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Varieties whose finitely generated members are free

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Abstract

We prove that a variety of algebras whose finitely generated members are free must be definitionally equivalent to the variety of sets, the variety of pointed sets, a variety of vector spaces over a division ring, or a variety of affine vector spaces over a division ring.

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References

  1. Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer, New York (1981)

    Book  MATH  Google Scholar 

  2. Campion, T (http://mathoverflow.net/users/2362/tim-campion).: Varieties where every algebra is free. http://mathoverflow.net/q/157974 (version: 2014-03-15). Accessed 7 July 2015

  3. Freese, R., McKenzie, R.: Commutator Theory for Congruence Modular Varieties, vol, 125. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1987)

  4. Givant, S.: Universal Horn classes categorical or free in power. Ann. Math. Logic 15, 1–53 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hobby, D., McKenzie, R.: The Structure of Finite Algebras, vol. 76. Contemporary Mathematics. American Mathematical Society, Providence (1988)

  6. Idziak, P., McKenzie, R., Valeriote, M.: The structure of locally finite varieties with polynomially many models. J. Am. Math. Soc. 22, 119–165 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kearnes, K.A.: Almost all minimal idempotent varieties are congruence modular. Algebra Universalis 44, 39–45 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kearnes, K.A., Kiss, E.W.: The shape of congruence lattices. Mem. Amer. Math. Soc. 222, no. 1046 (2013)

  9. Kearnes, K.A., Kiss, E.W., Valeriote, M.A.: Minimal sets and varieties. Trans. Am. Math. Soc. 350, 1–41 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kearnes, K., Szendrei, Á.: The relationship between two commutators. Int. J. Algebra Comput. 8, 497–531 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Szendrei, Á.: On closed classes of quasilinear functions. Czechoslov. Math. J. 30, 498–509 (1980)

    MathSciNet  MATH  Google Scholar 

  12. Szendrei, Á.: Strongly abelian minimal varieties. Acta Sci. Math. (Szeged) 59, 25–42 (1994)

    MathSciNet  MATH  Google Scholar 

  13. Szendrei, Á.: Maximal non-affine reducts of simple affine algebras. Algebra Universalis 34, 144–174 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Colorado, Boulder, CO, 80309-0395, USA

    Keith A. Kearnes & Ágnes Szendrei

  2. Department of Algebra and Number Theory, Loránd Eötvös University, Pázmány Péter sétány 1/c, Budapest, 1117, Hungary

    Emil W. Kiss

Authors
  1. Keith A. Kearnes
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  2. Emil W. Kiss
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  3. Ágnes Szendrei
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Corresponding author

Correspondence to Ágnes Szendrei.

Additional information

Presented by R. Willard.

Dedicated to E. Tamás Schmidt.

This article is part of the topical collection “In memory of E. Tamás Schmidt” edited by Robert W. Quackenbush.

This material is based upon work supported by the National Science Foundation Grant no. DMS 1500254 and the Hungarian National Foundation for Scientific Research (OTKA) Grant nos. K104251 and K115518.

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Kearnes, K.A., Kiss, E.W. & Szendrei, Á. Varieties whose finitely generated members are free. Algebra Univers. 79, 3 (2018). https://doi.org/10.1007/s00012-018-0480-5

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  • DOI: https://doi.org/10.1007/s00012-018-0480-5

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