ALGEBRAIC COMBINATORICS

no. 4
Garside combinatorics for Thompson’s monoid F + and a hybrid with the braid monoid B ∞ +
Dehornoy, Patrick1; Tesson, Emilie1
1 Université de Caen Laboratoire de Mathématiques Nicolas Oresme UMR 6139 14032 Caen, France
Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 683-709.

On the model of simple braids, defined to be the left divisors of Garside’s elements Δ n in the monoid B ∞ + , we investigate simple elements in Thompson’s monoid F + and in a larger monoid H + that is a hybrid of B ∞ + 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:
Accepted:
Published online:
DOI: 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
Dehornoy, Patrick 1; Tesson, Emilie 1

1 Université de Caen Laboratoire de Mathématiques Nicolas Oresme UMR 6139 14032 Caen, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2019__2_4_683_0,
     author = {Dehornoy, Patrick and Tesson, Emilie},
     title = {Garside combinatorics for {Thompson{\textquoteright}s} monoid $F^+$ and a hybrid with the braid monoid $B_{\infty }^{+}$},
     journal = {Algebraic Combinatorics},
     pages = {683--709},
     publisher = {MathOA foundation},
     volume = {2},
     number = {4},
     year = {2019},
     doi = {10.5802/alco.52},
     zbl = {1422.05106},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.52/}
}
TY  - JOUR
AU  - Dehornoy, Patrick
AU  - Tesson, Emilie
TI  - Garside combinatorics for Thompson’s monoid $F^+$ and a hybrid with the braid monoid $B_{\infty }^{+}$
JO  - Algebraic Combinatorics
PY  - 2019
SP  - 683
EP  - 709
VL  - 2
IS  - 4
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.52/
DO  - 10.5802/alco.52
LA  - en
ID  - ALCO_2019__2_4_683_0
ER  - 
%0 Journal Article
%A Dehornoy, Patrick
%A Tesson, Emilie
%T Garside combinatorics for Thompson’s monoid $F^+$ and a hybrid with the braid monoid $B_{\infty }^{+}$
%J Algebraic Combinatorics
%D 2019
%P 683-709
%V 2
%N 4
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.52/
%R 10.5802/alco.52
%G en
%F 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. https://alco.centre-mersenne.org/articles/10.5802/alco.52/

[1] Adyan, Sergey I. Fragments of the word Δ in the braid group, Mat. Zametki, Volume 36 (1984) no. 1, pp. 25-34 | MR | Zbl

[2] Bétréma, Jean; Penaud, Jean-Guy Animaux et arbres guingois, Theoret. Comput. Sci., Volume 117 (1993) no. 1-2, pp. 67-89 Conference on Formal Power Series and Algebraic Combinatorics (Bordeaux, 1991) | DOI | MR | Zbl

[3] Birman, Joan; Ko, Ki Hyoung; Lee, Sang Jin A new approach to the word and conjugacy problems in the braid groups, Adv. Math., Volume 139 (1998) no. 2, pp. 322-353 | DOI | MR | Zbl

[4] Brieskorn, Egbert; Saito, Kyoji Artin-Gruppen und Coxeter-Gruppen, Invent. Math., Volume 17 (1972), pp. 245-271 | DOI | MR | Zbl

[5] Brin, Matthew G. On the Zappa–Szép product, Comm. Algebra, Volume 33 (2005) no. 2, pp. 393-424 | DOI | MR | Zbl

[6] Brin, Matthew G. The algebra of strand splitting. II. A presentation for the braid group on one strand, Internat. J. Algebra Comput., Volume 16 (2006) no. 1, pp. 203-219 | DOI | MR | Zbl

[7] Brin, Matthew G. The algebra of strand splitting. I. A braided version of Thompson’s group V, J. Group Theory, Volume 10 (2007) no. 6, pp. 757-788 | DOI | MR | Zbl

[8] Cannon, James W.; Floyd, William. J.; Parry, Walter R. Introductory notes on Richard Thompson’s groups, Enseign. Math. (2), Volume 42 (1996) no. 3-4, pp. 215-256 | MR | Zbl

[9] Clifford, Alfred H.; Preston, Gordon B. The algebraic theory of semigroups. Vol. I, Mathematical Surveys and Monographs, 7, American Mathematical Society, Providence, R.I., 1961, xv+224 pages | MR | Zbl

[10] Dehornoy, Patrick Groups with a complemented presentation, J. Pure Appl. Algebra, Volume 116 (1997) no. 1-3, pp. 115-137 (Special volume on the occasion of the 60th birthday of Professor Peter J. Freyd) | DOI | MR | Zbl

[11] Dehornoy, Patrick The group of parenthesized braids, Adv. Math., Volume 205 (2006) no. 2, pp. 354-409 | DOI | MR | Zbl

[12] Dehornoy, Patrick The subword reversing method, Internat. J. Algebra Comput., Volume 21 (2011) no. 1-2, pp. 71-118 | DOI | MR | Zbl

[13] Dehornoy, Patrick Tamari lattices and the symmetric Thompson monoid, Associahedra, Tamari lattices and related structures (Prog. Math.), Volume 299, Birkhäuser/Springer, Basel, 2012, pp. 211-250 | DOI | MR | Zbl

[14] Dehornoy, Patrick A cancellativity criterion for presented monoids (2018) (Semigroup Forum, to appear, https://arxiv.org/abs/1802.04607) | DOI | Zbl

[15] Dehornoy, Patrick; Digne, François; Godelle, Eddy; Krammer, Daan; Michel, Jean Foundations of Garside theory, EMS Tracts in Mathematics, 22, European Mathematical Society (EMS), Zürich, 2015, xviii+691 pages | DOI | MR | Zbl

[16] Deligne, Pierre Les immeubles des groupes de tresses généralisés, Invent. Math., Volume 17 (1972) no. 4, pp. 273-302 | DOI | MR | Zbl

[17] Epstein, David B. A.; Cannon, James W.; Holt, Derek F.; Levy, Silvio V. F.; Paterson, Michael S.; Thurston, William P. Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992, xii+330 pages | MR | Zbl

[18] Garside, F. A. The braid group and other groups, Quart. J. Math. Oxford Ser. (2), Volume 20 (1969) no. 1, pp. 235-254 | DOI | MR | Zbl

[19] Gouyou-Beauchamps, Dominique; Viennot, Gérard Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem, Adv. in Appl. Math., Volume 9 (1988) no. 3, pp. 334-357 | DOI | MR | Zbl

[20] Holt, Derek F.; Rees, Sarah; Röver, Claas E. Groups, languages and automata, London Mathematical Society Student Texts, 88, Cambridge University Press, Cambridge, 2017, xi+294 pages | DOI | MR | Zbl

[21] Newman, M. H. A. On theories with a combinatorial definition of “equivalence.”, Ann. of Math. (2), Volume 43 (1942), pp. 223-243 | DOI | MR | Zbl

[22] Ore, Oystein Linear equations in non-commutative fields, Ann. of Math. (2), Volume 32 (1931) no. 3, pp. 463-477 | DOI | MR | Zbl

[23] Sloane, Neil J. A. The on-line encyclopedia of integer sequences, Notices Amer. Math. Soc., Volume 65 (2018) no. 9, pp. 1062-1074 | MR | Zbl

[24] Szép, Jenö On factorisable, not simple groups., Acta Sci. Math., Volume 13 (1950), pp. 239-241 | MR | Zbl

[25] Terese Term rewriting systems, Cambridge Tracts in Theoretical Computer Science, 55, Cambridge University Press, Cambridge, 2003, xxii+884 pages | MR

[26] Tesson, Émilie Un hybride du groupe de Thompson F et du groupe de tresses B ∞ , PhD Thesis, Université de Caen (France) (2018) (https://tel.archives-ouvertes.fr/tel-01871101)

[27] Thompson, R.J. Tranformation structure of algebraic logic, PhD Thesis, University of Berkeley (USA) (1979) (ProQuest LLC, Ann Arbor, MI, 345 pages, MR2628711)

[28] Viennot, Gérard Problèmes combinatoires posés par la physique statistique, Séminaire Bourbaki : volume 1983/84, exposés 615-632 (Astérisque), Société mathématique de France, 1985 no. 121-122, pp. 225-246 | Numdam | MR | Zbl

[29] Zappa, Guido Sulla costruzione dei gruppi prodotto di due dati sottogruppi permutabili tra loro, Atti Secondo Congresso Un. Mat. Ital., Bologna, 1940, Edizioni Cremonense, Rome, 1942, pp. 119-125 | MR | Zbl

Cited by Sources: