## 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)

### Citations:

Zbl 0837.20048; Zbl 0839.20051
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### References:

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