Free augmented LD-systems

Abstract

Define an augmented LD-system, or ALD-system, to be a set equipped with two binary operations, one satisfying the left self-distributivity law $x\ast (y\ast z)=(x\ast y)\ast (x\ast z)$ and the other satisfying the mixed laws $(xoy)\ast z=x\ast (y\ast z)$ and $x\ast (yoz)=(x\ast y)o(x\ast z)$. We solve the word problem of the ALD laws, and prove that every element in the parenthesized braid group $B\_\bullet$ of [Bri1, Dhb, Dhe] generates a free ALD-system of rank 1, thus getting a concrete realization of the latter structure.