Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-01T20:09:52.608Z Has data issue: false hasContentIssue false

On the Number of Conjugates of N-Ary Ouasigroups

Published online by Cambridge University Press:  20 November 2018

Mary McLeish*
Affiliation:
University of Alberta, Edmonton, Alberta
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.

Higher dimensional quasigroups (a set Q with a cancellative, n-ary operation 〈 〉, ([2]) have been studied by T. Evans ([3], [4]), A. Cruse [1], C. C. Lindner ([10], [11]) and also by many others under the guise of magic cubes, Graeco-latin cubes, etc. Conjugates or parastrophes have been discussed by S. K. Stein [18], A. Sade [17] and more recently by C. C. Lindner and D. Steedley in [14], where it is shown that ordinary quasigroups exist of every order ≧ 4 with a prescribed number of distinct conjugates. It is suggested that the problem be extended to n-ary quasigroups.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Cruse, A., On the finite completion of partial latin cubes, J. Combinatorial Theory (A. 17 (1974), 112119.Google Scholar
2. Denes, J. and Keedwell, A. D., Latin squares and their applications (Academic Press, New York, 1974).Google Scholar
3. Evans, T., The construction of orthogonal K-skeins and latin k-cubes, Aequationes Math. 14 (1976), 485491.Google Scholar
4. Evans, T., Latin cubes orthogonal to their transposes—a ternary analogue of Stein quasigroups, Aequationes Math. 9 (1973), 296297.Google Scholar
5. Ganter, B., Combinatorial designs and algebras, Preprint No. 270, Mai 1976, Technische Hochshule, Darmstadt.Google Scholar
6. Hall, M., Jr., The theory of groups (MacMillan, New York, 1959).Google Scholar
7. Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers (Oxford University Press, 1968).Google Scholar
8. Humbolt, L., Sur une extension de la notion de carrés latins, C. R. Acad. Se. Paris, Sér. A. 273 (1971), 795798.Google Scholar
9. Lederman, W., Introduction to the theory of finite groups (Oliver and Boyd, Edinburgh, 1967).Google Scholar
10. Lindner, C. C., Two finite embedding theorems for partial 3-quasigroups, to appear.Google Scholar
11. Lindner, C. C., A finite partial idempotent latin cube can be embedded in a finite idempotent latin cube, J. Combinatorial Theory (A). 21 (1976), 104109.Google Scholar
12. Lindner, C. C., Some remarks on the Steiner triple systems associated with Steiner quadruple systems, Colloquium Math. 32 (1975), 301306.Google Scholar
13. Linder, C. C. and Rosa, A., A survey of Steiner quadruple systems, Discrete Math., to appear.Google Scholar
14. Linder, C. C. and Steedley, D., On the number of conjugates of a quasigroup, Alg. Universali. 5 (1975), 191196.Google Scholar
15. McLeish, M., On the existence of ternary quasigroups with 2 or S conjugacy classes, J. Comb. Theory, Seria A, submitted.Google Scholar
16. Sade, A., Produit direct—singulier de quasigroups, orthogonaux et anti-abêliens, Ann. Soc. Sci. Bruxelles, Sér. I. 74 (1960), 9199.Google Scholar
17. Sade, A., Quasigroupes parastrophiques. Expressions et identités, Math. Nachr. 20 (1959), 73106.Google Scholar
18. Stein, S. K., On the foundations of quasigroups, Trans. Amer. Math. Soc. 85 (1957), 228256.Google Scholar