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On Closure Conditions

Published online by Cambridge University Press:  20 November 2018

Pl. Kannappan  and
M. A. Taylor
Pl. Kannappan
Affiliation:
Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada
M. A. Taylor
Affiliation:
Department of Mathematics, Acadia University, Wolfville, Nova Scotia, Canada
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Quasigroups and groupoids with one or other of the Reidemeister or Thomsen closure conditions, the relationship among them with emphasis on their relationship to associativity viz groups, Abelian groups, have been investigated in [2], [3], [4], [5], [6], [12], and others. In [10] R- and T-groupoids, (that is, groupoids possessing one of the first two closure conditions mentioned above) which are generalizations of groups and Abelian groups were investigated. In this paper, we show that groupoids with the given identities may be described in terms of R- and T-groupoids. These results and others are used to give another proof of theorems given in [1], [7], and [5] describing the variety of all groups and Abelian groups defined by single laws.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 3 , 01 September 1976 , pp. 291 - 296
Copyright
Copyright © Canadian Mathematical Society 1976

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