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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Equationally complete discriminator varieties of groupoids

Author: Robert W. Quackenbush
Journal: Proc. Amer. Math. Soc. 90 (1984), 203-206
MSC: Primary 08B05; Secondary 08B10
MathSciNet review: 727233
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Abstract: J. Kalicki proved that there are continuum many equationally complete varieties of groupoids. In this note we give a constructive proof of this by defining a countable partial groupoid which has continuum many completions such that each completion generates an equationally complete variety, and no two distinct completions generate the same variety. Moreover, the variety generated by all the completions is a discriminator variety, and every nontrivial groupoid in this variety is cancellative but not a quasigroup; this answers a question of R. Padmanabhan. A. D. Bol'bot proved a similar result for loops, but his computations are more difficult since his varieties are not discriminator varieties.

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Keywords: Groupoid, variety, equational completeness, discriminator variety, quasigroup
Article copyright: © Copyright 1984 American Mathematical Society