Construction of Self-Distributive Operations and Charged Braids

Starting from a certain monoid that describes the geometry of the left self-distributivity identity, we construct an explicit realization of the free left self-distributive system on any number of generators. This realization lives in the charged braid group, an extension of Artin’s braid group B∞ with a simple geometrical interpretation. AMS Classification: 20N02, 20F36, 08B20. Constructing examples of operations that satisfy a given identity and, in particular, constructing a concrete realization of the free objects in the associated equational variety is an obviously difficult task, for which no uniform method exists. Here we consider these questions in the case of the left self-distributivity identity x ∗ (y ∗ z) = (x ∗ y) ∗ (x ∗ z). (LD) Due to its connection with set theory [13] [14] [8] and knot theory [1] [9] [10], this identity has received much attention in the recent years. A binary system made of a set equipped with a left self-distributive operation will be called an LD-system. Thus the question we consider here is the construction of concrete realizations of free LD-systems. The first result in this direction has been obtained by R. Laver in [13]: if j is a non-trivial elementary embedding of a rank into itself, then the family of all iterations of j equipped with the operation of applying an embedding to another one is a free LD-system. This solution however is not completely satisfactory, as the existence of the object it relies upon, namely a non-trivial elementary embedding of a rank into itself, is an unprovable set-theoretical axiom, one for which even a relative consistency result cannot be proved. Subsequently, we have shown in [6] how to deduce from the general study of the identity (LD) the existence of a left self-distributive operation on Artin’s braid group B∞. This construction provides a concrete realization for the free LD-system on one generator inside B∞, leading in particular to an efficient solution for the word problem of the identity (LD)—and to new results about braids, such as the orderability of this group and a new efficient algorithm for its word problem. On the other hand, Larue has shown in [12] how to extend ‘by hand’ the braid group B∞ so as to obtain a realization for the free LD-system on any number of generators. In this paper, we show how to extend the analysis of [6] so as to include the case of several generators. This leads to introducing an extension of Artin’s braids that we call charged braids. The precise result is as follows.