The free one-generated left distributive algebra: basics and a simplified proof of the division algorithm

The left distributive law is the law a · ( b· c ) = ( a·b ) · ( a· c ). Left distributive algebras have been classically used in the study of knots and braids, and more recently free left distributive algebras have been studied in connection with large cardinal axioms in set theory. We provide a survey of results on the free left distributive algebra on one generator, A, and a new, simplified proof of the existence of a normal form for terms in A. Topics included are: the confluence of A, the linearity of the iterated left division ordering < L of A, the connections of A to the braid groups, and an extension P of A obtained by freely adding a composition operation. This is followed by a simplified proof of the division algorithm for P, which produces a normal form for terms in A and is a powerful tool in the study of A.