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Algebra universalis

, Volume 63, Issue 1, pp 17–36 | Cite as

Some results about neat reducts

  • Tarek Sayed Ahmed
Article

Abstract

This is a survey article on the concept of neat reducts. An old venerable idea in algebraic logic, in this paper we show why it is regaining momentum.

2000 Mathematics Subject Classification

Primary: 03G15 Secondary: 03CO5 

Key words and phrases

Algebraic logic cylindric algebras neat reducts amalgamation complete representations 

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References

  1. 1.
    Sayed Ahmed T.: The class of neat reducts is not elementary. Logic Journal of IGPL 9, 593–628 (2001)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Sayed Ahmed T.: The class of 2-dimensional neat reducts of polyadic algebras is not elementary. Fund. Math. 172, 61–81 (2002)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Sayed Ahmed T.: A Model-theoretic Solution to a problem of Tarski. Mathematical Logic Quaterly 48, 343–355 (2002)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Sayed Ahmed T.: Martin’s axiom, omitting types and complete representations in algebraic logic. Studia Logica 72, 1–25 (2002)MathSciNetGoogle Scholar
  5. 5.
    Sayed Ahmed T.: Neat embeddings, interpolation, and omitting types, an overview. Notre Dame Journal of formal logic 44, 157–173 (2003)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Sayed Ahmed T.: On amalgamation of reducts of polyadic algebras. Algebra Universalis 51, 301–359 (2004)MATHMathSciNetGoogle Scholar
  7. 7.
    Sayed Ahmed T.: Algebraic Logic, where does it stand today?. Bulletin of Symbolic Logic 11(4), 465–516 (2005)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Sayed Ahmed T.: The class of infinite dimensional neat reducts of quasi-polyadic algebras is not axiomatizable. Mathematical Logic Quarterly 52, 106–112 (2006)MATHCrossRefGoogle Scholar
  9. 9.
    Sayed Ahmed T.: An interpolation theorem for first order logic with infinitary predicates. Logic Journal of IGPL 15, 21–32 (2007)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Sayed Ahmed T.: A note on neat reducts. Studia Logica 85, 139–151 (2007)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Sayed Ahmed T.: Neat embedding is not sufficient for complete representations. Bulletin Section of Logic 36, 29–36 (2007)Google Scholar
  12. 12.
    Sayed Ahmed T.: The superamalgamation property via neat embeddings, and a problem of Henkin and Monk. International Journal of Algebra 2, 533–554 (2008)MATHMathSciNetGoogle Scholar
  13. 13.
    Sayed Ahmed T.: On complete representability of reducts of Polyadic algebras. Studia Logica 89, 325–332 (2008)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Sayed Ahmed, T.: A categorical approach to amalgamation theorems. To appear in Reports on Mathematical Logic.Google Scholar
  15. 15.
    Sayed Ahmed, T.: Confirming a Conjecture of Tarski in Algebraic Logic. To appear in Reports on Marthematical Logic.Google Scholar
  16. 16.
    Sayed Ahmed, T.: Omitting types algebraically. To appear in International Journal of AlgebraGoogle Scholar
  17. 17.
    Sayed Ahmed, T.: On relation algebras with quasi-projections. PreprintGoogle Scholar
  18. 18.
    Sayed Ahmed, T.: Algebras having the unique neat embedding property. Submitted.Google Scholar
  19. 19.
    Sayed Ahmed T., Andréka H., Németi I.: Omitting types for finite variable fragments and complete representations for algebras. Journal of Symbolic Logic 73, 65–89 (2008)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Sayed Ahmed T., Samir B.: A Neat embedding theorem for expansions of cylindric algebras. Logic Journal of IGPL 15, 41–51 (2007)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Sayed Ahmed T., Samir B.: Omitting types for first order logic with infinitary predicates. Mathematical Logic Quarterly 53(6), 564–576 (2007)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Andréka H: Complexity of equations valid in algebras of relations. Annals of Pure and Applied Logic 89, 149–209 (1997)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Andréka, H., Comer, C., Madarász, J., Németi, I., Sayed Ahmed, T.: Epimorphisms in cylindric algebras. Algebra Universalis (in press).Google Scholar
  24. 24.
    Andréka, H, Monk., J.D., Németi, I. (editors): Algebraic Logic. North-Holland, Amsterdam, (1991)Google Scholar
  25. 25.
    Andréka H., Thompson R.J.: A stone-type representation theorem for algebras of relation of higher rank. Trans. Amer. Math. Soc. 309, 671–682 (1988)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Amer M., Sayed Ahmed T.: Polyadic and cylindric algebras of sentences. Mathematical Logic Quarterly 52, 44–49 (2006)Google Scholar
  27. 27.
    Comer S.D.: Classes without the amalgamation property. Pacific journal of Mathematics 28(2), 309–318 (1969)MATHMathSciNetGoogle Scholar
  28. 28.
    Daigneault A., Monk J.D.: Representation Theory for Polyadic algebras. Fund. Math. 52, 151–176 (1963)MATHMathSciNetGoogle Scholar
  29. 29.
    Ellentuck E.: Categoricity regained. Journal of Symbolic Logic 41, 639–643 (1976)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Fremlin D, H.: Consequences of MA. Cambridge University press. (1984)Google Scholar
  31. 31.
    Hirsch R.: Relation algebra reducts of cylindric algebras and complete representations. Journal of Symbolic Logic 72(2), 673–703 (2007)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Hircsh R., Hodkinson I.: Complete representations in algebraic logic. Journal of Symbolic Logic 62, 816–847 (1997)CrossRefMathSciNetGoogle Scholar
  33. 33.
    Hirsch R., Hodkinson I., Maddux R.: Relation algebra reducts of cylindric algebras and an application to proof theory. Journal of Symbolic Logic 67, 197–213 (2002)MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Henkin, L., Monk, J.D., Tarski, A.: Cylindric Algebras Part I. North Holland, (1971)Google Scholar
  35. 35.
    Henkin, L., Monk, J.D., Tarski, A.: Cylindric Algebras Part II. North Holland, (1985)Google Scholar
  36. 36.
    Hodges, W.: Model Theory, vol 42 of Encyclopedia of mathematics and its applications. Cambridge University Press (1993)Google Scholar
  37. 37.
    Goldblatt R., Hodkinson I., Venema Y.: Erdos graphs resolve Fine’s canonicity problem. Bulletin of Symbolic Logic 10, 186–208 (2004)MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Madárasz J., Sayed Ahmed T.: Amalgamation, interpolation and epimorphisms. Algebra Universalis 56(2), 179–210 (2007)MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Madarász J.: Interpolation and Amalgamation, pushing the Limits. Part I. Studia Logica 61, 316–345 (1998)Google Scholar
  40. 40.
    Maksimova L.: Amalgamation and interpolation in normal modal logics. Studia Logica 50, 457–471 (1991)MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Martin D.A., Solovay R.M.: Internal Cohen extensions. Ann. Mathematical Logic 2, 143–178 (1970)MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Miller, A.: Covering 2ω with ω1 disjoint closed sets. The Kleene Symposuim (proceedings, Madison, Wisconsin, 1978). Studies in Logic and the Foundation of Mathematics, 101, North-Holland, Amsterdam, (1980), p.415–421.Google Scholar
  43. 43.
    Miller A.: Characterization of the least cardinal for which the Baire Category Theorem fails. Proc. Amer. Math. Soc. 86, 498–502 (1982)MATHMathSciNetGoogle Scholar
  44. 44.
    Németi I.: The Class of Neat Reducts of Cylindric Algebras is Not a Variety But is closed w.r.t. HP. Notre Dame Journal of Formal logic 24, 399–409 (1983)MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Németi, I.: Algebraisation of quantifier logics, an introductory overview. Math.Inst. Budapest, Preprint, No 13-1996. A shortened version appeared in Studia Logica 50, 465-569 (1991)Google Scholar
  46. 46.
    Pigozzi D.: Amalgamation, congruence extension, and interpolation properties in algebras. Algebra Universalis. 1, 289–349 (1971)CrossRefMathSciNetGoogle Scholar
  47. 47.
    Sagi G., Shelah S.: Weak and strong interpolation for algebraic logics. Journal of Symbolic Logic. 71, 104–118 (2006)MATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    Sagi, G.: On the Finitization problem in Algebraic logic. Ph.D Dissertation. Budapest (1999)Google Scholar
  49. 49.
    Sain, I.: Searching for a finitizable algebraization of first order logic. Logic Journal of IGPL. Oxford University Press, 4, 495–589 (2000)Google Scholar
  50. 50.
    Sagi G, Ferenszi M.: On some developments in the representation theory of cylindric-like algebras. Algebra Universalis 55, 345–353 (2006)MATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Simon A.: What the finitization problem is not. Algebraic methods in Logic and Computer Science, Banach Centre Publications 28, 95–116 (1996)Google Scholar
  52. 52.
    Simon, A.: Non-representable algebras of relations. Ph.D Dissertation. Budapest (1997).Google Scholar
  53. 53.
    Simon A.: Connections between quasi-projective relation algebras and cylindric algebras. Algebra universalis 56, 263–302 (2007)MATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Tarski, A., Givant, S.: A formalization of set theory without variables. Amer. Math. Soc. Colloquium Publications. 41(1987)Google Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceCairo UniversityGiza, CairoEgypt

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