Transfinite Braids and Left Distributive Operations

We complete Artin’s braid group B∞ with some limit points (with respect to a natural topology), thus obtaining an extended monoid where new left self-distributive operations are defined. This construction provides an effective realization for some free algebraic system involving a left distributive operation and a compatible associative product. AMS Subject classification: 20F36, 20N02 Here we investigate a rather natural extension of the usual notion of a braid, namely that obtained by considering two infinite series of strands rather than just one as in the case of Artin’s braid group B∞. The corresponding group is very large, but it turns out that a certain submonoid EB∞ of this group can be described very simply as a completion of B∞. This completion is obtained by adding upper bounds to some sequences that are increasing for a canonical linear ordering, or, equivalently, that are Cauchy sequences with respect to the associated topology. The basic study of the monoid EB∞ is the content of the first two sections. The sequel of the paper is devoted to the study of left self-distributive operations on the monoid EB∞, i.e., of binary operations ∗ that satisfy the algebraic identity x ∗ (y ∗ z) = (x ∗ y) ∗ (x ∗ z). (LD) There is nothing gratuitous in this task. Indeed it is well known that deep connections exist between braids (and knots) and self-distributive structures [2], [15], [16], [17], [9] — as well as between the latter and set theory [5], [20], [21], [13] (see also [3] for another relation between braid groups and distributivity). In particular we recall in Section 2 that each new example of an LD-system (defined as a set equipped with a left self-distributive operation) can potentially bring new information about braids using the formalism of braid colourings. So constructing “concrete” LD-systems is a natural aim. It is both easy and difficult. It is easy, since there are very common examples, like lattices, or groups equipped with the conjugacy operation x ∗ y = xyx−1. But these examples