The well-ordering of dual braid monoids

TL;DR

This paper characterizes the well-ordering of dual braid monoids by introducing a new ordering and proving its equivalence to the standard braid ordering, providing an inductive description and computing its ordinal type.

Contribution

It introduces a new ordering on dual braid monoids and proves its equivalence to the standard braid ordering, enabling an inductive characterization and ordinal computation.

Findings

The dual braid monoids are well-ordered under the new ordering.

The ordinal type of the dual braid monoids is computed.

The new ordering coincides with the standard Dehornoy ordering.

Abstract

We describe the restriction of the Dehornoy ordering of braids to the dual braid monoids introduced by Birman, Ko and Lee: we give an inductive characterization of the ordering of the dual braid monoids and compute the corresponding ordinal type. The proof consists in introducing a new ordering on the dual braid monoid using the rotating normal form of arXiv:0811.3902 [math.GR], and then proving that this new ordering coincides with the standard ordering of braids.