Archive for Mathematical Logic

, Volume 57, Issue 3–4, pp 239–272 | Cite as

l-Groups C(X) in continuous logic

  • Philip Scowcroft


In the context of continuous logic, this paper axiomatizes both the class \(\mathcal {C}\) of lattice-ordered groups isomorphic to C(X) for X compact and the subclass \(\mathcal {C}^+\) of structures existentially closed in \(\mathcal {C}\); shows that the theory of \(\mathcal {C}^+\) is \(\aleph _0\)-categorical and admits elimination of quantifiers; establishes a Nullstellensatz for \(\mathcal {C}\) and \(\mathcal {C}^+\); shows that \(C(X)\in \mathcal {C}\) has a prime-model extension in \(\mathcal {C}^+\) just in case X is Boolean; and proves that in a sense relevant to continuous logic, positive formulas admit in \(\mathcal {C}^+\) elimination of quantifiers to positive formulas.


Lattice-ordered group Continuous logic Existentially closed 

Mathematics Subject Classification

03C60 06F20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bankston, P.: Some applications of the ultrapower theorem to the theory of compacta. Appl. Categ. Struct. 8, 45–66 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ben Yaacov, I., Berenstein, A., Henson, C.W., Usvyatsov, A.: Model theory for metric structures. In: Chatzidakis, Z., Macpherson, D., Pillay, A., Wilkie, A. (eds.) Model Theory with Applications to Algebra and Analysis, vol. 2, pp. 315–427. Cambridge University Press, Cambridge (2008)Google Scholar
  3. 3.
    Bohnenblust, H.F., Kakutani, S.: Concrete representation of (M)-spaces. Ann. Math. 42, 1025–1028 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Comfort, W.W., Negrepontis, S.: The Theory of Ultrafilters. Springer-Verlag, New York (1974)CrossRefzbMATHGoogle Scholar
  5. 5.
    Eagle, C., Vignati, A.: Saturation and elementary equivalence of \(C^{*}\)-algebras. arXiv:1406.4875v4
  6. 6.
    Efimov, B., Engelking, R.: Remarks on dyadic spaces. II. Colloq. Math. 13, 181–197 (1964/1965)Google Scholar
  7. 7.
    Gillman, L., Jerison, M.: Rings of Continuous Functions. D. Van Nostrand Co., Princeton (1960)CrossRefzbMATHGoogle Scholar
  8. 8.
    Glass, A.M.W., Pierce, K.R.: Existentially complete abelian lattice-ordered groups. Trans. Am. Math. Soc. 261, 255–270 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Glass, A.M.W., Pierce, K.R.: Equations and inequations in lattice-ordered groups. In: Smith, J.E., Kenny, G.O., Ball, R.N. (eds.) Ordered Groups: Proceedings of the Boise State Conference, pp. 141–171. Marcel Dekker Inc, New York (1980)Google Scholar
  10. 10.
    Gleason, A.: Projective topological spaces. Ill. J. Math. 2, 482–489 (1958)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Hager, A.: Minimal covers of topological spaces. In: Kopperman, R., Misra, P., Reichman, J., Todd, A. (eds.) Papers on General Topology and Related Category Theory and Topological Algebra, pp. 44–59. New York Academy of Sciences, New York (1989)Google Scholar
  12. 12.
    Hodges, W.: Building Models by Games. Dover Publications, New York (2006)zbMATHGoogle Scholar
  13. 13.
    Kakutani, S.: Concrete representation of abstract (M)-spaces (A characterization of the space of continuous functions). Ann. Math. 42, 994–1024 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Levy, R.: Almost P-spaces. Can. J. Math. 29, 284–288 (1977)CrossRefzbMATHGoogle Scholar
  15. 15.
    Saracino, D., Wood, C.: Finitely generic abelian lattice-ordered groups. Trans. Am. Math. Soc. 277, 113–123 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Saracino, D., Wood, C.: An example in the model theory of abelian lattice-ordered groups. Algebra Universalis 19, 34–37 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Scowcroft, P.: Some model-theoretic correspondences between dimension groups and AF algebras. Ann. Pure Appl. Log. 162, 755–785 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Stone, M.H.: The generalized Weierstrass approximation theorem. Math. Mag. 21, 167–184 (1948)MathSciNetCrossRefGoogle Scholar
  19. 19.
    van den Dries, L.: Some applications of a model theoretic fact to (semi-) algebraic geometry. Indag. Math. 44, 397–401 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Weispfenning, V.: Model Theory of Lattice Products. Habilitationsschrift, U. Heidelberg, Heidelberg (1979)zbMATHGoogle Scholar
  21. 21.
    Weispfenning, V.: Model theory of abelian \(l\)-groups. In: Glass, A.M.W., Holland, W.C. (eds.) Lattice-Ordered Groups: Advances and Techniques, pp. 41–79. Kluwer Academic Publishers, Dordrecht (1989)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceWesleyan UniversityMiddletownUSA

Personalised recommendations