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; Tesson, Emilie

Algebraic Combinatorics, Volume 2 (2019) no. 4, p. 683-709

Abstract

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.

Received : 2018-03-08
Accepted : 2018-11-28
Published online : 2019-08-01
DOI : https://doi.org/10.5802/alco.52
Classification: 05E15, 20M05, 20E22, 68Q42
Keywords: presented monoid, divisibility relation, simple elements, Thompson’s group, braid group, normal form, Garside element, directed animal

@article{ALCO_2019__2_4_683_0,
     author = {Dehornoy, Patrick and Tesson, Emilie},
     title = {Garside combinatorics for Thompson's monoid $F^+$ and a hybrid with the braid monoid $B\_{\infty }^{+}$},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {2},
     number = {4},
     year = {2019},
     pages = {683-709},
     doi = {10.5802/alco.52},
     language = {en},
     url = {https://alco.centre-mersenne.org/item/ALCO_2019__2_4_683_0}
}

Dehornoy, Patrick; Tesson, Emilie. Garside combinatorics for Thompson’s monoid $F^+$ and a hybrid with the braid monoid $B_{\infty }^{+}$. Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 683-709. doi : 10.5802/alco.52. alco.centre-mersenne.org/item/ALCO_2019__2_4_683_0/

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