in mathematics, the Dehornoy order is a left-invariant total order

Total order

In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...

 on the braid group

Braid group

In mathematics, the braid group on n strands, denoted by Bn, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group Sn. Here, n is a natural number; if n > 1, then Bn is an infinite group...

, found by .

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, so that for some i the word contains σ_i but does not contain σ for j ≤ i. 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 ba⁻¹ ∈ 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.