Algebra Universalis February 14, 2002 13:45 1739u F01017 (1739u), pages 65–68 Page 65 Sheet 1 of 4 Algebra univers. 47 (2002) 65–68 0002-5240/02/010065 – 04 $1.50 + 0.20/0 c Birkh¨auser Verlag, Basel, 2002 Algebra Universalis Well-foundedness conditions connected with left-distributivity Richard Laver and John A. Moody Abstract. We state a problem about free left-distributive algebras. If u and w are mem- bers of such an algebra write u< L w if w = (((uu 0 )u 1 )...)un for some u 0 , ··· ,un. A conjecture about left-division in such algebras is given; it entails a normal form and that for every w the set of left divisors of w is well-ordered under < L . Let A k be the free left-distributive algebra on k-many generators, where k is a cardinal and the left-distributive law is the law a(bc)=(ab)(ac). The purpose of this paper is the describe a conjecture connected with left division in A k , and some propositions related to it. This problem is one of a number of well-foundedness questions about free left-distributive algebras. Here are some basic facts (see [4] and [8], section 2 for respectively an account and a summary). Fix a set S of cardinality k and let T be the set of S-terms in the language of one binary operation. For τ 0 ,τ 1 ∈ T , τ 1 is the result of a forward transformation on τ 0 if and only if τ 1 can be obtained from τ 0 by replacing a subterm of the form a(bc) by (ab)(ac). Then A k = T/≡, where τ ≡ σ iff σ can be obtained from τ by a sequence of forward transformations and/or their inverses. Write [τ ] for the equivalence class of τ . Let τ 0 → τ 1 mean that τ 1 is obtainable from τ 0 by a sequence of forward transformations. Then A k is confluent [1], that is, if τ 0 ≡ τ 1 then there’s a τ such that τ 0 → τ and τ 1 → τ . For a, b ∈A k , write a | b iff for some c, ac = b. Write a< L b if a is an iterated left divisor of b; b = ((aa 0 )a 1 ) ... )a n for some n ≥ 0. Then < L is a partial ordering: transitivity is immediate, and irreflexivity (a < L a), first shown in [6] as a corollary to a large cardinal axiom of set theory, was then shown in [3] without it—see [5] for a shorter proof of irreflexivity. And on A = A 1 , < L is a linear ordering (modulo irreflexivity this was shown in different ways in [3] and [6]). On A k (k> 1), < L is not linear, but any linear ordering of S naturally induces a linear ordering of A k which extends < L [2]. Finally, A k satisfies left cancellation. For k = 1 this follows from the linearity of < L and the fact that Presented by R. W. Quackenbush. Received March 25, 2001; accepted in final form April 10, 2001. 2000 Mathematics Subject Classification: 20F36, 20N02. Key words and phrases : Left-distributivity, braid groups. The work of the first author was supported by NSF Grant DMS 9972257. 65