Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-15T06:24:41.909Z Has data issue: false hasContentIssue false
  • English
  • Français

Remarks on Quasigroups Obeying a Generalized Associative Law

Published online by Cambridge University Press:  20 November 2018

Gary G. Ford
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.

T. Evans in [4] introduced a concept generalizing the associative law for groups. Certain very restricted cases of this generalization had occurred, somewhat as side results, in the considerations of A. K. Suškevič [7], D. C. Murdoch [5], [6], and R. H. Bruck [2] associated with quasigroups which obey theorems generalizing certain results in group theory.

In this paper we consider quasigroups obeying an instance of Evans' generalization. We show two means of constructing such quasigroups, and we demonstrate these constructions with several examples. The principal result of this paper is a theorem which reveals a peculiar trait of quasigroups obeying an instance of the generalized associative law we are considering. Such a quasigroup obeys a law expressed with permutations in which the nontrivial permutations involved either are both automorphisms of the quasigroup or else both fail to distribute over any product from the quasigroup.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 1 , March 1970 , pp. 17 - 21
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Albert, A. A., Quasigroups, Trans. Am. Math. Soc. 54 (1943), 507-519.Google Scholar
2. Bruck, R. H., Some results in the theory of quasigroups, Trans. Am. Math. Soc. 55 (1944), 19-52.Google Scholar
3. Bruck, R. H., A Survey of binary systems, Berlin-Goettingen-Heidelberg: Springer-Verlag, 1958.Google Scholar
4. Evans, T., A note on the associative law, J. London Math. Soc. 25 (1950), 196-201.Google Scholar
5. Murdoch, D. C., Quasigroups which satisfy certain generalized associative laws, Am. J. Math. 61(1939), 509-522.Google Scholar
6. Murdoch, D. C., Structure of abelian quasigroups, Trans. Am. Math. Soc. 49 (1941), 392-409.Google Scholar
7. Suškevič, A. K., On a generalization of the associative law, Trans. Am. Math. Soc. 31 (1929), 204-214.Google Scholar