Braids and Self-Distributivity (Relié)

The aim of this book is to present recently discovered connections between Artin's braid groups and left selfdistributise systems. which are sets equipped Midi a binary operation satisfying the identity x(yz) = (xy)(xz). Order properties are crucial. In the 1980s new examples of left self-distributive systems were discovered using unprovable axioms of set theory. and purely algebraic statements were deduced.
The quest for elementary proofs of these statements led to a general theory of selfdistributivity centered on a certain group that captures the geometrical properties of this identity. This group happens to be closely connected with Artin's braid groups, and new properties of the braids naturally arose as an application, in particular the existence of a left invariant linear order, which subsequently received alternative topological constructions.
The test proposes a first synthesis of this area of research. Three domains are considered here, namely braids, self-distributive systems, and set theory. Although not a comprehensive tonne on these subjects, the exposition is self-contained, and a number of basic results are established. In particular, the first chapters include a rather complete algebraic study of Artin's braid groups.