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Braid groups and left distributive operations. (English) Zbl 0837.20048
Summary: The decidability of the word problem for the free left distributive law is proved by introducing a structure group which describes the underlying identities. This group is closely connected with Artin’s braid group $$B_\infty$$. Braid colourings associated with free left distributive structures are used to show the existence of a unique ordering on the braids which is compatible with left translation and such that every generator $$\sigma_i$$ is preponderant over all $$\sigma_k$$ with $$k > i$$. This ordering is a linear ordering.

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