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2013 | 11 | 12 | 2150-2175
Tytuł artykułu

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

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EN
Abstrakty
EN
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
Wydawca
Czasopismo
Rocznik
Tom
11
Numer
12
Strony
2150-2175
Opis fizyczny
Daty
wydano
2013-12-01
online
2013-10-08
Twórcy
  • New York City College of Technology
Bibliografia
  • [1] Artin E., Theorie der Zöpfe, Abh. Math. Sem. Univ. Hamburg, 1925, 4(1), 47–72 http://dx.doi.org/10.1007/BF02950718
  • [2] Birman J.S., Braids, Links, and Mapping Class Groups, Ann. of Math. Stud., 82, Princeton University Press, Princeton, 1974
  • [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
  • [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)00072-2
  • [5] Dehornoy P., Braid groups and left distributive operations, Trans. Amer. Math. Soc., 1994, 345(1), 115–150 http://dx.doi.org/10.1090/S0002-9947-1994-1214782-4
  • [6] Dehornoy P., Braids and Self-Distributivity, Progr. Math., 192, Birkhäuser, Basel, 2000 http://dx.doi.org/10.1007/978-3-0348-8442-6
  • [7] Dehornoy P., Dynnikov I., Rolfsen D., Wiest B., Why are Braids Orderable?, Panor. Syntheses, 14, Société Mathématique de France, Paris, 2002
  • [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/S0218216592000203
  • [9] Hurwitz A., Ueber Riemann’sche Flächen wit gegebenen Verzweigungspunkten, Math. Ann., 1891, 39(1), 1–60 http://dx.doi.org/10.1007/BF01199469
  • [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)90077-9
  • [11] Kunen K., Elementary embeddings and infinitary combinatorics, J. Symbolic Logic, 1971, 36(3), 407–413 http://dx.doi.org/10.2307/2269948
  • [12] Larue D.M., Braid words and irreflexivity, Algebra Universalis, 1994, 31(1), 104–112 http://dx.doi.org/10.1007/BF01188182
  • [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
  • [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-E
  • [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.1995.1014
  • [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)00147-6
  • [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.2010.04.019
  • [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/s00012-002-8175-2
  • [19] Miller S.K., Free Left Distributive Algebras, PhD thesis, University of Colorado, Boulder, 2007
  • [20] Miller S.K., Free left distributive algebras on κ generators (in preparation)
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-013-0290-0
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