Braid group actions on left distributive structures, and well orderings in the braid groups (Q1917384)

Let \(B_n\) be the Artin braid group with standard generators \(\sigma_1,\dots,\sigma_{n-1}\), and set \(B_\infty=\varinjlim B_n\). \textit{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 \textit{E. A. Elrifai} and \textit{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 \(n\).