Varieties of Algebras with no Nontrivial Finite Members

  • Andrzej Kisielewicz

Abstract

New examples of varieties with no nontrivial finite members given in this paper improve some earlier results or answer some open questions in this area. In particular, a generalization of Marczewski’s problem presented at the International Algebra Conference in Lisbon, June 88, is shown to have a solution in the negative.

Keywords

Stein 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A.K. Austin, [1965], A note on models of identities, Proc. Amer. Math. Soc. 16, 522–523.MathSciNetMATHCrossRefGoogle Scholar
  2. A.K. Austin, [1966], Finite models for laws in two variables, Proc. Amer. Math. Soc. 17, 1410–1412.MathSciNetMATHCrossRefGoogle Scholar
  3. S. Burris, [1972], Models in equational theories of unary algebras, Algebra Universalis 1, 386–392.MathSciNetMATHCrossRefGoogle Scholar
  4. J. Dudek, [1982], On the variety V (+, 0), Math. Sem. Notes 10, 9–15.MathSciNetMATHGoogle Scholar
  5. J. Dudek, [1985], Polynomially infinite varieties of algebras I, Math. Inst. Univ. Wroclaw, Preprint no 30.Google Scholar
  6. J. Dudek, [1988], A note on models of identities, Algebra Universalis, 25, 400-401.Google Scholar
  7. J. Dudek and A. Kisielewicz, [a], On finite models of regular identities, Notre Dame J. Formal Logic, to appear.Google Scholar
  8. A. Goetz and C. Ryll-Nardzewski, [1960], On bases of abstract algebras, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astr. Phys. 8, 157–161.MathSciNetMATHGoogle Scholar
  9. G. Grätzer, [1967], On spectra of classes of universal algebras, Proc. Amer. Math. Soc. 18, 729–735.MathSciNetMATHCrossRefGoogle Scholar
  10. G. Grätzer, [1979], Universal Algebra, 2nd ed., Springer-Verlag, Berlin.MATHGoogle Scholar
  11. B. Jonsson and A. Tarski, [1961], On two properties of free algebras, Math. Scand. 9, 95–101.MathSciNetMATHGoogle Scholar
  12. A. Kisielewicz, [1988], Marczewsk’s problem on algebras with bases of different cardinalities — solution and generalization, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astr. Phys., to appear.Google Scholar
  13. A. Kisielewicz, [a], On algebras with bases of different cardinalities, Fund. Math., to appear.Google Scholar
  14. R. McKenzie, [1975], On spectra, and the negative solution of the decision problem for identities having a finite non-trivial model, J. Symbolic Logic 40, 186–196.MathSciNetMATHCrossRefGoogle Scholar
  15. S.K. Stein, [1965], Finite models of identities, Proc. Amer. Math. Soc. 14, 216–222.CrossRefGoogle Scholar
  16. W. Taylor, [1973], Characterizing Mal’cev conditions, Algebra Universalis 3, 351–397.MathSciNetMATHCrossRefGoogle Scholar
  17. W. Taylor, [1979], Equational Logic, Houston J. Math., survey iss.Google Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • Andrzej Kisielewicz
    • 1
  1. 1.Institute of MathematicsTechnical University of WroclawWroclawPoland

Personalised recommendations