Braid groups and left distributive operations

Dehornoy, Patrick

doi:10.1090/S0002-9947-1994-1214782-4

Braid groups and left distributive operations
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by Patrick Dehornoy

Trans. Amer. Math. Soc. 345 (1994), 115-150

DOI: https://doi.org/10.1090/S0002-9947-1994-1214782-4

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Abstract:

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.

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