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 * (y * z) = (x * y) * (x * z) and the other satisfying the mixed laws (x o y) * z = x * (y * z) and x * (y o z) = (x * y) o (x * z). We solve the word problem of the ALD laws, and prove that every element in the parenthesized braid group B\ of [Bri1, Dhb, Dhe] generates a free ALD-system of rank 1, thus getting a concrete realization of the latter structure.
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