On the model of simple braids, defined to be the left divisors of Garside’s elements in the monoid , we investigate simple elements in Thompson’s monoid and in a larger monoid that is a hybrid of and : in both cases, we count how many simple elements left divide the right lcm of the first atoms, and characterize their normal forms in terms of forbidden factors. In the case of , a generalized Pascal triangle appears.
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}, 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/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. https://alco.centre-mersenne.org/item/ALCO_2019__2_4_683_0/
[1] Fragments of the word in the braid group, Mat. Zametki, Volume 36 (1984) no. 1, pp. 25-34 | MR 757642 | Zbl 0599.20044
[2] 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)) | Article | MR 1235169 | Zbl 0780.68098
[3] A new approach to the word and conjugacy problems in the braid groups, Adv. Math., Volume 139 (1998) no. 2, pp. 322-353 | Article | MR 1654165 | Zbl 0937.20016
[4] Artin-Gruppen und Coxeter-Gruppen, Invent. Math., Volume 17 (1972), pp. 245-271 | Article | MR 0323910 | Zbl 0243.20037
[5] On the Zappa–Szép product, Comm. Algebra, Volume 33 (2005) no. 2, pp. 393-424 | Article | MR 2124335 | Zbl 1078.20062
[6] 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 | Article | MR 2217649 | Zbl 1170.20306
[7] The algebra of strand splitting. I. A braided version of Thompson’s group , J. Group Theory, Volume 10 (2007) no. 6, pp. 757-788 | Article | MR 2364825 | Zbl 1169.20021
[8] Introductory notes on Richard Thompson’s groups, Enseign. Math. (2), Volume 42 (1996) no. 3-4, pp. 215-256 | MR 1426438 | Zbl 0880.20027
[9] The algebraic theory of semigroups. Vol. I, Mathematical Surveys and Monographs, Volume 7, American Mathematical Society, Providence, R.I., 1961, xv+224 pages | MR 0132791 | Zbl 0111.03403
[10] 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) | Article | MR 1437615 | Zbl 0870.20023
[11] The group of parenthesized braids, Adv. Math., Volume 205 (2006) no. 2, pp. 354-409 | Article | MR 2258261 | Zbl 1160.20027
[12] The subword reversing method, Internat. J. Algebra Comput., Volume 21 (2011) no. 1-2, pp. 71-118 | Article | MR 2787454 | Zbl 1256.20053
[13] Tamari lattices and the symmetric Thompson monoid, Associahedra, Tamari lattices and related structures (Prog. Math.) Volume 299, Birkhäuser/Springer, Basel, 2012, pp. 211-250 | Article | MR 3221540 | Zbl 1284.06004
[14] A cancellativity criterion for presented monoids (2018) (Semigroup Forum, to appear, https://arxiv.org/abs/1802.04607) | Article | Zbl 07127651
[15] Foundations of Garside theory, EMS Tracts in Mathematics, Volume 22, European Mathematical Society (EMS), Zürich, 2015, xviii+691 pages | Article | MR 3362691 | Zbl 1370.20001
[16] Les immeubles des groupes de tresses généralisés, Invent. Math., Volume 17 (1972) no. 4, pp. 273-302 | Article | MR 0422673 | Zbl 0238.20034
[17] Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992, xii+330 pages | MR 1161694 | Zbl 0764.20017
[18] The braid group and other groups, Quart. J. Math. Oxford Ser. (2), Volume 20 (1969) no. 1, pp. 235-254 | Article | MR 0248801 | Zbl 0194.03303
[19] 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 | Article | MR 956559 | Zbl 0727.05036
[20] Groups, languages and automata, London Mathematical Society Student Texts, Volume 88, Cambridge University Press, Cambridge, 2017, xi+294 pages | Article | MR 3616303 | Zbl 06691444
[21] On theories with a combinatorial definition of “equivalence.”, Ann. of Math. (2), Volume 43 (1942), pp. 223-243 | Article | MR 0007372 | Zbl 0060.12501
[22] Linear equations in non-commutative fields, Ann. of Math. (2), Volume 32 (1931) no. 3, pp. 463-477 | Article | MR 1503010 | Zbl 0001.26601
[23] The on-line encyclopedia of integer sequences, Notices Amer. Math. Soc., Volume 65 (2018) no. 9, pp. 1062-1074 | MR 3822822 | Zbl 06989892
[24] On factorisable, not simple groups., Acta Sci. Math., Volume 13 (1950), pp. 239-241 | MR 48434 | Zbl 0039.25503
[25] Term rewriting systems, Cambridge Tracts in Theoretical Computer Science, Volume 55, Cambridge University Press, Cambridge, 2003, xxii+884 pages | MR 2007186
[26] Un hybride du groupe de Thompson et du groupe de tresses (2018) (PhD Thesis)
[27] Tranformation structure of algebraic logic (1979) (PhD Thesis)
[28] 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 768962 | Zbl 0563.60095
[29] 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 0019090 | Zbl 0026.29104