The structure of locally finite varieties with polynomially many models

Idziak, Paweł; McKenzie, Ralph; Valeriote, Matthew

doi:10.1090/S0894-0347-08-00614-0

The structure of locally finite varieties with polynomially many models
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by Paweł Idziak, Ralph McKenzie and Matthew Valeriote

J. Amer. Math. Soc. 22 (2009), 119-165

DOI: https://doi.org/10.1090/S0894-0347-08-00614-0

Published electronically: September 12, 2008

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Abstract:

We prove that a locally finite variety has at most polynomially many (in $k$) non-isomorphic $k$–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 $H$-sets, with constants, for various finite groups $H$.

References

George E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR 0557013
Joel Berman and PawełM. Idziak, Generative complexity in algebra, Mem. Amer. Math. Soc. 175 (2005), no. 828, viii+159. MR 2130585, DOI 10.1090/memo/0828
Joel Berman and Ralph McKenzie, Clones satisfying the term condition, Discrete Math. 52 (1984), no. 1, 7–29. MR 765281, DOI 10.1016/0012-365X(84)90100-6
Marcin Bilski, Generative complexity in semigroup varieties, J. Pure Appl. Algebra 165 (2001), no. 2, 137–149. MR 1865962, DOI 10.1016/S0022-4049(00)00189-4
David Hobby and Ralph McKenzie, The structure of finite algebras, Contemporary Mathematics, vol. 76, American Mathematical Society, Providence, RI, 1988. MR 958685, DOI 10.1090/conm/076
PawełM. Idziak and Ralph 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, DOI 10.4064/fm170-1-3
Keith A. Kearnes, Type preservation in locally finite varieties with the CEP, Canad. J. Math. 43 (1991), no. 4, 748–769. MR 1127028, DOI 10.4153/CJM-1991-043-1
Keith A. Kearnes, An order-theoretic property of the commutator, Internat. J. Algebra Comput. 3 (1993), no. 4, 491–533. MR 1250248, DOI 10.1142/S0218196793000299
Keith A. Kearnes, Locally solvable factors of varieties, Proc. Amer. Math. Soc. 124 (1996), no. 12, 3619–3625. MR 1343705, DOI 10.1090/S0002-9939-96-03501-0
K. A. Kearnes, A Hamiltonian property for nilpotent algebras, Algebra Universalis 37 (1997), no. 4, 403–421. MR 1465297, DOI 10.1007/s000120050025
Keith A. Kearnes and Emil W. Kiss, Modularity prevents tails, Proc. Amer. Math. Soc. 127 (1999), no. 1, 11–19. MR 1625765, DOI 10.1090/S0002-9939-99-04882-0
Keith A. Kearnes and Emil W. Kiss, Residual smallness and weak centrality, Internat. J. Algebra Comput. 13 (2003), no. 1, 35–59. MR 1970866, DOI 10.1142/S0218196703001237
Emil W. Kiss and Matthew A. Valeriote, Abelian algebras and the Hamiltonian property, J. Pure Appl. Algebra 87 (1993), no. 1, 37–49. MR 1222175, DOI 10.1016/0022-4049(93)90067-4
L. Lovász, Operations with structures, Acta Math. Acad. Sci. Hungar. 18 (1967), 321–328. MR 214529, DOI 10.1007/BF02280291
Ralph McKenzie and Matthew Valeriote, The structure of decidable locally finite varieties, Progress in Mathematics, vol. 79, Birkhäuser Boston, Inc., Boston, MA, 1989. MR 1033992, DOI 10.1007/978-1-4612-4552-0