View More View Less
  • 1 Cairo University, Giza, Egypt
Restricted access

Purchase article

USD  $25.00

1 year subscription (Individual Only)

USD  $800.00

Let α be an infinite ordinal. Let RCAα denote the variety of representable cylindric algebras of dimension α. Modifying Andréka’s methods of splitting, we show that the variety RQEAα of representable quasi-polyadic equality algebras of dimension α is not axiomatized by a set of universal formulas containing only finitely many variables over the variety RQAα of representable quasi-polyadic algebras of dimension α. This strengthens a seminal result due to Sain and Thompson, answers a question posed by Andréka, and lifts to the transfinite a result of hers proved for finite dimensions > 2. Using the modified method of splitting, we show that all known complexity results on universal axiomatizations of RCAα (proved by Andréka) transfer to universal axiomatizations of RQEAα. From such results it can be inferred that any algebraizable extension of Lω,ω is severely incomplete if we insist on Tarskian square semantics. Ways of circumventing the strong non-negative axiomatizability results hitherto obtained in the first part of the paper, such as guarding semantics, and /or expanding the signature of RQEAω by substitutions indexed by transformations coming from a finitely presented subsemigroup of (ωω, ○) containing all transpositions and replacements, are surveyed, discussed, and elaborated upon.

  • [1]

    Andréka, H., Complexity of equations valid in algebras of relations, Annals of Pure and Applied logic,, 89 (1997), p. 149209.

  • [2]

    Andréka, H., Ferenczi, M. and Németi, I. (Editors), Cylindric-like Algebras and Algebraic Logic, Bolyai Society Mathematical Studies (2013).

    • Search Google Scholar
    • Export Citation
  • [3]

    Andréka, H., Hodkinson, I. and Németi, I., Finite algebras of relations are representable on finite sets, Journal of Symbolic Logic,, 64 (1) (1999), p. 243267.

    • Search Google Scholar
    • Export Citation
  • [4]

    Andréka, H., Németi I. and Sayed Ahmed T., A non-representable quasipolyadic equality algebra with a representable cylindric reduct, Studia Math. Hungarica,, 50(1) (2013), p. 116.

    • Search Google Scholar
    • Export Citation
  • [5]

    Andréka, H. and Thompson R., A Stone type representation theorem for algebras of relations of higher rank, Transactions of the American Mathematical Society,, 309 (1988), p. 671682.

    • Search Google Scholar
    • Export Citation
  • [6]

    Benthem, V., Crs and guarded logic, a fruitful contact. In [2].

  • [7]

    Bulian, J. and Hodkinson, I., Bare canonicity of representable cylindric and polyadic algebras, Annals of Pure and Applied Logic,, 164 (2013), p. 884906.

    • Search Google Scholar
    • Export Citation
  • [8]

    Chang, C. and Keisler, J., Model Theory, Studies in Logic and the Foundation of Mathematics, 7, North Holland 1994.

  • [9]

    Ferenczi, M., The polyadic generalization of the Boolean axiomatization of fields of sets, Trans. of the Amer. Math. Society,, 364(2) (2011), p. 867886.

    • Search Google Scholar
    • Export Citation
  • [10]

    Ferenczi, M., A new representation theory for cylindric-like algebras. In [2], p. 106–135.

  • [11]

    Fremlin, D. H., Consequences of Martin’s axiom, Cambridge University Press, 1984.

  • [12]

    Henkin, L., Monk, J. D. and Tarski, A., Cylindric Algebras Part 1. North Holland, 1970.

  • [13]

    Henkin, L., Monk, J. D. and Tarski, A., Cylindric Algebras Part II. North Holland, 1985.

  • [14]

    Hirsch, R. and Hodkinson, I., Relation Algebras by Games. Studies In Logic. North Holland, 147 (2002).

  • [15]

    Hirsch, R. and Hodkinson, I., Complete representations in algebraic logic, Journal of Symbolic Logic,, 62(3) (1997) p. 816847.

  • [16]

    Hirsch, R. and Hodkinson, I., Completions and complete representations in algebraic logic. In [2].

  • [17]

    Hirsch, R. and Sayed Ahmed, T., The neat embedding problem for algebras other than cylindric algebras and for infinite dimensions, Journal of Symbolic Logic,, 79(1) (2014), p. 208222.

    • Search Google Scholar
    • Export Citation
  • [18]

    Kurucz, A., Representable cylindric algebras and many dimensional modal logics. In [2], pp. 185–204

  • [19]

    Marx, M., Algebraic relativization and arrow logic. Ph.D. thesis, 1995 ILLC dissertation Series.

  • [20]

    Khaled, M. and Sayed Ahmed, T., The Andréka–Resek–Thompson and Ferenczi results using games and more, Pre-print.

  • [21]

    Monk, J. D., Non–finite axiomatizability of classes of representable cylindric algebras, Journal of Symbolic Logic,, 34 (1969), p. 331–343.

    • Search Google Scholar
    • Export Citation
  • [22]

    Sain, I., Searching for a finitizable algebraization of first order logic, Logic Journal of IGPL. Oxford University Press,, 8(4) (2000), 495589.

    • Search Google Scholar
    • Export Citation
  • [23]

    Sain, I. and Gyuris, V., Finite Schematizable Algebraic Logic, Logic Journal of IGPL,, 5 (1997), p. 699–751.

  • [24]

    Sain, I. and Thompson, R. J., Strictly finite Schema Axiomatization of Quasi polyadic algebras. In: ‘Algebraic Logic’ North Holland, Editors Andréka, H., Monk, D., Németi, I., p. 539572.

    • Search Google Scholar
    • Export Citation
  • [25]

    Sági, G., On non-representable G-polyadic algebras with representable cylindric reducts, Logic Journal of IGPL,, 19(1) (2011), p. 105109.

    • Search Google Scholar
    • Export Citation
  • [26]

    Sági, G., Polyadic algebras. In [2].

  • [27]

    Sági, G. and Sziráki, D., Vaught’s conjecture from the perspective of algebraic logic, Logic Journal of IGPL (2012). First published on line January 5, 2012.

    • Search Google Scholar
    • Export Citation
  • [28]

    Sayed Ahmed, T., Amalgamation for reducts of polyadic algebras, Algebra Universalis,, 51 (2004), p. 301359.

  • [29]

    Sayed Ahmed, T., Algebraic Logic, where does it stand today? Bulletin of Symbolic Logic,, 11(4) (2005), p. 465516.

  • [30]

    Sayed Ahmed, T., On finite axiomatizability of expansions of cylindric algebras, Journal of Algebra, Number Theory, Advances and Applications,, 1 (2010), p. 1940.

    • Search Google Scholar
    • Export Citation
  • [31]

    Sayed Ahmed, T., On the complexity of axiomatizations of the class of representable quasi-polyadic equality algebras, Mathematical Logic Quarterly,, 4 (2011), p. 384394.

    • Search Google Scholar
    • Export Citation
  • [32]

    Sayed Ahmed, T., Three interpolation theorems for typeless logics, Logic Journal of IGPL,, 20(6) (2012), p. 10011037.

  • [33]

    Sayed Ahmed, T., Neat reducts and neat embeddings in cylindric algebras. In [2].

  • [34]

    Sayed Ahmed, T., Completions, Complete representations and Omitting types. In [2].

  • [35]

    Sayed Ahmed, T., On notions of representability for cylindric-polyadic algebras and a solution to the finitizability problem for first order logic with equality, Mathematical Logic Quarterly,, 61(6) (2015), p. 418447.

    • Search Google Scholar
    • Export Citation
  • [36]

    Sayed Ahmed, T., Non existence of finite variable universal axiomatizations for representable diagonal free cylindric algebras of dimension>2. Pre-print.