Skip to main content
SpringerLink
Log in

Cylindric algebras and finite polyadic algebras

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

In this survey paper the short history of cylindric and finitary polyadic algebras (term-definitionally equivalent to quasi-polyadic algebras) is sketched, and the two concepts are compared. Roughly speaking, finitary polyadic algebras constitute a subclass of cylindric algebras that include a transposition operator being strong enough. We discuss the following question: should the definition of cylindric algebras include a transposition operator? Results confirm that the existence of a transposition operator ensures representability (by relativised set algebras). The different variants of cylindric algebras including a transposition operator play an important role in the theory of cylindric-like algebras.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ahmed, T.S.: A note on substitutions in representable cylindric algebras. Math. Log. Q. 55, 280–287 (2009)

    Article  MathSciNet  Google Scholar 

  2. Andréka, H.: Complexity of equations valid in algebras of relations. Ann. Pure Appl. Log. 89, 149–209 (1977)

    Article  MathSciNet  Google Scholar 

  3. Andréka, H.: A finite axiomatization of locally square cylindric-relativised set algebras. Stud. Sci. Math. Hung. 38, 1–11 (2001)

    MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  Google Scholar 

  5. Andréka, H., van Benthem, J., Németi, I.: Modal languages and bounded fragements of predicate logic. J. Philos. Log. 27, 217–274 (1998)

    Article  Google Scholar 

  6. Andréka, H., Németi, I.: How many varieties of cylindric algebras are there. Trans. Am. Math. Soc. 369, 8903–8937 (2017)

    Article  MathSciNet  Google Scholar 

  7. Andréka, H., Ferenczi, M., Németi, I.: Cylindric-like Algebras and Algebraic Logic. Springer, Kluwer, New York (2012)

    MATH  Google Scholar 

  8. Andréka, H., Németi, I., Sain, I.: Algebraic Logic. In: Gabbay, D.M., Guenther, F. (eds.) Handbook of Philosophical Logic, vol. 2, pp. 133–247. Kluwer, Dordrecht (2001)

    Chapter  Google Scholar 

  9. Andréka, H., Németi, I., Sayed Tarek, A.: A non representable infinite dimensional quasi-polyadic equality algebra with a representable cylindric reduct. Stud. Sci. Math. Hung. 50, 1–16 (2013)

    MathSciNet  MATH  Google Scholar 

  10. Daigneault, A., Monk, J.D.: Representation theory for polyadic algebras. Fundam. Math. 52, 151–176 (1963)

    Article  MathSciNet  Google Scholar 

  11. Ferenczi, M.: Finitary polyadic algebras from cylindric algebras. Stud. Log. 87, 1–11 (2007)

    Article  MathSciNet  Google Scholar 

  12. Ferenczi, M.: Existence of partial transposition means representability in cylindric algebras. Math. Log. Q. 57, 87–94 (2011)

    Article  MathSciNet  Google Scholar 

  13. Ferenczi, M.: The polyadic generalization of the Boolean axiomatization of fields of sets. Trans. Am. Math. Soc. 364, 867–886 (2012)

    Article  MathSciNet  Google Scholar 

  14. Ferenczi, M.: Representations of polyadic-like equality algebras. Algebra Univers. 75, 107–125 (2016)

    Article  MathSciNet  Google Scholar 

  15. Halmos, P.R.: Algebraic Logic IV. Equality in polyadic algebras. Trans. Am. Math. Soc. 86, 1–27 (1957)

    MathSciNet  Google Scholar 

  16. Halmos, P.R.: Algebraic Logic. Chelsea Publishing, New York (1962)

    MATH  Google Scholar 

  17. Henkin, L.: Relativization with repect to formulas and its use in proofs of independence. Compos. Math. 20, 88–106 (1968)

    MATH  Google Scholar 

  18. Henkin, L., Monk, J.D., Tarski, A.: Cylindric Algebras I. North Holland (1971)

  19. Henkin, L., Monk, J. D., Tarski, A.: Cylindric Algebras II. North Holland (1985)

  20. Jónsson, B., Tarski, A.: Boolean algebras with operators I. Am. J. Math. 73, 891–939 (1951)

    Article  MathSciNet  Google Scholar 

  21. Jónsson, B., Tarski, A.: Boolean algebras with operators II. Am. J. Math. 74, 127–162 (1952)

    Article  MathSciNet  Google Scholar 

  22. Jónsson, B.: Varieties of relation algebras. Algebra Univers. 15, 273–298 (1982)

    Article  MathSciNet  Google Scholar 

  23. Jónsson, B.: The theory of binary relations. In: Algebraic Logic. Proc. Conf. Budapest, 1988., Colloq. Math. Soc. J. Bolyai 54, pp. 245–292. North Holland (1991)

  24. Keisler, H.J.: A complete first order logic with infinitary predicates. Fundam. Math. 52, 177–203 (1963)

    Article  MathSciNet  Google Scholar 

  25. Maddux, R.: Topics in Relation Algebra. PhD thesis, University of California, Berkeley (1978)

  26. Németi, I.: Algebraization of quantifier logic and cylindric-relativized set algebras. In: Logic Colloquium, vol. 92, pp. 177–241. CSLI Publications (1995)

  27. Németi, I.: Free algebras and decidability in algebraic logic. PhD thesis, in Hungarian (1986)

  28. Németi, I., Sági, G.: On the equational theory of representable polyadic equality algebras. J. Symb. Log. 65, 1143–1167 (2000)

    Article  MathSciNet  Google Scholar 

  29. Németi, I., Simon, A.: Relation algebras from cylindric-and polyadic algebras. Log. J. IGPL 5, 575–588 (1997)

    Article  MathSciNet  Google Scholar 

  30. Pinter, C.: Cylindric algebras and algebras of substitutions. Trans. Am. Math. Soc. 175, 167–179 (1973)

    Article  MathSciNet  Google Scholar 

  31. Resek, D.: Some results on relativized cylindric algebras. PhD thesis, Berkeley (1975)

  32. Rieger, L.: Algebraic Methods of Algebraic Logic. Elsivier Academia, Amsterdam (1967)

    MATH  Google Scholar 

  33. Sain, I., Thompson, R.: Strictly finite schema axiomatization of quasi-polyadic algebras. In: Algebraic Logic, Colloq. Math. Soc. J. Bolyai 54, pp. 539–571. North Holland (1991)

  34. Sági, G.: Non-computability of the equational theory of polyadic algebras. Bull. Sect. Log. 30, 155–164 (2001)

    MathSciNet  MATH  Google Scholar 

  35. Sági, G.: A note on algebras of substitutions. Stud. Log. 72(2002), 265–284 (2002)

    Article  MathSciNet  Google Scholar 

  36. Sági, G.: On nonrepresentable G-polyadic algebras with representable cylindric reducts. Log. J. IGPL. 19, 105–109 (2011)

    Article  MathSciNet  Google Scholar 

  37. Sági, G.: Polyadic algebras. In: Andréka, H., Ferenczi, M., Németi, I. (eds.) Cylindric-like Algebras and Algebraic Logic, pp. 367–393. Kluwer, Bolyai J. Math. Soc. (2012)

Download references

Author information

Authors and Affiliations

  1. Department of Algebra, Budapest University of Technology and Economics, Budapest, Egry J. u. 1. bld. H, 1111, Hungary

    Miklós Ferenczi

Authors
  1. Miklós Ferenczi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miklós Ferenczi.

Additional information

In memory of Bjarni Jónsson.

This article is part of the topical collection “In memory of Bjarni Jónsson” edited by J. B. Nation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ferenczi, M. Cylindric algebras and finite polyadic algebras. Algebra Univers. 79, 60 (2018). https://doi.org/10.1007/s00012-018-0546-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00012-018-0546-4

Mathematics Subject Classification

Keywords

Search

Navigation