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Varieties that make one Cross

Part of: General nonassociative rings

Published online by Cambridge University Press:  09 April 2009

Sheila Oates MacDonald  and
M. R. Vaughan-Lee
Sheila Oates MacDonald
Affiliation:
Department of Mathematics University of Queensland St. Lucia 4067, Australia
M. R. Vaughan-Lee
Affiliation:
Christ Church Oxford OX1 1DP, United Kingdom
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Abstract

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An example is constructed of a locally finite variety of non-associative algebras which satisfies the maximal condition on subvarieties but not the minimal condition. Based on this, counterexamples to various conjectures concerning varieties generated by finite algebras are constructed. The possibility of finding a locally finite variety of algebras which satisfies the minimal condition on subvarieties but not the maximal is also investigated.

MSC classification

Secondary: 17A30: Algebras satisfying other identities
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 26 , Issue 3 , November 1978 , pp. 368 - 382
Copyright
Copyright © Australian Mathematical Society 1978

References

Birkhoff, G. (1935), “On the structure of abstract algebras”, Proc. Cambridge Phil. Soc. 31, 433–454.CrossRefGoogle Scholar
Higman, G. (1952), “Ordering by divisibility in abstract algebras”, Proc. London Math. Soc. 2, 326–336.CrossRefGoogle Scholar
Macdonald, Sheila Oates (1973), “Various varieties”, J. Austral. Math. Soc. 16, 363–367.CrossRefGoogle Scholar
Murskii, V. L. (1965), “The existence in three-valued logic of a closed class with finite basis not having a finite complete system of identities”, Soviet Math. Doklady 6, 1020–1024.Google Scholar
Neumann, Hanna (1967), Varieties of Groups (Ergebnisse der Mathematik und ihrer Grenzgebebiete, Bd. 37, Springer-Verlag, Berlin).CrossRefGoogle Scholar
Park, R. E. (1976), “Equational classes of non-associative ordered algebras” (Ph.D. dissertation, UCLA).Google Scholar
Polin, S. V. (1976), “The identities of finite algebras”, Sib. Math. J. 17, 1356–1366.Google Scholar