On braid words and irreflexivity

Abstract

A left-distributive algebra is a set B equipped with a binary operation (here written as concatenation) such that a(bc) = (ab)(ac) for all a, b, c ∈ B. The free left-distributive algebra on n letters is denoted An, and we write A for A1 If P,Q ∈ B write P <L Q iff one can write P as a strict prefix of Q, i.e., Q = ((PQ1) . . .)Qk for some Q1, . . . , Qk, k ≥ 1. Then a proof that <L is irreflexive (that is, that P 6= ((PQ1) . . .)Qk for all P,Q1, . . . , Qk ∈ A) on A was found by R. Laver ([Lav 92]), under large cardinal assumptions, as part of a theorem that A is isomorphic to a certain algebra of elementary embeddings from set theory. It was also proved in [Lav 92] that <L linearly orders A, the part that for all P,Q ∈ A at least one of P <L Q,P = Q,Q <L P holds being proved independently and by a different method by P. Dehornoy ([Deh 89a, Deh 89b]). The linear ordering of A gives left cancellation, the solvability of the word problem, and other consequences. Left open was whether irreflexivity, and hence the linear ordering, can be proved in ZFC. Recently, Dehornoy ([Deh 92]) has found such a proof, involving an extension of the infinite braid group but without invoking axioms extending ZFC. The purpose of this note is to prove the result without the additional machinery of this extended group, and at shorter length.