Left-distributive algebras

Thomas Jech
The Pennsylvania State University

A left-distributive algebra is a set with one binary operation satisfying the left distributive law
a(bc)=(ab)(ac)
(the operation of conjugation in a group is an example).

The subject of my lectures is the free left-distributive algebra on one generator. This algebra arises in the study of elementary embeddings in set theory, and the word problem for it was first solved by R. Laver using the theory of large cardinals. Subsequently, P. Dehornoy found an elementary solution of the problem by giving a representation in the braid group.

The free algebra can also be expressed as a limit of finite algebras. These cyclic algebras have been studied by R. Dougherty and myself. The statement that the limit algebra is free is equivalent to a simple statement of elementary number theory, and follows from Laver's work (using large cardinals). It is however unprovable by elementary methods (viz. in primitive recursive arithmetic), and it is an open problem whether the large cardinal assumptions are necessary.

The study of cyclic left-distributive algebras leads to some interesting questions on computability and complexity.