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By R. Göbel, C. Metelli, A. Orsatti, L. Salce

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Extra info for Abelian Groups and Modules: Proceedings of the Udine Conference Udine April 9–14, 1984

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L Suppose For n=O player I has no choice. play so 'V\) far. - 1 ('\t l' (n-1))""); LJ ~~ f1 . 11 (3), condition (ii) has whereas IXll <~for l player I uses the strategy defined above}. is it disjoint? solutions, by induction on n for so ~ ~ many ordinals are excluded. 1 <"' Why (l many ~ is a play of ~~ Trivially W is If,_ is a branch of f for (f ,N) above we can reconstruct the play from Now ~ a let in which barrier.

As we may assume But then = B 0 by [] EXAMPLE 3. 3. w C • o. B(n) 2 p -bounded countable valuated group A G of length that is not a direct sum of cyclics. order p be of order x1 Let 2 shall n th and value and value sequence . 0 The p • 1 • 2 • 3 • 2 • 3 • 4 Ulm invariant of show that G (n-l,n). G is has no finite for 0. For Let n '1- 1, n G > let x n have be the subgroup of and summand with a 1 G(p) c G(2). nonzero 0 th We Ulm Subgroups of Bounded Abelian Groups invariant. A Suppose that is nonzero.

Been such [Sh 7] for more results and [Sh 1] , We will continue with the combinatorics of [ Sh 4], utilized GBbel and Shelah in [GS ]. Dugas and GBbel in [DG 1], [DG 2] and by by The nicest feature of these fact that they were carried out in ZFC. VIII, Th. which characterization proofs was the Their main drawbacks were: The algebraic objects had strong limit singular cardinal numbers of not small cofinality. (ii) The combinatorics was not separated from the proof; so analogous proofs have to repeat it.

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