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Braid group actions on left distributive structures, and well orderings in the braid groups. (English) Zbl 0859.20029
Let \(B_n\) be the Artin braid group with standard generators \(\sigma_1,\dots,\sigma_{n-1}\), and set \(B_\infty=\varinjlim B_n\). P. Dehornoy [Trans. Am. Math. Soc. 345, No. 1, 115-150 (1994; Zbl 0837.20048)] introduced a linear ordering \(<\) in \(B_\infty\) characterized by the property that \(\alpha<\beta\) if and only if \(\alpha^{-1}\beta\) can be represented as a nonempty braid word \(w=\sigma^{\pm 1}_{i_1}\dots\sigma^{\pm1}_{i_m}\) such that the generator with smallest subscript appearing in \(w\) occurs only positively. In this paper, the author proves that the rule \(x^{\sigma_i}_i=x_{i-1}\), \(x^{\sigma_i}_{i-1}=x_{i-1}x_i\), \(x^{\sigma_i}_j=x_j\) (\(j\neq i,i-1\)) defines a “partial action” of \(B_\infty\) on a certain subset of the “decreasing division forms” in the free left distributive algebra on \(\{x_0,x_1,x_2,\dots\}\). As consequences, the linear order \(<\) of \(B_\infty\) turns out to extend the partial ordering of E. A. Elrifai and H. R. Morton [Q. J. Math., Oxf. II. Ser. 45, No. 180, 479-497 (1994; Zbl 0839.20051)] and to give a well ordering on the set \(B^+_n\) of positive braids (where all generators occur only positively) for any fixed \(n\).

MSC:
20F36 Braid groups; Artin groups
06A05 Total orders
08A50 Word problems (aspects of algebraic structures)
17A50 Free nonassociative algebras
17A30 Nonassociative algebras satisfying other identities
20N02 Sets with a single binary operation (groupoids)
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