Dehornoy's original discovery of the order on the braid group used huge cardinals, but there are now several more elementary constructions of it.
Suppose that σ1, ..., σn−1 are the usual generators of the braid group Bn on n strings.
The set P of positive elements in the Dehornoy order is defined to be the elements that can be written as word in the elements σ1, ..., σn−1 and their inverses, such that for some i the word contains σi but does not contain
j±1 for j < i nor σ
The set P has the properties PP ⊆ P, and the braid group is a disjoint union of P, 1, and P−1. These properties imply that if we define a < b to mean a−1b ∈ P then we get a left-invariant total order on the braid group.
The Dehornoy order is a well-ordering when restricted to the monoid generated by σ1, ..., σn−1.
- Dehornoy, Patrick (1994), "Braid groups and left distributive operations" (PDF), Transactions of the American Mathematical Society, 345 (1): 115–150, doi:10.2307/2154598, ISSN 0002-9947, MR 1214782
- Dehornoy, Patrick (1995), "From large cardinals to braids via distributive algebra", Journal of Knot Theory and its Ramifications, 4 (1): 33–79, doi:10.1142/S0218216595000041, ISSN 0218-2165, MR 1321290
- Dehornoy, Patrick; Dynnikov, Ivan; Rolfsen, Dale; Wiest, Bert (2002), Why are braids orderable? (PDF), Panoramas et Synthèses, 14, Paris: Société Mathématique de France, ISBN 978-2-85629-135-1, MR 1988550
- Dehornoy, Patrick; Dynnikov, Ivan; Rolfsen, Dale; Wiest, Bert (2008), Ordering braids, Mathematical Surveys and Monographs, 148, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4431-1, MR 2463428