On the weak distributivity game
The game \({\mathcal G}_{\text{fin}} (\kappa)\) of \textit{T. J. Jech} [ibid. 26, 11-29 (1984; Zbl 0555.03025)] is studied. The implication: if \({\mathcal B}\) is weakly \((\omega,\kappa)\)-distributive then player I does not have a winning strategy in the game \({\mathcal G}_{\text{fin}} (\kappa)\) is considered. It is shown that the implication is true for a proper complete Boolean algebra [\textit{S. Shelah}, Proper forcing (Lect. Notes Math. 940) (1982; Zbl 0495.03035)], but is not true in general. Also, a weakly \((\omega, \infty)\)-distributive Boolean algebra is obtained such that player I has a winning strategy in the game \({\mathcal G}_ \omega (\omega_ 2)\), solving problems 3 and 4 of the paper of Jech.
scientific article; zbMATH DE number 513339
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