A Normal Form for the Free Left Distributive Law

We construct a new normal form for one variable terms up to left distributivity. The proof that this normal form exists for every term is considerably simpler than the corresponding proof for the forms previously introduced by Richard Laver. In particular the determination of the present normal form can be made in a primitive recursive way. Throughout the paper W denotes the set of all wellformed terms constructed using a single variable a and a single binary operator •, i.e. the free algebra generated by a. Practically we shall use right Polish notation, thus writing PQ• for the product of P and Q. Now we denote by =LD the least congruence on W which forces the left distributivity identity PQR•• =LD PQ•PR•• (LD) The quotient W/=LD is the free left distributive algebra (LD-algebra) generated by a. The study of free LD-algebras has revealed interesting connections both with the set theory of large cardinals (see [10], [12], [4], [7], [8]) and with the topology of braids (see [5], [9], [6]). In particular the decidability of the relation =LD, i.e. the word problem for the standard presentation of the free LD-algebra with one generator, proved to be a rather delicate question. It has been solved independently in [3] and [10] assuming some auxiliary assumption which Laver in [10] deduced from a very strong logical assumption. Subsequently this assumption was eliminated in [5], and the proof was completed within elementary arithmetic resulting in an exponential complexity for the relation =LD.