Finite Thurston type orderings on dual braid monoids

TL;DR

This paper introduces a new normal form for dual positive braids called the C-normal form, extending previous forms, and characterizes the order type of the restriction of finite Thurston type orderings to dual braid monoids.

Contribution

It presents the C-normal form for dual positive braids and describes the order structure of the dual braid monoids under Thurston type orderings.

Findings

The C-normal form extends Fromentin's rotating normal form.

The restriction of the ordering to dual positive monoids is well-ordered.

The order type of the restriction is ^{^{n-2}}.

Abstract

For a finite Thurston type ordering < of the braid group B_{n}, we introduce a new normal form of a dual positive braid which we call the C-normal form. This normal form extends Fromentin's rotating normal form and the author's C-normal form of positive braids. Using the C-normal form, we give a combinatorial description of the restriction of the ordering < to the dual braid monoids B_{n}^{+*}. We prove that the restriction to the dual positive monoid (B_{n}^{+*},<) is a well-ordered set of order type \omega^{\omega^{n-2}}.