In the mathematical area of braid theory, the Dehornoy order is a left-invariant total order on the braid group, found by Patrick Dehornoy.
[1][2] Dehornoy's original discovery of the order on the braid group used huge cardinals, but there are now several more elementary constructions of it.
σ
are the usual generators of the braid group
σ
-positive word to be a braid that admits at least one expression in the elements
σ
σ
and their inverses, such that the word contains
of positive elements in the Dehornoy order is defined to be the elements that can be written as a
We have: These properties imply that if we define
then we get a left-invariant total order on the braid group.
because the braid word
-positive, but, by the braid relations, it is equivalent to the
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
, 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
{\displaystyle LD:x(yz)=(xy)(xz),}
a property that is a priori unconnected with large cardinals.
[4][5] In 1992, Dehornoy produced an example of an acyclic shelf by introducing a certain groupoid
that captures the geometrical aspects of the
As a result, an acyclic shelf was constructed on the braid group
, and this implies the existence of the braid order directly.
[2] 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.