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 [special characters omitted] = [special characters omitted] on one generator is representable as the span of an elementary embedding j : V_λ → V _λ (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 [special characters omitted], and give results in the direction of a generalization of this proof to [special characters omitted], the free left distributive algebra on κ generators. The algorithm takes place not in [special characters omitted] but in [special characters omitted]---the result of enlarging [special characters omitted] to freely add a composition operation.

Let D_q = {p ∈ [special characters omitted] : for some r, pr = q} denote the set of left divisors of q. Under the assumption that the division form algorithm for [special characters omitted] terminates, we prove a conjecture of J. Moody: If u, v, p, r ∈ [special characters omitted] with uv = pr and D _u ∩ D_v = ∅ = D_p ∩ D_r, then u = p and v = r.