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Two finitely generated varieties having no infinite simple members

Author(s): Ross Willard
Journal: Proc. Amer. Math. Soc. 126 (1998), 629-635.
MSC (1991): Primary 08B26
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Abstract: Using a method of R. McKenzie, we construct a finitely generated semisimple variety of infinite type, and a finitely generated nonsemisimple variety of finite type, both having arbitrarily large finite but no infinite simple members. This amplifies M. Valeriote's negative solution to Problem 11 from Hobby and McKenzie, The Structure of Finite Algebras.

References:

1.: D. Hobby and R. McKenzie, The Structure of Finite Algebras, Contemp. Math., vol. 76, Amer. Math. Soc. (Providence, R.I.), 1988. MR 89m:08001
2.: R. McKenzie, The residual bounds of finite algebras, Int. J. Algebra and Computation 6 (1996), 1-28. MR 97e:08002a
3.: R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties. Vol. I, Wadsworth & Brooks/Cole (Monterey, CA), 1987. MR 88e:08001
4.: M. Valeriote, A residually small, finitely generated, semi-simple variety which is not residually finite, Int. J. Algebra and Computation 6 (1996), 563-569. MR 97h:08005
5.: R. Willard, On McKenzie's method, Periodica Math. Hungarica 32 (1996), 149-165. CMP 97:01

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Additional Information:

Ross Willard
Affiliation: Department of Pure Mathematics University of Waterloo Waterloo, Ontario, Canada N2L 3G1
Email: rdwillar@gillian.math.uwaterloo.ca

DOI: 10.1090/S0002-9939-98-04521-3
PII: S 0002-9939(98)04521-3
Keywords: Variety, finitely generated, simple, semisimple
Received by editor(s): October 26, 1995
Additional Notes: The support of the NSERC of Canada is gratefully acknowledged.
Communicated by: Lance W. Small
Copyright of article: Copyright 1998, American Mathematical Society


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