Geodesic
On braid groups
Sbornik. Mathematics, Tome 192 (2001) no. 5, pp. 693-703

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Artin's braid groups are studied from the viewpoint of right-ordered groups. A right order is constructed such that the cone of elements $\geqslant 1$ is finitely generated as a monoid. The structure of ideals of this cone is determined, and it turns out to be quite specific and impossible for linearly ordered groups. It is also proved that no linear order on the pure braid subgroup can be extended to a right order on the whole of the braid group. 
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T. V. Dubrovina; N. I. Dubrovin. On braid groups. Sbornik. Mathematics, Tome 192 (2001) no. 5, pp. 693-703. http://geodesic.mathdoc.fr/item/SM_2001_192_5_a2/