×

zbMATH — the first resource for mathematics

Braid groups and left distributive operations. (English) Zbl 0837.20048
Summary: The decidability of the word problem for the free left distributive law is proved by introducing a structure group which describes the underlying identities. This group is closely connected with Artin’s braid group \(B_\infty\). Braid colourings associated with free left distributive structures are used to show the existence of a unique ordering on the braids which is compatible with left translation and such that every generator \(\sigma_i\) is preponderant over all \(\sigma_k\) with \(k > i\). This ordering is a linear ordering.

MSC:
20F36 Braid groups; Artin groups
06A05 Total orders
08A50 Word problems (aspects of algebraic structures)
17A30 Nonassociative algebras satisfying other identities
17A50 Free nonassociative algebras
20N02 Sets with a single binary operation (groupoids)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Joan S. Birman, Erratum: ”Braids, links, and mapping class groups” (Ann. of Math. Studies, No. 82, Princeton Univ. Press, Princeton, N. J., 1974), Princeton University Press, Princeton, N. J.; University of Tokyo Press, Toyko, 1975. Based on lecture notes by James Cannon.
[2] E. Brieskorn, Automorphic sets and braids and singularities, Braids (Santa Cruz, CA, 1986) Contemp. Math., vol. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 45 – 115.
[3] P. Cartier, Développements récents sur les groupes de tresses, applications ’a la topologie et à l’algèbre, Séminaire Bourbaki, exposé 716, 1989. · Zbl 0928.20030
[4] Patrick Dehornoy, Algebraic properties of the shift mapping, Proc. Amer. Math. Soc. 106 (1989), no. 3, 617 – 623. · Zbl 0679.20058
[5] Patrick Dehornoy, Free distributive groupoids, J. Pure Appl. Algebra 61 (1989), no. 2, 123 – 146. · Zbl 0686.20041
[6] Patrick Dehornoy, Sur la structure des gerbes libres, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 3, 143 – 148 (French, with English summary). · Zbl 0688.20038
[7] Patrick Dehornoy, Problème de mots dans les gerbes libres, Theoret. Comput. Sci. 94 (1992), no. 2, 199 – 213 (French, with English summary). Discrete mathematics and applications to computer science (Marseille, 1989). · Zbl 0752.08006
[8] Patrick Dehornoy, Structural monoids associated to equational varieties, Proc. Amer. Math. Soc. 117 (1993), no. 2, 293 – 304. · Zbl 0776.08006
[9] -, A canonical ordering for free left distributive magmas, Proc. Amer. Math. Soc. (to appear).
[10] Patrick Dehornoy, A normal form for the free left distributive law, Internat. J. Algebra Comput. 4 (1994), no. 4, 499 – 528. · Zbl 0827.20079
[11] -, Reduction of braid words, preprint, 1993.
[12] Randall Dougherty, Critical points in an algebra of elementary embeddings, Ann. Pure Appl. Logic 65 (1993), no. 3, 211 – 241. · Zbl 0791.03028
[13] Randall Dougherty and Thomas Jech, Finite left-distributive algebras and embedding algebras, Adv. Math. 130 (1997), no. 2, 201 – 241. · Zbl 0884.08006
[14] Elsayed A. El-Rifai and H. R. Morton, Algorithms for positive braids, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 479 – 497. · Zbl 0839.20051
[15] D. B. Epstein et al., Word processing in groups, Jones and Barlett, 1992. · Zbl 0764.20017
[16] R. Fenn and C. Rourke, Racks and links in codimension 2, J. Knot Theory and its Ramifications (to appear). · Zbl 0787.57003
[17] F. A. Garside, The braid group and other groups, Quart. J. Math. Oxford Ser. (2) 20 (1969), 235 – 254. · Zbl 0194.03303
[18] T. Kepka, Notes on left-distributive groupoids, Acta Univ. Carolin. — Math. Phys. 22 (1981), no. 2, 23 – 37 (English, with Russian and Czech summaries). · Zbl 0517.20048
[19] Robert M. Solovay, William N. Reinhardt, and Akihiro Kanamori, Strong axioms of infinity and elementary embeddings, Ann. Math. Logic 13 (1978), no. 1, 73 – 116. · Zbl 0376.02055
[20] David Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982), no. 1, 37 – 65. · Zbl 0474.57003
[21] David M. Larue, On braid words and irreflexivity, Algebra Universalis 31 (1994), no. 1, 104 – 112. · Zbl 0793.08007
[22] R. Laver, Elementary embeddings of a rank into itself, Abstracts Amer. Math. Soc. 7 (1986), 6.
[23] Richard Laver, The left distributive law and the freeness of an algebra of elementary embeddings, Adv. Math. 91 (1992), no. 2, 209 – 231. · Zbl 0822.03030
[24] Richard Laver, A division algorithm for the free left distributive algebra, Logic Colloquium ’90 (Helsinki, 1990) Lecture Notes Logic, vol. 2, Springer, Berlin, 1993, pp. 155 – 162. · Zbl 0809.08004
[25] -, Braid group actions on left distributive structures and well-orderings in the braid group, preprint, 1993.
[26] Saunders Mac Lane, Natural associativity and commutativity, Rice Univ. Studies 49 (1963), no. 4, 28 – 46. · Zbl 0244.18008
[27] Handbook of mathematical logic, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Edited by Jon Barwise; With the cooperation of H. J. Keisler, K. Kunen, Y. N. Moschovakis and A. S. Troelstra; Studies in Logic and the Foundations of Mathematics, Vol. 90.
[28] W. Thurston, Finite state algorithms for the braid group, preprint, 1988.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.