Advertisement

The Non-Slender Rank of an Abelian Group

  • Burkhard Wald
Conference paper
Part of the International Centre for Mechanical Sciences book series (CISM, volume 287)

Abstract

For a family (Ai)i∈I of Abelian groups and a cardinal K we define the K-product \(\mathop \Pi \limits_{i \in I} {A_i}\) to be the subgroup of the cartesian product \({\mathop \Pi \limits_I ^{(K)}}A\) consisting of all elements which support is less than K. Let us write AI(K) instead of \({A^{I(w)}} = \mathop \oplus \limits_I A\), A(I) instead of (math) and A[I] instead of AI(W1) . We are going to use the groups Z[K] to introduce a new cardinal invariant for an abelian group.

Keywords

Abelian Group Springer Lecture Note Measurable Cardinal Inaccessible Cardinal Infinite Cardinal 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Bc]
    S. Balcerzyk, On groups of functions defined on Boolean algebras, Fund. Math. 50 (1962) 347–367.MATHMathSciNetGoogle Scholar
  2. [Bg]
    J.E. Baumgardner, Almost disjoint sets, the dence set problem and the partition calculus, Ann. Math. Logic 9 (1976) 401–439.CrossRefMathSciNetGoogle Scholar
  3. [Ch]
    S.U. Chase, On direct sums and products of modules, Pacific J.Math. 12 (1962) , 847–854.CrossRefMATHMathSciNetGoogle Scholar
  4. [D]
    H.D. Donder, in preparationGoogle Scholar
  5. [DG]
    M. Dugas and R. Göbel, On radicals and products, to appear in Pacific J. Math.Google Scholar
  6. [DZ]
    M. Dugas and B. Zimmermann-Huisgen, Iterated direct sums and products of moduls, in Abelian Group Theory. Proceedings, Oberwolfach 1981, Springer Lecture Notes 874 (1981) 179–173.CrossRefGoogle Scholar
  7. [F]
    L. Fuchs, Infinite Abelian Groups II, Academic Press, New York 1974.Google Scholar
  8. [GW]
    R. Göbel and B. Wald, Wachstumstypen und schlanke Gruppen, Symp. Math. 23 (1979) 201–239.Google Scholar
  9. [GWW]
    R. Göbel, B. Wald and P. Westphal, Groups of integer-valuated functions, in Abelian Group Theory, Proceedings, Oberwolfach 1981, Springer Lecture Notes, 874 (1981) 161–178.Google Scholar
  10. [I]
    A.V. Ivanov, Direct sums and complete direct sums of abelian groups (Russian), In Abelian Groups and Modules, Tomsk. Gos. Univ., (1) 70–90, 136–137.Google Scholar
  11. [J]
    T. Jech, Set Theory, Academic Press, New York, London (1978).Google Scholar
  12. [JP]
    T. Jech and K. Prikry, Ideals over uncountable sets: application of almost disjoint functions and generic ultrapowers, Memoirs Amer. Math.Soc. 18 (1979) no. 214.Google Scholar
  13. [K1]
    K. Kunen, Some application of iterated ultrapowers in set theory, Ann.Math.Logic 1 (1970) 179–227.CrossRefMATHMathSciNetGoogle Scholar
  14. [K2]
    K. Kunen, Saturated ideals, J. Symbolic Logic 43 (1978) 65–76.CrossRefMATHMathSciNetGoogle Scholar
  15. [KM]
    A. Kanamori and M. Magidor, The evolution of large cardinal axioms in set theory, in Higher Set Theory, Proceedings, Oberwolfach 1977, Springer Lecture Notes 669 (1978) 99–275.CrossRefGoogle Scholar
  16. [N]
    R.J. Nunke, Slender groups, Bull. Amer. Math. Soc. 67 (1961) 274–275; Acta Sci. Math Szeged 23 (1962) 67–73.CrossRefMATHMathSciNetGoogle Scholar
  17. [S]
    R.M. Soloway, Real-valued measurable cardinals, in Axiomatic Set Theory (D. Scott, ed.), Proc. Symp. Pure Math. 13 I (1971) 397–428.CrossRefGoogle Scholar
  18. [T]
    A. Tarski, Ideale in vollständigen Mengen-Körpern I, Fund. Math. 32 (1939) 45–63Google Scholar
  19. [U]
    S. Ulam, Zur Maßtheorie in der allgemeinen Mengenlehre, Fund. Math. 16 (1930) 140–150.MATHGoogle Scholar
  20. [W1]
    B. Wald, Martinaxiom und die Beschreibung gewisser Homomorphismen in der Theorie der N1-freien abelschen Gruppen, Manuscripta Math. 42 (1983) 297–309.CrossRefMATHMathSciNetGoogle Scholar
  21. [W2]
    B. Wald, On K-products modulo µ-products, in Abelian Group Theory, Proceedings, Honolulu. 1982/83, Springer Lecture Notes 1006 (1983) 362–370.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 1984

Authors and Affiliations

  • Burkhard Wald
    • 1
  1. 1.Freie Universität BerlinGermany

Personalised recommendations