Free Left Distributive Algebras and a Canonical Extension

TL;DR

This paper explores the structure of free left distributive algebras under large cardinal assumptions, extending Laver's representation and establishing new elementary embedding properties.

Contribution

It proves elementary equivalence results for finitely-generated free LDAs and constructs a canonical extension with embedding properties, linking large cardinals to algebraic structures.

Findings

Finitely-generated free LDAs with different numbers of generators are $ ext{Σ}_1$-elementarily equivalent.

Constructed an extension of the monogenerated free LDA with elementary embedding properties.

Demonstrated algebraic properties provable from large cardinals without standard set theory proofs.

Abstract

Assuming a large cardinal hypothesis, Laver gave a representation of the monogenerated free left distributive algebra (LDA) using elementary embeddings and used this representation to prove many algebraic results. Some of these results were later proved by Dehornoy in ZFC, without the large cardinal hypotheses. However, there is an important algebraic result whose consistency strength is unknown. (See Laver (1995) and Dougherty & Jech (1997).) Recent results [arXiv:2508.02244] extend the connection between elementary embeddings of set theory and free LDAs to the many-generated case. Assuming large cardinals, we prove two results. First, we prove that finitely-generated free LDAs with distinct numbers of generators are $\Sigma_1$-elementarily equivalent but not $\Sigma_2$-elementarily equivalent. We also prove a partial structural analogue to Laver's representation of LDAs. We construct an extension of the monogenerated free LDA where application by any fixed element is an elementary embedding of LDAs. We argue that this extension is canonical by demonstrating homogeneity and universality properties. These results also provide additional examples of algebraic properties provable from large cardinals without known proofs from the standard axioms of set theory.