Free augmented LD-systems

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 ◦ y) ∗ z = x ∗ (y ∗ z) and x ∗ (y ◦ z) = (x ∗ y) ◦ (x ∗ z). We solve the word problem of the ALD laws, and prove that every element in the parenthesized braid group B• of [2, 3, 5, 6] generates a free ALD-system of rank 1, thus getting a concrete realization of the latter structure. Define an LD-system to be an algebraic system made of a set S equipped with a binary operation ∗ that satisfies the left self-distributivity law (LD) x ∗ (y ∗ z) = (x ∗ y) ∗ (x ∗ z). Classical examples include groups equipped with their conjugacy operation x ∗ y = xyx−1, and lattices with their inf or sup operation. Less classical examples have appeared in Set Theory with the iterations of elementary embeddings [12], and in Low Dimensional Topology where (LD) provides an algebraic translation of Reidemeister move III [10, 13, 9]. A rich theory has been developed for LD-systems [4]. In particular, it is known that there exists on Artin’s braid group B∞ an LD-operation ∗ such that the ∗-closure of any braid is a free LD-system of rank 1—which provides a concrete realization of the latter structure. Many examples of LD-systems turn out to be equipped with a second operation connected in various ways with the self-distributive operation. In the typical case of group conjugacy, using ◦ for the group product, the following mixed identities are satisfied x ∗ (y ∗ z) = (x ◦ y) ∗ z, (ALD1) x ∗ (y ◦ z) = (x ∗ y) ◦ (x ∗ z). (ALD2) When we add the identity x ◦ y = (x ∗ y) ◦ x, the associativity of ◦ and the existence of a unit, one obtains the structure of an LD-monoid, which is investigated in Chapter XI of [4] (and in [7, 8] under the name of LD-algebra). It is easy to verify that all LD-systems cannot be enriched into LD-monoids. In particular, this is the case for the above mentioned LD-structure on B∞, for which there can exist no second operation verifying (ALD1). In [6], building on earlier approaches of [2, 3, 5], a new group B• extending both Artin’s braid group B∞ and R.Thompson’s group F is investigated. This group is called the parenthesized braid group, as its elements can be naturally interpreted using braid diagrams in which the strands come grouped into blocks that can be encoded in parenthesized words. It is shown that the LD-structure of B∞ extends to B• and that the latter can be completed with a second operation that satisfies the above identities (ALD1) and (ALD2)—but none of the further 1991 Mathematics Subject Classification. 20N02, 20F36.