Still another approach to the braid ordering

We develop a new approach to the linear ordering of the braid group $B\mathrm{\_}n$, based on investigating its restriction to the set $\text{Div}(\mathrm{\Delta }\mathrm{\_}{n}^d)$ of all divisors of $\mathrm{\Delta }\mathrm{\_}{n}^d$ in the monoid $B\mathrm{\_}{\mathrm{\infty }}^{+}$, i.e., to positive $n$-braids whose normal form has length at most $d$. In the general case, we compute several numerical parameters attached with the finite orders $(\text{Div}(\mathrm{\Delta }\mathrm{\_}{n}^d),<)$. In the case of 3 strands, we moreover give a complete description of the increasing enumeration of $(\text{Div}(\mathrm{\Delta }\mathrm{\_}{3}^d),<)$. We deduce a new and specially direct construction of the ordering on $B\mathrm{\_}3$, and a new proof of the result that its restriction to $B\mathrm{\_}{3}^{+}$ is a well-ordering of ordinal type ${\omega }^{\omega }$.