The wellordering on positive braids Original Research Article

This paper studies Artinʹs braid monoids using combinatorial methods. More precisely, we investigate the linear ordering defined by Dehornoy. Laver has proved that the restriction of this ordering to positive braids is a wellordering. In order to study this order, we develop a natural wellordering much less-than on the free monoid on infinitely many generators by representing words as trees. Our construction leads to a (new) normal form for (positive) braids. Our main result is that the restriction of our order much less-than to the normal braid words coincides with the restriction of Dehornoyʹs ordering to positive braids. Our method gives an alternative proof of Laverʹs result using purely combinatorial arguments and gives the order type, namely ωωω.