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. 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 formula_1 are the usual generators of the braid group formula_2 on formula_3 strings. Define a formula_4-positive word to be a braid that admits at least one expression in the elements formula_1 and their inverses, such that the word contains formula_4, but does not contain formula_7 nor formula_8 for formula_9 of positive elements in the Dehornoy order is defined to be the elements that can be written as a formula_4-positive word for some formula_11. We have:
These properties imply that if we define formula_17 as formula_18 then we get a left-invariant total order on the braid group. For example, formula_19 because the braid word formula_20 is not formula_21-positive, but, by the braid relations, it is equivalent to the formula_21-positive word formula_23, which lies in formula_24.
History.
Set theory introduces the hypothetical existence of various "hyper-infinity" notions such as large cardinals. In 1989, it was proved that one such notion, axiom formula_25, implies the existence of an algebraic structure called an acyclic shelf which in turn implies the decidability of the word problem for the left self-distributivity law formula_26 a property that is a priori unconnected with large cardinals.
In 1992, Dehornoy produced an example of an acyclic shelf by introducing a certain groupoid formula_27 that captures the geometrical aspects of the formula_28 law. As a result, an acyclic shelf was constructed on the braid group formula_29, which happens to be a quotient of formula_27, and this implies the existence of the braid order directly. Since the braid order appears precisely when the large cardinal assumption is eliminated, the link between the braid order and the acyclic shelf was only evident via the original problem from set theory.