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The structure of locally finite varieties with polynomially many models
Authors:
Paweł Idziak, Ralph McKenzie and Matthew Valeriote
Journal:
J. Amer. Math. Soc. 22 (2009), 119-165
MSC (2000):
Primary 08A05; Secondary 03C45
Posted:
September 12, 2008
MathSciNet review:
2449056
Full-text PDF
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Additional Information
Abstract: We prove that a locally finite variety has at most polynomially many (in  ) non-isomorphic  -generated algebras if and only if it decomposes into a varietal product of an affine variety over a ring of finite representation type, and a sequence of strongly Abelian varieties equivalent to matrix powers of varieties of  -sets, with constants, for various finite groups  .
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- George E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley, Reading, Mass. 1976. MR 0557013 (58:7738)
- 2.
- Joel Berman and Paweł M. Idziak, Generative complexity in algebra, Mem. Amer. Math. Soc. 175 (2005), no. 828, viii+159. MR 2130585 (2006a:08001)
- 3.
- Joel Berman and Ralph McKenzie, Clones satisfying the term condition, Discrete Math. 52 (1984), no. 1, 7-29. MR 765281 (86m:08005)
- 4.
- Marcin Bilski, Generative complexity in semigroup varieties, J. Pure Appl. Algebra 165 (2001), no. 2, 137-149. MR 1865962 (2002h:20082)
- 5.
- David Hobby and Ralph McKenzie, The structure of finite algebras, Contemporary Mathematics, vol. 76, American Mathematical Society, Providence, RI, 1988. Revised edition: 1996. MR 958685 (89m:08001)
- 6.
- P. Idziak and R. McKenzie, Varieties with polynomially many models. I, Fund. Math. 170 (2001), no. 1-2, 53-68. Dedicated to the memory of Jerzy Łoś. MR 1881368 (2003e:08002)
- 7.
- Keith A. Kearnes, Type-preservation in locally finite varieties with the CEP, Canadian J. Math. 43 (1991) no. 4, 748-769. MR 1127028 (92m:08005)
- 8.
- -, An order-theoretic property of the commutator, Internat. J. Algebra Comput. 3 (1993), no. 4, 491-533. MR 1250248 (95c:08002)
- 9.
- -, Locally solvable factors of varieties, Proc. Amer. Math. Soc. 124 (1996), no. 12, 3619-3625. MR 1343705 (97b:08007)
- 10.
- -, A Hamiltonian property for nilpotent algebras, Algebra Universalis 37 (1997), no. 4, 403-421. MR 1465297 (98k:08001)
- 11.
- Keith A. Kearnes and Emil W. Kiss, Modularity prevents tails, Proc. Amer. Math. Soc. 127 (1999), no. 1, 11-19. MR 1625765 (99m:08003)
- 12.
- -, Residual smallness and weak centrality, Internat. J. Algebra Comput. 13 (2003), no. 1, 35-59. MR 1970866 (2004c:08012)
- 13.
- E. W. Kiss, M. Valeriote, Abelian algebras and the Hamiltonian property, Journal of Pure and Applied Algebra 87 (1993), 37-49. MR 1222175 (94d:08002)
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Additional Information
Paweł Idziak
Affiliation:
Department of Theoretical Computer Science, Jagiellonian University, Kraków, Poland
Email:
idziak@tcs.uj.edu.pl
Ralph McKenzie
Affiliation:
Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email:
ralph.n.mckenzie@vanderbilt.edu
Matthew Valeriote
Affiliation:
Department of Mathematics & Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1
Email:
matt@math.mcmaster.ca
DOI:
http://dx.doi.org/10.1090/S0894-0347-08-00614-0
PII:
S 0894-0347(08)00614-0
Received by editor(s):
June 26, 2006
Posted:
September 12, 2008
Article copyright:
© Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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