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Algebra universalis

, 80:41 | Cite as

Atoms in infinite dimensional free sequence-set algebras

  • Mohamed KhaledEmail author
  • István Németi
Article
  • 51 Downloads

Abstract

A. Tarski proved that the m-generated free algebra of \(\mathrm {CA}_{\alpha }\), the class of cylindric algebras of dimension \(\alpha \), contains exactly \(2^m\) zero-dimensional atoms, when \(m\ge 1\) is a finite cardinal and \(\alpha \) is an arbitrary ordinal. He conjectured that, when \(\alpha \) is infinite, there are no more atoms other than the zero-dimensional atoms. This conjecture has not been confirmed or denied yet. In this article, we show that Tarski’s conjecture is true if \(\mathrm {CA}_{\alpha }\) is replaced by \(\mathrm {D}_{\alpha }\), \(\mathrm {G}_{\alpha }\), but the m-generated free \(\mathrm {Crs}_{\alpha }\) algebra is atomless.

Keywords

Free algebras Cylindric-like algebras Atoms and zero-dimensional elements 

Mathematics Subject Classification

03G15 03B20 06E25 

Notes

References

  1. 1.
    Henkin, L., Monk, J. D., Tarski, A.: Cylindric Algebras Part I. vol 64 of studies in logic and the foundation of mathematics. North Holland (1971)Google Scholar
  2. 2.
    Sankappanavar, H.P., Burris, S.: A course in universal algebra. In: Graduate Texts in Mathematics, vol 78. Springer, New York (1981)Google Scholar
  3. 3.
    Henkin, L., Monk, J. D., Tarski, A.: Cylindric Algebras Part II. vol 115 of studies in logic and the foundation of mathematics. North Holland (1985)Google Scholar
  4. 4.
    Németi, I.: Free algebras and decidability in algebraic logic. Academic Doctoral Dissertation (in Hungarian), Hungarian Academy of Sciences, Budapest (1986) http://www.renyi.hu/~nemeti/NDis/NDis86.pdf
  5. 5.
    Andréka, H., Thompson, R.J.: A Stone type representation theorem for algebras of relations of higher rank. Trans. Am. Math. Soc. 309, 671–682 (1988)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Andréka, H., Jónsson, B., Németi, I.: Free algebras in discriminator varieties. Algebra Univ. 28, 401–447 (1991)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Németi, I.: Decidable versions of first order logic and cylindric-relativized set algebras. In: Logic colloquium’92, Cisirma, L., Gabbay, D.M., de Rijke, M. (eds) CSLI Publications, Stanford, California, and European Association for Logic, Language and Information, pp. 177–241 (1995)Google Scholar
  8. 8.
    Németi, I.: Cylindric-relativized set algebras have strong amalgamation. J. Symb. Logic 50, 689–700 (1985)CrossRefGoogle Scholar
  9. 9.
    Mikulás, Sz: Taming first-order logic. Log. J. IGPL 6(2), 305–316 (1998)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Andréka, H., Hodkinson, I., Németi, I.: Finite algebras of relations are represntable on finite sets. J. Symb. Logic 64, 243–267 (1999)CrossRefGoogle Scholar
  11. 11.
    Monk, J.D.: An introduction to cylindric set algebras. Log. J. IGPL 8(4), 451–492 (2000)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Andréka, H.: A finite axiomatization of locally square cylindric-relativized set algebras. Stud. Sci. Math. Hungar. 38, 1–11 (2001)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Madarász, J., Németi, I.: Free Boolean algebras with closure operators and a conjecture of Henkin, Monk, and Tarski. Stud. Sci. Math. Hungar. 38, 273–278 (2001)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Marx, M.J.: Computing with cylindric modal logics and arrow logics, lower bounds. Stud. Log. 72(2), 233–252 (2002)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gyenis, Z.: On atomicity of free algebras in certain cylindric-like varieties. Log. J. IGPL 19(1), 44–52 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Andréka, H., Németi, I.: Reducing First-order Logic to \(\rm Df_3\), free algebras. In: Andréka, H., Ferenczi, M., Németi, I. (eds) (2013). Cylindric-like Algebras and Algebraic Logic, vol 22 of Bolyai Society Mathematical Studies, Springer (2013)Google Scholar
  17. 17.
    Khaled, M.: Gödel’s incompleteness properties and the guarded fragment: an algebraic approach. PhD thesis. Central European University. (2016) https://mathematics.ceu.edu/sites/mathematics.ceu.hu/files/attachment/basicpage/27/phdthesis.pdf
  18. 18.
    Khaled, M.: The free non-commutative cylindric algebras are not atomic. Log. J. IGPL 25(5), 673–685 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Banerjee, A., Khaled, M.: First order logic without equality on relativized semantics. Ann. Pure Appl. Log. 169(11), 1227–1242 (2018)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Khaled, M.: The finitely axiomatizable complete theories of non-associative arrow frames. Adv. Math. 346, 194–218 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Engineering and Natural SciencesBahçeşehir UniversityIstanbulTurkey
  2. 2.Alfréd Rényi Institute of MathematicsBudapestHungary

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