On finite Thurston type orderings of braid groups

We prove that for any finite Thurston-type ordering ${<}_T$ on the braid group\ $B_n$, the restriction to the positive braid monoid $(B_n^{+},{<}_T)$ is a\ well-ordered set of order type ${\omega }^{{\omega }^{n-2}}$. The proof uses a combi\ natorial description of the ordering ${<}_T$. Our combinatorial description is \ based on a new normal form for positive braids which we call the $\text{C}$-normal fo\ rm. It can be seen as a generalization of Burckel’s normal form and Dehornoy’s \ $\mathrm{\Phi }$-normal form (alternating normal form).