Braid groups and left distributive operations

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 ∞ {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 σ i {\sigma _i} is preponderant over all σ k {\sigma _k} with k > i k > i . This ordering is a linear ordering.