December 2008 Power-collapsing games
Miloš S. Kurilić, Boris Šobot
J. Symbolic Logic 73(4): 1433-1457 (December 2008). DOI: 10.2178/jsl/1230396930


The game 𝔖ls (κ) is played on a complete Boolean algebra 𝔹, 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∈𝔹. In the α-th move White chooses pα ∈(0,p)𝔹 and Black responds choosing iα ∈{0,1}. White wins the play iff \bigwedgeβ ∈κ α ≥ β pα iα =0, where pα ⁰=pα and pα ¹=p∖ pα . The corresponding game theoretic properties of c.B.a.'s are investigated. So, Black has a winning strategy (w.s.) if κ ≥ π (𝔹) or if 𝔹 contains a κ⁺-closed dense subset. On the other hand, if White has a w.s., then κ ∈ [𝔥₂(𝔹), π(𝔹)). The existence of w.s. is characterized in a combinatorial way and in terms of forcing. In particular, if 2 < κ =κ ∈ Reg and forcing by 𝔹 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. 𝔹 such that White (respectively, Black) has a w.s. for each infinite cardinal κ∈ S (resp. κ∉ S). Also it is shown consistent that for each κ ∈ Reg there is a c.B.a. on which the game 𝔖ls(κ) is undetermined.


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Miloš S. Kurilić. Boris Šobot. "Power-collapsing games." J. Symbolic Logic 73 (4) 1433 - 1457, December 2008.


Published: December 2008
First available in Project Euclid: 27 December 2008

zbMATH: 1159.03035
MathSciNet: MR2467228
Digital Object Identifier: 10.2178/jsl/1230396930

Primary: 03E05 , 03E35 , 03E40 , 03G05 , 06E10 , 91A44

Keywords: Boolean algebras , Forcing , games , Suslin trees

Rights: Copyright © 2008 Association for Symbolic Logic


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Vol.73 • No. 4 • December 2008
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