Left division in the free left distributive algebra on many generators

Left distributive algebras arise in the study of classical structures such as groups, knots, and braids, as well as more exotic objects like large cardinals. A long-standing open question is whether the set of left divisors of every term in the free left distributive algebra on any number of generators is well-ordered. A conjecture of J. Moody describes a halting condition for descending sequences of left divisors in the free left distributive algebra on an arbitrary number of generators. In this paper we present progress toward an affirmative answer to the open question mentioned above, namely we prove that the many generator form of Moody’s conjecture holds if each member of the free left distributive algebra has a “division form” representation in an expanded algebra.