Dehornoy order

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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, ..., 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 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, ..., σn−1.

References

Further reading

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