We shall investigate several classes of left-distributive algebras that behave like algebras of elementary embeddings including permutative LD-systems, locally Laver-like LD-systems, and generalized Laver tables. In these algebras, there is a notion of a critical point, a composition operation, and the notion of equivalence up to a certain critical point. Furthermore, the locally Laver-like LD-systems are used to generate and classify generalized Laver tables. After discussing the general theory of these algebras, we shall show that there exists generalized Laver tables which cannot arise from algebras of elementary embeddings. We shall then give a framework that allows us to construct from rank-into-rank embeddings finite algebras that satisfy the distributivity identity $x*f(x_{1},…,x_{n})=f(x*x_{1},…,x*x_{n})$ where $(X,*)$ is a left-distributive algebra.

We show that the isomorphism problems for left distributive algebras, racks, quandles, and keis are as complex as possible in the sense of Borel reducibility. These various kinds of algebraic structure are important for their connections with the theory of knots, links and braids, and in particular, Joyce showed that quandles could be used as complete invariants for tame knots. However, quandles have heuristically seemed to be unsatisfactory invariants. Our result confirms this view, showing that from a set-theoretic perspective, classifying tame knots by quandles replaces one problem with (a special case of) a harder problem.

The Laver tables are finite self-distributive algebras generated by one element that approximate the free left-distributive algebra on one generator if a rank-into-rank cardinal exists. We shall generalize the notion of a Laver table to a class of locally finite self-distributive algebraic structures with an arbitrary number of generators. These generalized Laver tables emulate algebras of rank-into-rank embeddings with an arbitrary number of generators modulo some rank. Furthermore, if there exists a rank-into-rank cardinal, then the free left-distributive algebras on any number of generators can be embedded in a canonical way into inverse limits of generalized Laver tables. As with the classical Laver tables, the reduced generalized Laver tables can be given an associative operation that is analogous to the composition of elementary embeddings and satisfies the same identities that algebras of elementary embeddings are known to satisfy. Furthermore, the notion of the critical point also holds in these generalized Laver tables as well even though generalized Laver tables are locally finite or finite. While the only classical Laver tables are the tables of cardinality $2^{n}$, the finite generalized laver tables occur much more frequently and many generalized Laver tables can be constructed from the classical Laver tables. We shall give some results that allow one to quickly compute the self-distributive operation in a certain class of generalized Laver tables.

Here are the slides.

The speaker will continue to discuss the properties of rank-into-rank embeddings and their connections to the study of the tower of finite left-distributive algebras known as Laver Tables.

In the early 1990’s Richard Laver discovered a deep and striking correspondence between critical sequences of rank-to-rank embeddings and finite left distributive algebras on integers. Each $A_n$ in the tower of finite algebras (commonly called the Laver Tables) can be defined purely algebraically, with no reference to the elementary embeddings, and yet there are facts about the Laver tables that have only been proven from a large cardinal assumption. We present here some of Laver’s foundational work on the algebra of critical sequences of rank-to-rank embeddings and related algebraic work of Laver’s and the author’s, describe how the finite algebras arise from the large cardinal embeddings, and mention several related open problems.