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Archive for Mathematical Logic

, Volume 55, Issue 7–8, pp 939–954 | Cite as

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

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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.

Keywords

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

Mathematics Subject Classification

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

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References

  1. 1.
    Baer, R.: Abelian groups without elements of finite order. Duke Math. J. 3, 68–122 (1937)MathSciNetCrossRefMATHGoogle Scholar
  2. 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. 3.
    Barwise, J., Eklof, P.: Infinitary properties of abelian torsion groups. Ann. Math. Log. 2, 25–68 (1970)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Fuchs, L.: Infinite Abelian Groups, vol. II. Academic Press, New York (1973)MATHGoogle Scholar
  5. 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)CrossRefMATHGoogle Scholar
  6. 6.
    Hill, P.: On the classification of abelian groups, Preprint (1967)Google Scholar
  7. 7.
    Hunter, R., Richman, F.: Global Warfield groups. Trans. Am. Math. Soc. 266(1), 555–572 (1981)Google Scholar
  8. 8.
    Hunter, R., Richman, F., Walker, E.: Warfield modules. In: Abelian Group Theory, Lecture Notes in Mathematics, vol. 616, pp. 87–123. Springer (1977)Google Scholar
  9. 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)Google Scholar
  10. 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)Google Scholar
  11. 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)Google Scholar
  12. 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)MathSciNetMATHGoogle Scholar
  13. 13.
    Jacoby, C., Loth, P.: Partial decomposition bases and Warfield modules. Comm. Algebra 42, 4333–4349 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Jacoby, C., Loth, P.: Partial decomposition bases and global Warfield groups, Comm. Algebra 44, 3262–3277 (2016)Google Scholar
  15. 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) Google Scholar
  16. 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)Google Scholar
  17. 17.
    Kaplansky, I.: Infinite Abelian Groups. University of Michigan Press, Ann Arbor (1968)MATHGoogle Scholar
  18. 18.
    Kaplansky, I.: Modules over Dedekind rings and valuation rings. Trans. Am. Math. Soc. 72, 327–340 (1952)MathSciNetCrossRefMATHGoogle Scholar
  19. 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. 20.
    Nunke, R.: Homology and direct sums of countable abelian groups. Math. Zeit 101, 182–212 (1967)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Richman, F.: A guide to valuated groups. In: Abelian Group Theory. Lecture Notes in Mathematics, vol. 616, pp. 73–86. Springer (1977)Google Scholar
  22. 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)Google Scholar
  23. 23.
    Ulm, H.: Zur Theorie der abzahlbar-unendlichen abelschen Gruppen. Math. Ann. 107, 774–803 (1933)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Warfield, R.B.: Classification theory of abelian groups I, balanced projectives. Trans. Am. Math. Soc. 222, 33–63 (1976)MathSciNetMATHGoogle Scholar
  25. 25.
    Warfield, R. B.: Classification theory of abelian groups II, local theory. Lecture Notes in Mathematics, vol. 874, pp. 322–349. Springer (1981)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Jacoby ConsultingLong BeachUSA
  2. 2.Department of MathematicsSacred Heart UniversityFairfieldUSA

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