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-sys...

• Type: Journal
• Year: 2000
• Where: IJAC
• Authors: Patrick Dehornoy
• Keywords: Concrete Realization, Free Ld-system, IJAC 2000, Left Self-distributivity Identity