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Power-collapsing games

  • Miloš S. Kurilić (a1) and Boris Šobot (a2)

The game is played on a complete Boolean algebra , by two players. White and Black, in κ-many moves (where κ is an infinite cardinal). At the beginning White chooses a non-zero element p. In the α-th move White chooses pα ∈ (0, p) and Black responds choosing iα ∈{0, 1}. White winsthe play iff . where and .

The corresponding game theoretic properties of c.B.a.'s are investigated. So, Black has a winning strategy (w.s.) if κ ≥ π() or if contains a κ-closed dense subset. On the other hand, if White has a w.s., then κ. The existence of w.s. is characterized in a combinatorial way and in terms of forcing. In particular, if 2 = κ ∈ Reg and forcing by preserves the regularity of κ, then White has a w.s. iff the power 2κ is collapsed to κ in some extension. It is shown that, under the GCH, for each set S ⊆ Reg there is a c.B.a. such that White (respectively. Black) has a w.s. for each infinite cardinal κS (resp. κS). Also it is shown consistent that for each κ ∈ Reg there is a c.B.a. on which the game is undetermined.

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[3] T. Jech , More game-theoretic properties of Boolean algebras, Annals of Pure and Applied Logic, vol. 26 (1984), pp. 1129.

[4] T. Jech , Set theory, 2nd corr. ed., Springer, Berlin, 1997.

[7] J. Zapletal , More on the cut and choose game, Annals of Pure and Applied Logic, vol. 76 (1995), pp. 291301.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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