Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access October 8, 2013

The free one-generated left distributive algebra: basics and a simplified proof of the division algorithm

  • Richard Laver EMAIL logo and Sheila Miller
From the journal Open Mathematics

Abstract

The left distributive law is the law a· (b· c) = (a·b) · (a· c). Left distributive algebras have been classically used in the study of knots and braids, and more recently free left distributive algebras have been studied in connection with large cardinal axioms in set theory. We provide a survey of results on the free left distributive algebra on one generator, A, and a new, simplified proof of the existence of a normal form for terms in A. Topics included are: the confluence of A, the linearity of the iterated left division ordering <L of A, the connections of A to the braid groups, and an extension P of A obtained by freely adding a composition operation. This is followed by a simplified proof of the division algorithm for P, which produces a normal form for terms in A and is a powerful tool in the study of A.

[1] Artin E., Theorie der Zöpfe, Abh. Math. Sem. Univ. Hamburg, 1925, 4(1), 47–72 http://dx.doi.org/10.1007/BF0295071810.1007/BF02950718Search in Google Scholar

[2] Birman J.S., Braids, Links, and Mapping Class Groups, Ann. of Math. Stud., 82, Princeton University Press, Princeton, 1974 10.1515/9781400881420Search in Google Scholar

[3] Brieskorn E., Automorphic sets and braids and singularities, In: Braids, Santa Cruz, July 13–26, 1986, Contemp. Math., 78, American Mathematical Society, Providence, 1988, 45–115 10.1090/conm/078/975077Search in Google Scholar

[4] Burckel S., The wellordering on positive braids, J. Pure Appl. Algebra, 1997, 120(1), 1–17 http://dx.doi.org/10.1016/S0022-4049(96)00072-210.1016/S0022-4049(96)00072-2Search in Google Scholar

[5] Dehornoy P., Braid groups and left distributive operations, Trans. Amer. Math. Soc., 1994, 345(1), 115–150 http://dx.doi.org/10.1090/S0002-9947-1994-1214782-410.1090/S0002-9947-1994-1214782-4Search in Google Scholar

[6] Dehornoy P., Braids and Self-Distributivity, Progr. Math., 192, Birkhäuser, Basel, 2000 http://dx.doi.org/10.1007/978-3-0348-8442-610.1007/978-3-0348-8442-6Search in Google Scholar

[7] Dehornoy P., Dynnikov I., Rolfsen D., Wiest B., Why are Braids Orderable?, Panor. Syntheses, 14, Société Mathématique de France, Paris, 2002 Search in Google Scholar

[8] Fenn R., Rourke C., Racks and links in codimension two, J. Knot Theory Ramifications, 1992, 1(4), 343–406 http://dx.doi.org/10.1142/S021821659200020310.1142/S0218216592000203Search in Google Scholar

[9] Hurwitz A., Ueber Riemann’sche Flächen wit gegebenen Verzweigungspunkten, Math. Ann., 1891, 39(1), 1–60 http://dx.doi.org/10.1007/BF0119946910.1007/BF01199469Search in Google Scholar

[10] Joyce D., A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra, 1982, 23(1), 37–65 http://dx.doi.org/10.1016/0022-4049(82)90077-910.1016/0022-4049(82)90077-9Search in Google Scholar

[11] Kunen K., Elementary embeddings and infinitary combinatorics, J. Symbolic Logic, 1971, 36(3), 407–413 http://dx.doi.org/10.2307/226994810.2307/2269948Search in Google Scholar

[12] Larue D.M., Braid words and irreflexivity, Algebra Universalis, 1994, 31(1), 104–112 http://dx.doi.org/10.1007/BF0118818210.1007/BF01188182Search in Google Scholar

[13] Laver R., A division algorithm for the free left distributive algebra, In: Logic Colloquium’ 90, Helsinki, July 15–22, 1990, Lecture Notes Logic, 2, Springer, Berlin, 1993, 155–162 10.1017/9781316718254.012Search in Google Scholar

[14] Laver R., The left distributive law and the freeness of an algebra of elementary embeddings, Adv. Math., 1992, 91(2), 209–231 http://dx.doi.org/10.1016/0001-8708(92)90016-E10.1016/0001-8708(92)90016-ESearch in Google Scholar

[15] Laver R., On the algebra of elementary embeddings of a rank into itself, Adv. Math., 1995, 110(2), 334–346 http://dx.doi.org/10.1006/aima.1995.101410.1006/aima.1995.1014Search in Google Scholar

[16] Laver R., Braid group actions on left distributive structures, and well orderings in the braid groups, J. Pure Appl. Algebra, 1996, 108(1), 81–98 http://dx.doi.org/10.1016/0022-4049(95)00147-610.1016/0022-4049(95)00147-6Search in Google Scholar

[17] Laver R., Miller S.K., Left division in the free left distributive algebra on one generator, J. Pure Appl. Algebra, 2010, 215(3), 276–282 http://dx.doi.org/10.1016/j.jpaa.2010.04.01910.1016/j.jpaa.2010.04.019Search in Google Scholar

[18] Laver R., Moody J.A., Well-foundedness conditions connected with left-distributivity, Algebra Univsersalis, 2002, 47(1), 65–68 http://dx.doi.org/10.1007/s00012-002-8175-210.1007/s00012-002-8175-2Search in Google Scholar

[19] Miller S.K., Free Left Distributive Algebras, PhD thesis, University of Colorado, Boulder, 2007 Search in Google Scholar

[20] Miller S.K., Free left distributive algebras on κ generators (in preparation) Search in Google Scholar

Published Online: 2013-10-8
Published in Print: 2013-12-1

© 2013 Versita Warsaw

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 28.3.2024 from https://www.degruyter.com/document/doi/10.2478/s11533-013-0290-0/html
Scroll to top button