On finite Thurston-type orderings of braid groups

We give an combinatorial description of a finite Thurston type ordering <T of the positive braid monoid B + n by introducing the C-normal form. The C-normal form can be seen as a generalization of Burckel’s normal form and Dehornoy’s Φ-normal form. Using this description, we determine the order type of well-ordered set (Bn, <T ) for all finite Thurston type orderings <T .