| We investigate the structure of free left distributive algebras. The left distributive algebras (algebras satisfying the LD-law a( bc) = (ab)(ac)) which occur in classical math are not free. The free left distributive algebra A1 = A on one generator is representable as the span of an elementary embedding j : Vlambda → V lambda (Laver) and as a subset of the braid group under a certain operation (Dehornoy). We present a new, direct proof of a division theorem for the elements of A , and give results in the direction of a generalization of this proof to Ak , the free left distributive algebra on kappa generators. The algorithm takes place not in Ak but in Pk ---the result of enlarging Ak to freely add a composition operation.; Let Dq = {lcub}p ∈ Ak : for some r, pr = q{rcub} denote the set of left divisors of q. Under the assumption that the division form algorithm for Pk terminates, we prove a conjecture of J. Moody: If u, v, p, r ∈ Ak with uv = pr and D u ∩ Dv = empty = Dp ∩ Dr, then u = p and v = r. |
