| Title:
|
A scoop from groups: equational foundations for loops (English) |
| Author:
|
Phillips, J. D. |
| Author:
|
Vojtěchovský, Petr |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
49 |
| Issue:
|
2 |
| Year:
|
2008 |
| Pages:
|
279-290 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Groups are usually axiomatized as algebras with an associative binary operation, a two-sided neutral element, and with two-sided inverses. We show in this note that the same simplicity of axioms can be achieved for some of the most important varieties of loops. In particular, we investigate loops of Bol-Moufang type in the underlying variety of magmas with two-sided inverses, and obtain ``group-like'' equational bases for Moufang, Bol and C-loops. We also discuss the case when the inverses are only one-sided and/or the neutral element is only one-sided. (English) |
| Keyword:
|
inverse property loop |
| Keyword:
|
Bol loop |
| Keyword:
|
Moufang loop |
| Keyword:
|
C-loop |
| Keyword:
|
equational basis |
| Keyword:
|
magma with inverses |
| MSC:
|
03C05 |
| MSC:
|
20A05 |
| MSC:
|
20N05 |
| idZBL:
|
Zbl 1192.20058 |
| idMR:
|
MR2426892 |
| . |
| Date available:
|
2009-05-05T17:11:18Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119722 |
| . |
| Reference:
|
[1] Bates G.E., Kiokemeister F.: A note on homomorphic mappings of quasigroups into multiplicative systems.Bull. Amer. Math. Soc. 54 (1948), 1180-1185. Zbl 0034.29801, MR 0027768, 10.1090/S0002-9904-1948-09146-7 |
| Reference:
|
[2] Bruck R.H.: A Survey of Binary Systems, third printing, corrected.Ergebnisse der Mathematik und ihrer Grenzgebiete, New Series 20, Springer, Berlin, 1971. MR 0093552 |
| Reference:
|
[3] Colbourn C.J., Rosa A.: Triple Systems.Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1999. Zbl 1030.05017, MR 1843379 |
| Reference:
|
[4] Conway J.H.: A simple construction for the Fischer-Griess monster group.Invent. Math. 79 (1985), 513-540. Zbl 0564.20010, MR 0782233, 10.1007/BF01388521 |
| Reference:
|
[5] Dénes J., Keedwell A.D.: Latin Squares and their Applications.Akadémiai Kiadó, Budapest, 1974. MR 0351850 |
| Reference:
|
[6] Doro S.: Simple Moufang loops.Math. Proc. Cambridge Philos. Soc. 83 (1978), 377-392. Zbl 0381.20054, MR 0492031, 10.1017/S0305004100054669 |
| Reference:
|
[7] Evans T.: Homomorphisms of non-associative systems.J. London Math. Soc. 24 (1949), 254-260. MR 0032664, 10.1112/jlms/s1-24.4.254 |
| Reference:
|
[8] Fenyves F.: Extra loops II. On loops with identities of Bol-Moufang type.Publ. Math. Debrecen 16 (1969), 187-192. MR 0262409 |
| Reference:
|
[9] Hall M.: The Theory of Groups.The Macmillan Co., New York, 1959. Zbl 0919.20001, MR 0103215 |
| Reference:
|
[10] Goodaire E.G., Jespers E., Polcino Milies C.: Alternative Loop Rings.North-Holland Mathematics Studies 184, North-Holland Publishing Co., Amsterdam, 1996. Zbl 0878.17029, MR 1433590 |
| Reference:
|
[11] Kiechle H.: Theory of K-loops.Lecture Notes in Mathematics 1778, Springer, Berlin, 2002. Zbl 0997.20059, MR 1899153, 10.1007/b83276 |
| Reference:
|
[12] Kinyon M.K., Phillips J.D., Vojtěchovský P.: C-loops: Extensions and constructions.J. Algebra Appl. 6 (2007), 1 1-20. Zbl 1129.20043, MR 2302693, 10.1142/S0219498807001990 |
| Reference:
|
[13] Kunen K.: Quasigroups, loops, and associative laws.J. Algebra 185 (1996), 1 194-204. Zbl 0860.20053, MR 1409983, 10.1006/jabr.1996.0321 |
| Reference:
|
[14] Mann H.B.: On certain systems which are almost groups.Bull. Amer. Math. Soc. 50 (1944), 879-881. Zbl 0063.03769, MR 0011313, 10.1090/S0002-9904-1944-08256-6 |
| Reference:
|
[15] McCune W.W.: Prover9 and Mace, download at http://www.prover9.org.. |
| Reference:
|
[16] Nagy G.P.: A class of proper simple Bol loops.submitted, available at arXiv:math/0703919. |
| Reference:
|
[17] Ormes N., Vojtěchovský P.: Powers and alternative laws.Comment. Math. Univ. Carolin. 48 1 (2007), 25-40. Zbl 1174.20343, MR 2338827 |
| Reference:
|
[18] Pflugfelder H.O.: Quasigroups and Loops: Introduction.Sigma Series in Pure Mathematics 7, Heldermann Verlag, Berlin, 1990. Zbl 0715.20043, MR 1125767 |
| Reference:
|
[19] Pflugfelder H.O: Historical notes on loop theory.Comment. Math. Univ. Carolin. 41 2 (2000), 359-370. Zbl 1037.01010, MR 1780877 |
| Reference:
|
[20] Phillips J.D., Vojtěchovský P.: C-loops: An introduction.Publ. Math. Debrecen 2006 1-2 115-137. MR 2213546 |
| Reference:
|
[21] Phillips J.D., Vojtěchovský P.: The varieties of loops of Bol-Moufang type.Algebra Universalis 54 (2005), 3 259-271. Zbl 1102.20054, MR 2219409, 10.1007/s00012-005-1941-1 |
| Reference:
|
[22] Phillips J.D., Vojtěchovský P.: The varieties of quasigroups of Bol-Moufang type: An equational reasoning approach.J. Algebra 293 (2005), 17-33. Zbl 1101.20046, MR 2173964, 10.1016/j.jalgebra.2005.07.011 |
| Reference:
|
[23]
: Problem 10888.American Mathematical Monthly 110, no. 4 (April 2003), 347. 10.2307/3647897 |
| Reference:
|
[24] Robinson D.A.: Bol loops.Trans. Amer. Math. Soc. 123 (1966), 341-354. Zbl 0163.02001, MR 0194545, 10.1090/S0002-9947-1966-0194545-4 |
| Reference:
|
[25] Schafer R.D.: An Introduction to Nonassociative Algebras.Pure and Applied Mathematics 22, Academic Press, New York-London, 1966. Zbl 0145.25601, MR 0210757 |
| Reference:
|
[26] Sharma B.L.: Left loops which satisfy the left Bol identity.Proc. Amer. Math. Soc. 61 (1976), 2 189-195. MR 0422480, 10.1090/S0002-9939-1976-0422480-4 |
| Reference:
|
[27] Sharma B.L.: Left loops which satisfy the left Bol identity II.Ann. Soc. Sci. Bruxelles Sér. I 91 (1977), 2 69-78. Zbl 0385.20044, MR 0444826 |
| Reference:
|
[28] Springer T.A., Veldkamp F.D.: Octonions, Jordan Algebras and Exceptional Groups.Springer Monographs in Mathematics, Springer, Berlin, 2000. Zbl 1087.17001, MR 1763974 |
| Reference:
|
[29] Smith W.D.: Inclusions among diassociativity-related loop properties.preprint. |
| Reference:
|
[30] Tits J., Weiss R.M.: Moufang polygons.Springer Monographs in Mathematics, Springer, Berlin, 2002. Zbl 1010.20017, MR 1938841 |
| Reference:
|
[31] Ungar A.A.: Beyond Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces.Kluwer Academic Publishers, Dordrecht-Boston-London, 2001. MR 1978122 |
| . |
