assuming I2, advancing the understanding of algebraic structures derived from set-theoretic embeddings.

Findings

Constructed a two-generated free LD algebra of embeddings.

Extended previous single-generator results to two generators.

Relied on the large cardinal assumption I2.

Abstract

The set-theoretic large cardinal axiom known as I3 posits the existence of a non-trivial rank-to-rank embedding from an initial segment of the universe of sets into itself. Laver showed that the algebra generated by a single such embedding under the operation of application is in fact the free left distributive (LD) algebra on one generator. This and associated theorems using the set-theoretic structure of the embeddings yielded numerous results about general LD algebras under the assumption of I3, only some of which have since been proven without the use of such a strong axiom. A natural question is whether, under the assumption of I3, one can obtain a free LD algebra of embeddings on more than one generator. Here we show that, under an assumption only just above I3 in the large cardinal hierarchy (namely, I2), we indeed obtain a two-generated free left distributive algebra of rank-to-rank embeddings.