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

Open Mathematics (2013)

- Volume: 11, Issue: 12, page 2150-2175
- ISSN: 2391-5455

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topRichard Laver, and Sheila Miller. "The free one-generated left distributive algebra: basics and a simplified proof of the division algorithm." Open Mathematics 11.12 (2013): 2150-2175. <http://eudml.org/doc/269137>.

@article{RichardLaver2013,

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},

author = {Richard Laver, Sheila Miller},

journal = {Open Mathematics},

keywords = {Free left distributive algebra; Division algorithm; Normal form; Braid; Word; free left distributive algebra; division algorithm; normal form; braid; word},

language = {eng},

number = {12},

pages = {2150-2175},

title = {The free one-generated left distributive algebra: basics and a simplified proof of the division algorithm},

url = {http://eudml.org/doc/269137},

volume = {11},

year = {2013},

}

TY - JOUR

AU - Richard Laver

AU - Sheila Miller

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

JO - Open Mathematics

PY - 2013

VL - 11

IS - 12

SP - 2150

EP - 2175

AB - 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

LA - eng

KW - Free left distributive algebra; Division algorithm; Normal form; Braid; Word; free left distributive algebra; division algorithm; normal form; braid; word

UR - http://eudml.org/doc/269137

ER -

## References

top- [1] Artin E., Theorie der Zöpfe, Abh. Math. Sem. Univ. Hamburg, 1925, 4(1), 47–72 http://dx.doi.org/10.1007/BF02950718 Zbl51.0450.01
- [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 Zbl0958.20032
- [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 Zbl0837.20048
- [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 Zbl1048.20021
- [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 Zbl0787.57003
- [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 Zbl0474.57003
- [11] Kunen K., Elementary embeddings and infinitary combinatorics, J. Symbolic Logic, 1971, 36(3), 407–413 http://dx.doi.org/10.2307/2269948 Zbl0272.02087
- [12] Larue D.M., Braid words and irreflexivity, Algebra Universalis, 1994, 31(1), 104–112 http://dx.doi.org/10.1007/BF01188182 Zbl0793.08007
- [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 Zbl0809.08004
- [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 Zbl0822.03030
- [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 Zbl0822.03031
- [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 Zbl0859.20029
- [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 Zbl1208.08003
- [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 Zbl1058.20055
- [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)

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