Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-29T04:47:56.369Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note on Cayley Loops

Published online by Cambridge University Press:  20 November 2018

A. Ginzburg
A. Ginzburg*
Affiliation:
Technion, Israel Institute of Technology Haifa, Israel
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In § 1 the existence of two non-isotopic loops satisfying the same system of relations is proved: C1 is isomorphic to the well-known Cayley loop (see 1 or 6); C2 seems not to have been mentioned in the literature. In § 2 a characterization of the second type is given in terms of Tamari's generalized normal multiplication table (7; 8) as a complete symmetric quasiregular partition, i.e., a quasi-associative loop (4, 5), or a so-called near-group (9). In § 3 these distinctions are followed by comparing the corresponding algebras.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 16 , 1964 , pp. 77 - 81
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Bruck, R. H., A survey of binary systems (Göttingen, 1958).Google Scholar
2. Bruck, R. H. and Kleinfeld, E., The structure of alternative division rings, Proc. Amer. Math. Soc, 2 (1951), 878–890.Google Scholar
3. Dickson, L. E., Algebras and their arithmetics (New York, 1938).Google Scholar
4. Ginzburg, A., Multiplicative systems as homomorphic images of square sets (Thesis, Haifa, 1959).Google Scholar
5. Ginzburg, A., Systèmes multiplicatifs de relations. Boucles quasi-associatives, C.R. Acad. Sci. Paris, 250 (1960), 1413–1416.Google Scholar
6. Norton, D. A., Hamiltonian loops, Proc. Amer. Math. Soc, 3 (1952), 56–65.Google Scholar
7. Tamari, D., Les images homomorphes des groupoïdes de Brandt et Vimmersion des semi-groupes, C. R. Acad. Sci. Paris, 229 (1949), 1291–1293.Google Scholar
8. Tamari, D., Représentations isomorphes par des systèmes de relations. Systèmes associatifs, C. R. Acad. Sci. Paris, 232 (1951), 1332–1334.Google Scholar
9. Tamari, D., Near-groups” as generalized normal multiplication tables, Notices Amer. Math. Soc, 7 (1960), 77.Google Scholar