Algebra universalis

, Volume 75, Issue 3, pp 257–300 | Cite as

Algebraically closed and existentially closed Abelian lattice-ordered groups

  • Philip ScowcroftEmail author


If \({\mathcal{G}}\) is an Abelian lattice-ordered (l-) group, then \({\mathcal{G}}\) is algebraically (existentially) closed just in case every finite system of l-group equations (equations and inequations), involving elements of \({\mathcal{G}}\), that is solvable in some Abelian l-group extending \({\mathcal{G}}\) is solvable already in \({\mathcal{G}}\). This paper establishes two systems of axioms for algebraically (existentially) closed Abelian l-groups, one more convenient for modeltheoretic applications and the other, discovered by Weispfenning, more convenient for algebraic applications. Among the model-theoretic applications are quantifierelimination results for various kinds of existential formulas, a new proof of the amalgamation property for Abelian l-groups, Nullstellensätze in Abelian l-groups, and the display of continuum-many elementary-equivalence classes of existentially closed Archimedean l-groups. The algebraic applications include demonstrations that the class of algebraically closed Abelian l-groups is a torsion class closed under arbitrary products, that the class of l-ideals of existentially closed Abelian l-groups is a radical class closed under binary products, and that various classes of existentially closed Abelian l-groups are closed under bounded Boolean products.

Key words and phrases

lattice-ordered group algebraically closed existentially closed finitely generic infinitely generic 

2010 Mathematics Subject Classification

Primary: 03C60 Secondary: 06F20 03C25 


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  1. 1.
    Barwise, J. (ed.): Handbook of Mathematical Logic. Studies in Logic and the Foundations of Mathematics, vol. 90. North-Holland, Amsterdam (1978)Google Scholar
  2. 2.
    Conrad, P., Darnel, M.: Generalized Boolean algebras in lattice-ordered groups. Order 14, 295–319 (1997/98)Google Scholar
  3. 3.
    Darnel M.: Theory of Lattice-ordered Groups. Monographs and Textbooks in Pure and Applied Mathematics, vol. 187. Marcel Dekker, New York (1995)Google Scholar
  4. 4.
    Gillman L., Henriksen M.: Rings of continuous functions in which every finitely generated ideal is principal. Trans. Amer. Math. Soc. 82, 366–391 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gillman L., Jerison M.: Rings of Continuous Functions. The University Series in Higher Mathematics. Van Nostrand, Princeton (1960)CrossRefzbMATHGoogle Scholar
  6. 6.
    Glass A.M.W., Pierce K.R.: Existentially complete Abelian lattice-ordered groups. Trans. Amer. Math. Soc. 261, 255–270 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    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 (Proc. Conf., Boise State Univ., Boise, Idaho, 1978). Lecture Notes in Pure and Appl. Math., vol. 62, pp. 141–171. Marcel Dekker, New York (1980)Google Scholar
  8. 8.
    Hodges W.: Model Theory. Encyclopedia of Mathematics and its Applications, no. 42. Cambridge University Press, Cambridge (1993)Google Scholar
  9. 9.
    Hodges W.: Building Models by Games. Dover, New York (2006)zbMATHGoogle Scholar
  10. 10.
    Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces, vol. I. North-Holland Mathematical Library. North-Holland, Amsterdam (1971)Google Scholar
  11. 11.
    Mal’cev, A.I.: Regular products of models. In: Wells, B.F. (ed.) The Metamathematics of Algebraic Systems. Studies in Logic and the Foundations of Mathematics, vol. 66, pp. 95–113. North-Holland, Amsterdam (1971)Google Scholar
  12. 12.
    Marker D.: Model Theory: An Introduction. Graduate Texts in Math., vol. 217. Springer, New York (2002)Google Scholar
  13. 13.
    Martinez J.: Torsion theory for lattice-ordered groups. Czechoslovak Math. J. 25, 284–299 (1975)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Pierce K.R.: Amalgamations of lattice ordered groups. Trans. Amer. Math. Soc. 172, 249–260 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Robinson, A.: Forcing in model theory. In: Symposia Math. V (INDAM, Rome, 1969–1970), pp. 69–82. Academic Press, London (1971)Google Scholar
  16. 16.
    Robinson, A.: Infinite forcing in model theory. In: Fenstad, J.E. (ed.) Proceedings of the Second Scandinavian Logic Symposium. Studies in Logic and the Foundations of Mathematics, vol. 63, pp. 317–340. North-Holland, Amsterdam (1971)Google Scholar
  17. 17.
    Saracino D., Wood C.: Finitely generic Abelian lattice-ordered groups. Trans. Amer. Math. Soc. 277, 113–123 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Saracino D., Wood C.: An example in the model theory of abelian lattice-ordered groups. Algebra Universalis 19, 34–37 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Scowcroft P.: Existentially closed dimension groups. Trans. Amer. Math. Soc. 364, 1933–1974 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Scowcroft P.: More on generic dimension groups. Notre Dame J. Form. Log. 56, 511–553 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Volger H.: Filtered and stable boolean powers are relativized full boolean powers. Algebra Universalis 19, 399–402 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Weispfenning, V.: Model theory of Abelian l-groups. In: Glass, A.M.W., Holland, W.C. (eds.) Lattice-ordered Groups: Advances and Techniques. Math. Appl., vol. 48, pp. 41–79. Kluwer, Dordrecht (1989)Google Scholar

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© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceWesleyan UniversityMiddletownUSA

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