Does the braid group have a presentation based entirely upon the Dehornoy order?
Let $B_n$ be the braid group $B_3$ for now.
Let $\leq$ be the Dehornoy order on the group
The let $P_i(x,y,z,\ldots)$ be $i$ distinct predicates of the form $ x_i^{p_i}\cdot y_i^{q_i}\ldots\leq z_i^{r_i}\ldots$ where $x,y,z\ldots\in B_n$
Does the Braid group $B_3$ have a presentation $B_n=\langle x,y,z,\ldots\mid P_0\land\ldots P_i \rangle$
More generally I am asking whether the presentation of a group can be entirely order-theoretical.