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On braid words and irreflexivity. (English) Zbl 0793.08007
A left-distributive algebra is a set \({\mathcal B}\) equipped with a binary operation such that \(a(bc)= (ab)(ac)\) for all \(a,b,c\in {\mathcal B}\). The free left-distributive algebra on \(n\) generators is denoted \({\mathcal A}_ n\), and we write \({\mathcal A}\) for \({\mathcal A}_ 1\).
if \(P,Q\in {\mathcal B}\) write \(P<_ L Q\) iff one can write \(P\) as a strict prefix of \(Q\), i.e., \(Q= ((PQ_ 1)\dots)Q_ k\) for some \(Q_ 1,\dots, Q_ k\), \(k\geq 1\). Then a proof that \(<_ L\) is irreflexive (that is, that \(P\neq ((PQ_ 1)\dots)Q_ k\) for all \(P\), \(Q_ 1,\dots,Q_ k\in{\mathcal A})\) on \({\mathcal A}\) was found by R. Laver [Adv. Math. 91, 209-231 (1992)], under large cardinal assumptions, as part of a theorem that \({\mathcal A}\) is isomorphic to a certain algebra of elementary embeddings from set theory.
It was also proved by Laver [loc. cit.] that \(<_ L\) linearly orders \({\mathcal A}\), the part that for all \(P,Q\in {\mathcal A}\) at least one of \(P<_ L Q\), \(P=Q\), \(Q<_ L P\) holds being proved independently and by a different method by P. Dehornoy [J. Pure Appl. Algebra 61, No. 2, 123-146 (1989; Zbl 0686.20041); C. R. Acad. Sci., Paris, Sér. I 309, No. 3, 143-148 (1989; Zbl 0688.20038)]. The linear ordering of \({\mathcal A}\) 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, P. Dehornoy [“Braid groups and left distributive structures”, Preprint (1992)] 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.

MSC:
08B20 Free algebras
03E35 Consistency and independence results
08A50 Word problems (aspects of algebraic structures)
06A05 Total orders
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References:
[1] J.Birman,Braids, Links, and Mapping Class Groups, Annals of Mathematics Studies82 (1975). · Zbl 0305.57013
[2] P. Dehornoy,Free Distributive Groupoids, Journal of Pure and Applied Algebra61 (1989), 123-146. · Zbl 0686.20041 · doi:10.1016/0022-4049(89)90009-1
[3] P. Dehornoy,Sur la structure des gerbes libres, Comptes-Rendus de l’Acad. des Sciences de Paris 309-I (1989), 143-148. · Zbl 0688.20038
[4] P.Dehornoy,Braid Groups and Left Distributive Structures, preprint (1992). · Zbl 0752.08005
[5] R. Laver,The Left Distributive Law and the Freeness of an Algebra of Elementary Embeddings, Advances in Mathematics91 (1992), 209-231. · Zbl 0822.03030 · doi:10.1016/0001-8708(92)90016-E
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