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\(\kappa \)-stationary subsets of \(\mathcal P_{\kappa ^{+}}\lambda \), infinitary games, and distributive laws in Boolean algebras. (English) Zbl 1142.03031

Jech began the investigation of the the relationships between games and distributivity laws in Boolean algebras. He could show that the \((\omega,\infty)\)-d.l. holds iff player I does not have a winning strategy in the descending sequence game of length \(\omega\).
Here the author characterizes the \((\kappa,\lambda, <\mu)\)-distributivity law in Boolean algebras in terms of the cut and choose games \({\mathcal G}_{<\mu}^{\kappa}(\lambda)\) when \(\mu \leq \kappa \leq \lambda\) and \(\kappa^{<\kappa} = \kappa\). This continues previous work of the author and gives game-theoretic characterizations of distributivity laws for almost all triples of cardinals \(\kappa\), \(\lambda\), \(\mu\) with \(\mu \leq \lambda\), under GCH. To obtain the results, it is necessary to consider whether the \(\kappa\)-stationarity of \({\mathcal P}_{\kappa^+} \lambda\) in the ground model is preserved by \(\mathbb B\).
The author develops the theory of \(\kappa\)-club and \(\kappa\)-stationary subsets of \({\mathcal P}_{\kappa^+} \lambda\). Further on she constructs a Boolean algebra in which player I has a winning strategy for \({\mathcal G}_{\kappa}^{\kappa}(\kappa^+)\) but the \((\kappa, \infty, \kappa)\)-d.l. holds. Assuming GCH, she constructs Boolean algebras in which many games are undetermined.

MSC:

03E40 Other aspects of forcing and Boolean-valued models
03E05 Other combinatorial set theory
06E05 Structure theory of Boolean algebras
91A44 Games involving topology, set theory, or logic
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