Skip to main content
Log in

Construction of existentially closed Abelian lattice-ordered groups using Fraïssé limits

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

New examples of existentially closed Abelian lattice-ordered groups, possessing most of the known properties of infinitely generic Abelian lattice-ordered groups, are constructed using Fraïssé limits.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Conrad, P.F.: Embedding theorems for Abelian groups with valuations. Am. J. Math. 75, 1–29 (1953)

    Article  MathSciNet  Google Scholar 

  2. Darnel, M.R.: Theory of lattice-ordered groups. Monographs and Textbooks in Pure and Applied Mathematics, vol. 187. Marcel Dekker, New York (1995)

  3. Gillman, L., Henriksen, M.: Rings of continuous functions in which every finitely generated ideal is principal. Trans. Am. Math. Soc. 82, 366–391 (1956)

    Article  MathSciNet  Google Scholar 

  4. Gillman, L., Jerison, M.: Rings of Continuous Functions. The University Series in Higher Mathematics. Van Nostrand, Princeton (1960)

    Book  Google Scholar 

  5. Glass, A.M.W., Pierce, K.R.: Existentially complete Abelian lattice-ordered groups. Trans. Am. Math. Soc. 261, 255–270 (1980)

    Article  MathSciNet  Google Scholar 

  6. Hodges, W.: Model Theory. Encyclopedia of Mathematics and its Applications, vol. 42. Cambridge University Press, Cambridge (1993)

  7. Pierce, K.R.: Amalgamating Abelian ordered groups. Pac. J. Math. 43, 711–723 (1972)

    Article  MathSciNet  Google Scholar 

  8. Pierce, K.R.: Amalgamated sums of Abelian \(\ell \)-groups. Pac. J. Math. 65, 167–173 (1976)

    Article  MathSciNet  Google Scholar 

  9. Rothmaler, P.: Introduction to Model Theory. Algebra, Logic and Applications Series, vol. 15. Gordon and Breach Science Publishers, Amsterdam (2000)

  10. Saracino, D., Wood, C.: Finitely generic Abelian lattic-ordered groups. Trans. Am. Math. 277, 113–123 (1983)

    Article  Google Scholar 

  11. Saracino, D., Wood, C.: An example in the model theory of Abelian lattice-ordered groups. Algebra Universalis 19, 34–37 (1984)

    Article  MathSciNet  Google Scholar 

  12. Scowcroft, P.: Algebraically closed and existentially closed Abelian lattice-ordered groups. Algebra Universalis 3, 257–300 (2016)

    Article  MathSciNet  Google Scholar 

  13. Scowcroft, P.: Model-completions for Abelian lattice-ordered groups with finitely many disjoint elements. Ann. Pure Appl. Logic 170, 637–698 (2019)

    MathSciNet  MATH  Google Scholar 

  14. Weispfenning, V.: Model theory of Abelian \(\ell \)-groups. In: Glass, A.M.W., Holland, W.C. (eds.) Lattice-ordered Groups: Advances and Techniques. Math. Appl., vol. 48, pp. 41–79. Kluwer Academic Publishers, Dordrecht (1989)

  15. Wynne, B.: Construction of existentially closed Abelian lattice-ordered groups using upper extensions. Algebra Universalis 79(3), Art. 51 (2018)

Download references

Acknowledgements

The author wishes to thank Philip Scowcroft for discussing his work on \(\ell \)-groups and for his comments on an early draft of this paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Brian Wynne.

Additional information

Presented by W. Wm. McGovern.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wynne, B. Construction of existentially closed Abelian lattice-ordered groups using Fraïssé limits. Algebra Univers. 82, 18 (2021). https://doi.org/10.1007/s00012-020-00706-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00012-020-00706-1

Mathematics Subject Classification

Keywords

Navigation