Garside combinatorics for Thompson’s monoid $F^+$ and a hybrid with the braid monoid $B_{\infty }^{+}$

Dehornoy, Patrick; Tesson, Emilie

doi:10.5802/alco.52

Garside combinatorics for Thompson’s monoid

F^{+}

and a hybrid with the braid monoid

B_{\infty}^{+}

Dehornoy, Patrick ¹ ; Tesson, Emilie ¹

¹ Université de Caen Laboratoire de Mathématiques Nicolas Oresme UMR 6139 14032 Caen, France

Algebraic Combinatorics, Tome 2 (2019) no. 4, pp. 683-709.

Résumé

On the model of simple braids, defined to be the left divisors of Garside’s elements $Δ_{n}$ in the monoid $B_{\infty}^{+}$ , we investigate simple elements in Thompson’s monoid $F^{+}$ and in a larger monoid $H^{+}$ that is a hybrid of $B_{\infty}^{+}$ and $F^{+}$ : in both cases, we count how many simple elements left divide the right lcm of the first $n - 1$ atoms, and characterize their normal forms in terms of forbidden factors. In the case of $H^{+}$ , a generalized Pascal triangle appears.

Reçu le : 2018-03-08
Accepté le : 2018-11-28
Publié le : 2019-08-01

Zbl

DOI : 10.5802/alco.52

Classification : 05E15, 20M05, 20E22, 68Q42
Mots clés : presented monoid, divisibility relation, simple elements, Thompson’s group, braid group, normal form, Garside element, directed animal

Affiliations des auteurs :

Dehornoy, Patrick ¹ ; Tesson, Emilie ¹

¹ Université de Caen Laboratoire de Mathématiques Nicolas Oresme UMR 6139 14032 Caen, France

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     title = {Garside combinatorics for {Thompson{\textquoteright}s} monoid $F^+$ and a hybrid with the braid monoid $B_{\infty }^{+}$},
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     pages = {683--709},
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Dehornoy, Patrick; Tesson, Emilie. Garside combinatorics for Thompson’s monoid $F^+$ and a hybrid with the braid monoid $B_{\infty }^{+}$. Algebraic Combinatorics, Tome 2 (2019) no. 4, pp. 683-709. doi : 10.5802/alco.52. http://www.numdam.org/articles/10.5802/alco.52/

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