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Set Theory and Structure Theorems

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1006)

Abstract

The purpose of this survey is to provide an introduction to the interesting role which set-theoretic methods have begun to play in the problem of determining the structure and classification of certain classes of abelian groups, in particular, the class of ω1-separable groups, i.e., groups such that every countable subset is contained in a Σ -cyclic summand.

Keywords

  • Abelian Group
  • Test Problem
  • Structure Theorem
  • Countable Subset
  • Pure Subgroup

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Research supported by NSF Grant No. MCS80–03591E

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References

  1. Dugas, M., Fast freie abelsche Gruppen mit Endomorphismen-ring Z, J. Alg. 71 (1981), 314–321.

    CrossRef  Google Scholar 

  2. Eklof, P., Set Theoretic Methods in Homological Algebra and Abelian Groups, les Presses de l’Université de Montréal (1980).

    Google Scholar 

  3. Eklof, P., The structure of W1-separable groups, Trans. Amer. Math. Soc., to appear.

    Google Scholar 

  4. Eklof, P., Methods of logic in abelian group theory, in: Abelian Group Theory, Springer-Verlag Lecture Notes in Math., No. 616 (1977), 251–269.

    Google Scholar 

  5. Eklof, P. and Mekler, A., On constructing indecomposable groups in L, J. Alg. 49 (1977), 96–103.

    CrossRef  Google Scholar 

  6. Eklof, P. and Mekler, A., On endomorphism rings of (xi-separable primary groups, this volume.

    Google Scholar 

  7. Eklof, P., Mekler, A., and Shelah, S., Almost disjoint abelian groups, Israel J. Math., to appear.

    Google Scholar 

  8. Huber, M., Methods of set theory and the abundance of separable abelian p-groups, this volume

    Google Scholar 

  9. Kunen, K., Set Theory, North Holland Pub. Co. (1980).

    Google Scholar 

  10. Megibben, C., W1-separable p-groups, preprint.

    Google Scholar 

  11. Mekler, A., On Shelah’s Whitehead groups and CH, Rocky Mt. J. Math. 12 (1982), 271–278.

    Google Scholar 

  12. Mekler, A., Proper forcing and abelian groups, this volume.

    Google Scholar 

  13. Mekler, A., c.c.c. forcing without combinatorics, preprint.

    Google Scholar 

  14. Mekler, A., Structure theory for w1-separable groups, to appear.

    Google Scholar 

  15. Mekler, A., How to construct almost free groups, Can. J. Math. 32 (1980), 1206–1228.

    Google Scholar 

  16. Shelah, S., On endo-rigid strongly ti -free abelian groups in til, Israel J. Math. 40 (1981), 291–295.

    Google Scholar 

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© 1983 Springer-Verlag Berlin Heidelberg

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Eklof, P.C. (1983). Set Theory and Structure Theorems. In: Göbel, R., Lady, L., Mader, A. (eds) Abelian Group Theory. Lecture Notes in Mathematics, vol 1006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21560-9_13

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  • DOI: https://doi.org/10.1007/978-3-662-21560-9_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12335-4

  • Online ISBN: 978-3-662-21560-9

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