Skip to main content

The classification of \({\mathbb {Z}}_p\)-modules with partial decomposition bases in \(L_{\infty \omega }\)

Archive for Mathematical Logic volume 55, pages 939–954 (2016)Cite this article

Abstract

Ulm’s Theorem presents invariants that classify countable abelian torsion groups up to isomorphism. Barwise and Eklof extended this result to the classification of arbitrary abelian torsion groups up to \(L_{\infty \omega }\)-equivalence. In this paper, we extend this classification to a class of mixed \({\mathbb {Z}}_p\)-modules which includes all Warfield modules and is closed under \(L_{\infty \omega }\)-equivalence. The defining property of these modules is the existence of what we call a partial decomposition basis, a generalization of the concept of decomposition basis. We prove a complete classification theorem in \(L_{\infty \omega }\) using invariants deduced from the classical Ulm and Warfield invariants.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  1. Baer, R.: Abelian groups without elements of finite order. Duke Math. J. 3, 68–122 (1937)

    MathSciNet  Article  MATH  Google Scholar 

  2. Barwise, J.: Back and forth through infinitary logic. In: Morley, M.D. (ed.) Studies in Model Theory. Studies in Mathematics, p. 2. Mathematical Association of America, Washington (1973)

    Google Scholar 

  3. Barwise, J., Eklof, P.: Infinitary properties of abelian torsion groups. Ann. Math. Log. 2, 25–68 (1970)

    MathSciNet  Article  MATH  Google Scholar 

  4. Fuchs, L.: Infinite Abelian Groups, vol. II. Academic Press, New York (1973)

    MATH  Google Scholar 

  5. Göbel, R., Leistner, K., Loth, P., Strüngmann, L.: Infinitary equivalence of \(\mathbb{Z}_p\)-modules with nice decomposition bases. J. Comm. Algebra 3, 321–348 (2011)

    Article  MATH  Google Scholar 

  6. Hill, P.: On the classification of abelian groups, Preprint (1967)

  7. Hunter, R., Richman, F.: Global Warfield groups. Trans. Am. Math. Soc. 266(1), 555–572 (1981)

  8. Hunter, R., Richman, F., Walker, E.: Warfield modules. In: Abelian Group Theory, Lecture Notes in Mathematics, vol. 616, pp. 87–123. Springer (1977)

  9. Jacoby, C.: The classification in \(L_{\infty \omega }\) of groups with partial decomposition bases. Ph.D. thesis, University of California, Irvine (1980), revised version (2010)

  10. Jacoby, C.: Undefinability of local Warfield groups in \(L_{\infty \omega }\). In: Groups and Model Theory: A Conference in Honor of Rüdiger Göbel’s 70th Birthday, Contemp. Math., vol. 576, pp. 151–162. American Mathematical Society, Providence, RI (2012)

  11. Jacoby, C. and Loth, P.: Abelian groups with partial decomposition bases in \(L_{\infty \omega }^\delta \), Part II. In: Groups and Model Theory: A Conference in Honor of Rüdiger Göbel’s 70th Birthday, Contemp. Math., vol. 576, pp. 177–185. American Mathematical Society, Providence, RI (2012)

  12. Jacoby, C., Loth, P.: \(\mathbb{Z}_p\)-modules with partial decomposition bases in \(L_{\infty \omega }^\delta \). Houst. J. Math. 40(4), 1007–1019 (2014)

    MathSciNet  MATH  Google Scholar 

  13. Jacoby, C., Loth, P.: Partial decomposition bases and Warfield modules. Comm. Algebra 42, 4333–4349 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  14. Jacoby, C., Loth, P.: Partial decomposition bases and global Warfield groups, Comm. Algebra 44, 3262–3277 (2016)

  15. Jacoby, C., Loth, P.: The classification of infinite abelian groups with partial decomposition bases in \(L_{\infty \omega }\) Rocky Mountain J. Math. (to appear)

  16. Jacoby, C., Leistner, K., Loth, P., Strüngmann, L.: Abelian groups with partial decomposition bases in \(L_{\infty \omega }^\delta \), Part I. In: Groups and Model Theory: A Conference in Honor of Rüdiger Göbel’s 70th Birthday, Contemp. Math., vol. 576, pp. 163–175. American Mathematical Society, Providence, RI (2012)

  17. Kaplansky, I.: Infinite Abelian Groups. University of Michigan Press, Ann Arbor (1968)

    MATH  Google Scholar 

  18. Kaplansky, I.: Modules over Dedekind rings and valuation rings. Trans. Am. Math. Soc. 72, 327–340 (1952)

    MathSciNet  Article  MATH  Google Scholar 

  19. Karp, C.: Finite quantification equivalence. In: Addison, J.W., Henkin, L., Tarski, A. (eds.) The Theory of Models, pp. 407–412. North-Holland, Amsterdam (1965)

    Google Scholar 

  20. Nunke, R.: Homology and direct sums of countable abelian groups. Math. Zeit 101, 182–212 (1967)

    MathSciNet  Article  MATH  Google Scholar 

  21. Richman, F.: A guide to valuated groups. In: Abelian Group Theory. Lecture Notes in Mathematics, vol. 616, pp. 73–86. Springer (1977)

  22. Stanton, R. O.: Decomposition bases and Ulm’s Theorem. In: Abelian Group Theory. Lecture Notes in Mathematics, vol. 616, pp. 39–56. Springer (1977)

  23. Ulm, H.: Zur Theorie der abzahlbar-unendlichen abelschen Gruppen. Math. Ann. 107, 774–803 (1933)

    MathSciNet  Article  MATH  Google Scholar 

  24. Warfield, R.B.: Classification theory of abelian groups I, balanced projectives. Trans. Am. Math. Soc. 222, 33–63 (1976)

    MathSciNet  MATH  Google Scholar 

  25. Warfield, R. B.: Classification theory of abelian groups II, local theory. Lecture Notes in Mathematics, vol. 874, pp. 322–349. Springer (1981)

Download references

Author information

Authors and Affiliations

  1. Jacoby Consulting, Long Beach, CA, USA

    Carol Jacoby

  2. Department of Mathematics, Sacred Heart University, Fairfield, CT, 06825, USA

    Peter Loth

Authors
  1. Carol Jacoby
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Loth
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Loth.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jacoby, C., Loth, P. The classification of \({\mathbb {Z}}_p\)-modules with partial decomposition bases in \(L_{\infty \omega }\) . Arch. Math. Logic 55, 939–954 (2016). https://doi.org/10.1007/s00153-016-0506-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-016-0506-7

Keywords

  • Partial decomposition basis
  • Partial isomorphism
  • Infinitary equivalence
  • Ulm-Kaplansky invariants
  • Warfield invariants

Mathematics Subject Classification

  • Primary 03C52
  • 13C05
  • 20K21
  • Secondary 03E10
  • 20K25
  • 20K35