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The wellordering on positive braids. (English) Zbl 0958.20032
Summary: This paper studies Artin’s 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 \(\ll\) 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 \(\ll\) to the normal braid words coincides with the restriction of Dehornoy’s ordering to positive braids. Our method gives an alternative proof of Laver’s result using purely combinatorial arguments and gives the order type, namely \(\omega\).

MSC:
20F36 Braid groups; Artin groups
20M05 Free semigroups, generators and relations, word problems
68R15 Combinatorics on words
06A05 Total orders
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References:
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