| Title:
|
Ternary quasigroups and the modular group (English) |
| Author:
|
Smith, Jonathan D. H. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
49 |
| Issue:
|
2 |
| Year:
|
2008 |
| Pages:
|
309-317 |
| . |
| Category:
|
math |
| . |
| Summary:
|
For a positive integer $n$, the usual definitions of $n$-quasigroups are rather complicated: either by combinatorial conditions that effectively amount to Latin $n$-cubes, or by $2n$ identities on $n+1$ different $n$-ary operations. In this paper, a more symmetrical approach to the specification of $n$-quasigroups is considered. In particular, ternary quasigroups arise from actions of the modular group. (English) |
| Keyword:
|
quasigroup |
| Keyword:
|
ternary quasigroup |
| Keyword:
|
$n$-quasigroup |
| Keyword:
|
heterogeneous algebra |
| Keyword:
|
hyperidentity |
| Keyword:
|
modular group |
| Keyword:
|
conjugate |
| Keyword:
|
parastrophe |
| Keyword:
|
time reversal |
| MSC:
|
08A68 |
| MSC:
|
20N05 |
| MSC:
|
20N15 |
| MSC:
|
20N20 |
| idZBL:
|
Zbl 1192.20064 |
| idMR:
|
MR2426895 |
| . |
| Date available:
|
2009-05-05T17:11:33Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119725 |
| . |
| Reference:
|
[1] Chein O. et al.: Quasigroups and Loops: Theory and Applications.Heldermann, Berlin, 1990. Zbl 0719.20036, MR 1125806 |
| Reference:
|
[2] Coxeter H.S.M., Moser W.O.J.: Generators and Relations for Discrete Groups.Springer, Berlin, 1957. Zbl 0487.20023, MR 0088489 |
| Reference:
|
[3] 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:
|
[4] Higgins P.J.: Algebras with a scheme of operators.Math. Nachr. 27 (1963), 115-132. Zbl 0117.25903, MR 0163940, 10.1002/mana.19630270108 |
| Reference:
|
[5] James I.M.: Quasigroups and topology.Math. Z. 84 (1964), 329-342. Zbl 0124.16104, MR 0165524 |
| Reference:
|
[6] Lugowski H.: Grundzüge der Universellen Algebra.Teubner, Leipzig, 1976. Zbl 0485.08001, MR 0441819 |
| Reference:
|
[7] Sade A.: Quasigroupes obéissant à certaines lois.Rev. Fac. Sci. Univ. Istanbul, Ser. A 22 (1957), 151-184. MR 0106253 |
| Reference:
|
[8] Serre, J.-P.: Cours d'Arithmétique.Presses Universitaires de France, Paris, 1970. Zbl 0432.10001, MR 0255476 |
| Reference:
|
[9] Smith J.D.H.: Axiomatization of quasigroups.Discuss. Math. Gen. Algebra Appl. 27 (2007), 21-33. Zbl 1135.20048, MR 2319330, 10.7151/dmgaa.1116 |
| . |
