A well-ordering of dual braid monoids

Jean Fromentin

doi:10.1016/j.crma.2008.05.001

Group Theory

A well-ordering of dual braid monoids

Presented by: Étienne Ghys

Jean Fromentin ¹

¹ Laboratoire de mathématiques Nicolas-Oresme, UMR 6139 CNRS, Université de Caen BP 5186, 14032 Caen, France

Comptes Rendus. Mathématique, Volume 346 (2008) no. 13-14, pp. 729-734.

Abstract
Résumé

Let $B_{n}^{+ *}$ denote the dual braid monoid on n strands, i.e., the submonoid of the braid group $B_{n}$ consisting of the braids that can be expressed as positive words in the Birman–Ko–Lee generators. We introduce a new normal form on $B_{n}^{+ *}$ , which is based on expressing every braid of $B_{n}^{+ *}$ in terms of a certain finite sequence of braids of $B_{n - 1}^{+ *}$ . We deduce an inductive characterization of the Dehornoy ordering of dual braid monoids, and explicitly compute the associated order types.

Soit $B_{n}^{+ *}$ le monoïde de tresses dual sur n brins, c'est-à-dire le sous-monoïde du groupe de tresses $B_{n}$ formé par les tresses ayant une expression positive en les générateurs de Birman–Ko–Lee. Nous introduisons une nouvelle forme normale, dite cyclante, sur $B_{n}^{+ *}$ . Cette forme normale est basée sur une décomposition de chaque tresse de $B_{n}^{+ *}$ en termes d'une suite de tresses de $B_{n - 1}^{+ *}$ . Nous en déduisons une caractérisation inductive de l'ordre de Dehornoy sur les monoïdes de tresses duaux, et calculons explicitement les types d'ordre associés.

Received: 2008-01-24
Accepted: 2008-04-30
Published online: 2008-06-12

DOI: 10.1016/j.crma.2008.05.001

Author's affiliations:

Jean Fromentin ¹

¹ Laboratoire de mathématiques Nicolas-Oresme, UMR 6139 CNRS, Université de Caen BP 5186, 14032 Caen, France

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     title = {A well-ordering of dual braid monoids},
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Jean Fromentin. A well-ordering of dual braid monoids. Comptes Rendus. Mathématique, Volume 346 (2008) no. 13-14, pp. 729-734. doi : 10.1016/j.crma.2008.05.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.05.001/

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