Equationally complete discriminator varieties of groupoids

Author:
Robert W. Quackenbush

Journal:
Proc. Amer. Math. Soc. **90** (1984), 203-206

MSC:
Primary 08B05; Secondary 08B10

DOI:
https://doi.org/10.1090/S0002-9939-1984-0727233-X

MathSciNet review:
727233

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Abstract | References | Similar Articles | Additional Information

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.

**[1]**A. D. Bol'bot,*Varieties of quasigroups*, Sibirsk. Mat. Zh.**13**(1972), 252-271. MR**0297913 (45:6965)****[2]**A. Foster and A. Pixley,*Semi-categorical algebras*. II, Math. Z.**85**(1964), 169-184. MR**0168509 (29:5771)****[3]**J. Froemke and R. Quackenbush,*The spectrum of an equational class of groupoids*, Pacific J. Math.**58**(1975), 381-386. MR**0384649 (52:5522)****[4]**B. Jonsson,*Algebras whose congruence lattices are distributive*, Math. Scand.**21**(1967), 110-121. MR**0237402 (38:5689)****[5]**J. Kalicki,*The number of equationally complete classes of equations*, Nederl. Akad Wetensch. Proc. Ser. A**58**(1955), 660-662. MR**0074351 (17:571b)****[6]**V. L. Murskii,*The existence of a finite basis of identities, and other properties of "almost all" finite algebras*, Problemy Kibernet.**30**(1975), 43-56. (Russian) MR**0401606 (53:5433)****[7]**D. Pigozzi,*Equational logic and equational theories of algebras*, Seminar report, Purdue Univ., Comp. Sci. Dept., 1973.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1984-0727233-X

Keywords:
Groupoid,
variety,
equational completeness,
discriminator variety,
quasigroup

Article copyright:
© Copyright 1984
American Mathematical Society