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.
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References
S. Balcerzyk, On groups of functions defined on Boolean algebras, Fund. Math. 50 (1962) 347–367.
J.E. Baumgardner, Almost disjoint sets, the dence set problem and the partition calculus, Ann. Math. Logic 9 (1976) 401–439.
S.U. Chase, On direct sums and products of modules, Pacific J.Math. 12 (1962) , 847–854.
H.D. Donder, in preparation
M. Dugas and R. Göbel, On radicals and products, to appear in Pacific J. Math.
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.
L. Fuchs, Infinite Abelian Groups II, Academic Press, New York 1974.
R. Göbel and B. Wald, Wachstumstypen und schlanke Gruppen, Symp. Math. 23 (1979) 201–239.
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.
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.
T. Jech, Set Theory, Academic Press, New York, London (1978).
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.
K. Kunen, Some application of iterated ultrapowers in set theory, Ann.Math.Logic 1 (1970) 179–227.
K. Kunen, Saturated ideals, J. Symbolic Logic 43 (1978) 65–76.
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.
R.J. Nunke, Slender groups, Bull. Amer. Math. Soc. 67 (1961) 274–275; Acta Sci. Math Szeged 23 (1962) 67–73.
R.M. Soloway, Real-valued measurable cardinals, in Axiomatic Set Theory (D. Scott, ed.), Proc. Symp. Pure Math. 13 I (1971) 397–428.
A. Tarski, Ideale in vollständigen Mengen-Körpern I, Fund. Math. 32 (1939) 45–63
S. Ulam, Zur Maßtheorie in der allgemeinen Mengenlehre, Fund. Math. 16 (1930) 140–150.
B. Wald, Martinaxiom und die Beschreibung gewisser Homomorphismen in der Theorie der N1-freien abelschen Gruppen, Manuscripta Math. 42 (1983) 297–309.
B. Wald, On K-products modulo µ-products, in Abelian Group Theory, Proceedings, Honolulu. 1982/83, Springer Lecture Notes 1006 (1983) 362–370.
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Wald, B. (1984). The Non-Slender Rank of an Abelian Group. In: Göbel, R., Metelli, C., Orsatti, A., Salce, L. (eds) Abelian Groups and Modules. International Centre for Mechanical Sciences, vol 287. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2814-5_15
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DOI: https://doi.org/10.1007/978-3-7091-2814-5_15
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