Dehornoy order

In the mathematical area of braid theory, the Dehornoy order is a left-invariant total order on the braid group, found by Patrick Dehornoy (1994, 1995).

Dehornoy's original discovery of the order on the braid group used huge cardinals, but there are now several more elementary constructions of it.

Definition

Suppose that σ₁, ..., σ_n−1 are the usual generators of the braid group B_n 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, ..., σ_n−1 and their inverses, such that for some i the word contains σ_i but does not contain σ
j^±1 for j < i nor σ
i⁻¹.

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.

Properties

The Dehornoy order is a well-ordering when restricted to the monoid generated by σ₁, ..., σ_n−1.

References

• Dehornoy, Patrick (1994), "Braid groups and left distributive operations", 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 

Further reading

• 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