The group of self-distributivity is bi-orderable

Abstract

We prove that the group of left self-distributivity, a cousin of Thompson's group F and of Artin's braid group $B_{\infty }$ that describes the geometry of the identity x(yz) = (xy)(xz), admits a bi-invariant linear ordering. To this end, we define a partial action of this group on finite binary trees that preserves a convenient linear ordering.