## $$\kappa$$-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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### References:

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