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

Purchase article

USD  $25.00

1 year subscription (Individual Only)

USD  $800.00

Abstract

For β an ordinal, let PEAβ (SetPEAβ) denote the class of polyadic equality (set) algebras of dimension β. We show that for any infinite ordinal α, if APEAα is atomic, then for any n < ω, the n-neat reduct of A, in symbols rnAB, is a completely representable PEAn (regardless of the representability of A). That is to say, for all non-zero arnA, there is a BaSetPEAn and a homomorphism fa:rnAB such that fa(a) ≠ 0 and fa(X)=xXfa(x) for any X=A for which X exists. We give new proofs that various classes consisting solely of completely representable algebras of relations are not elementary; we further show that the class of completely representable relation algebras is not closed under ≡∞,ω. Various notions of representability (such as ‘satisfying the Lyndon conditions’, weak and strong) are lifted from the level of atom structures to that of atomic algebras and are further characterized via special neat embeddings. As a sample, we show that the class of atomic CAns satisfying the Lyndon conditions coincides with the class of atomic algebras in ElScNrnCAω, where El denotes ‘elementary closure’ and Sc is the operation of forming complete subalgebras.

  • [1]

    Andréka, H., Ferenczi, M. and Né meti, I. (Editors), Cylindric-like Algebras and Algebraic Logic, Bolyai Society Mathematical Studies 22 (2013).

    • Search Google Scholar
    • Export Citation
  • [2]

    Andréka, H., Németi, I. and Sayed-Ahmed, T., A non-representable qausi-polyadic equality algebra with a representable cylindric reduct Stud. Math. Hungerica, 50(1) (2013), 116.

    • Search Google Scholar
    • Export Citation
  • [3]

    Daigneault, A. and Monk, J. D. Representation Theory for Polyadic algebras, Fundamenta Mathematica, 52 (1963), 151176.

  • [4]

    Ferenczi, M., A new representation theory for cylindric-like algebras, in: Cylindric-like Algebras and Algebraic Logic, Bolyai Society Mathematical Studies 22 (2013), 106135.

    • Search Google Scholar
    • Export Citation
  • [5]

    Ferenczi, M., A new representation theory: Representing cylindric-like algebras by relativized set algebras, in: Cylindric-like Algebras and Algebraic Logic, Bolyai Society Mathematical Studies 22 (2013), 135162.

    • Search Google Scholar
    • Export Citation
  • [6]

    Henkin, L., Monk, J. D. and Tarski, A., Cylindric Algebras Part I, II. North Holland, 1971, 1985.

  • [7]

    Hirsch, R., Relation algebra reducts of cylindric algebras and complete representations, Journal of Symbolic Logic, 72(2) (2007), 673703.

    • Search Google Scholar
    • Export Citation
  • [8]

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

  • [9]

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

  • [10]

    Hirsch, R. and Hodkinson, I., Completions and complete representations, in: Cylindric-like Algebras and Algebraic Logic, Bolyai Society Mathematical Studies 22 (2013), 6190.

    • Search Google Scholar
    • Export Citation
  • [11]

    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), 208222.

    • Search Google Scholar
    • Export Citation
  • [12]

    Németi, I. and Sági, G., On the equational theory of representable Polyadic Equality algebras Journal of Symbolic Logic, 65(5) (2000), 11431167.

    • Search Google Scholar
    • Export Citation
  • [13]

    Sági, G., Non-computability of Equational Theory of Polyadic Algebra, Bulletin Section of Logic, 3 (2001), 155165.

  • [14]

    Sági, G., On non-representable G polyadic algebras with representable cylindric reduct, Logic Journal of IGPL, 10(1) (2008), 105109.

    • Search Google Scholar
    • Export Citation
  • [15]

    Sayed Ahmed, T., Neat embedding is not sufficient for complete representations, Bulletin Section of Logic, 36(1) (2007), 2936.

  • [16]

    Sayed Ahmed, T., The class of neat reducts is not elementary, Logic Journal of IGPL, 9 (2001), 593628.

  • [17]

    Sayed Ahmed, T., Yet some more non finite axiomatizability results for algebras of relations and ways to avoid them, Stud. Math. Hungarica, 53(3) (2016), 322378.

    • Search Google Scholar
    • Export Citation
  • [18]

    Sayed Ahmed, T., Neat reducts and neat embeddings in cylindric algebras, in: Cylindric-like Algebras and Algebraic Logic, Bolyai Society Mathematical Studies 22 (2013), 90105.

    • Search Google Scholar
    • Export Citation
  • [19]

    Sayed Ahmed, T., Completions, Complete representations and Omitting types, in: Cylindric-like Algebras and Algebraic Logic, Bolyai Society Mathematical Studies 22 (2013), 186205.

    • Search Google Scholar
    • Export Citation
  • [20]

    Sayed Ahmed, T. and Németi, I., On neat reducts of algebras of logic, Studia Logica, 68(2) (2001), 229262.

  • [21]

    Khaled, M. and Sayed Ahmed, T., On complete representations of algebrs of logic, IGPL, 17 (2009), 267272.

  • [22]

    Sayed Ahmed, T., The class of completely representable polyadic algebras of infinite dimensions is elementary Algebra universalis, 72(1) (2014), 371390.

    • Search Google Scholar
    • Export Citation
  • [23]

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

    • Search Google Scholar
    • Export Citation