previous :: next

### Power-collapsing games

Miloš S. Kurilić and Boris Šobot
Source: J. Symbolic Logic Volume 73, Issue 4 (2008), 1433-1457.

#### Abstract

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.

First Page:
Primary Subjects: 91A44, 03E40, 03E35, 03E05, 03G05, 06E10
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.jsl/1230396930
Digital Object Identifier: doi:10.2178/jsl/1230396930
Mathematical Reviews number (MathSciNet): MR2467228
Zentralblatt MATH identifier: 1159.03035

### References

N. Dobrinen, Games and generalized distributive laws in Boolean algebras, Proceedings of the American Mathematical Society, vol. 131 (2003), no. 1, pp. 309--318.
Mathematical Reviews (MathSciNet): MR1929051
Digital Object Identifier: doi:10.1090/S0002-9939-02-06501-2
--------, Errata to Games and generalized distributive laws in Boolean algebras'', Proceedings of the American Mathematical Society, vol. 131 (2003), no. 9, pp. 2967--2968.
Mathematical Reviews (MathSciNet): MR1974356
Digital Object Identifier: doi:10.1090/S0002-9939-03-07197-1
T. Jech, More game-theoretic properties of Boolean algebras, Annals of Pure and Applied Logic, vol. 26 (1984), pp. 11--29.
Mathematical Reviews (MathSciNet): MR739910
Digital Object Identifier: doi:10.1016/0168-0072(84)90038-1
Zentralblatt MATH: 0555.03025
--------, Set theory, 2nd corr. ed., Springer, Berlin, 1997.
Mathematical Reviews (MathSciNet): MR1492987
K. Kunen, Set theory, An introduction to independence proofs, North-Holland, Amsterdam, 1980.
Mathematical Reviews (MathSciNet): MR597342
M. S. Kurilić and B. Šobot, A game on Boolean algebras describing the collapse of the continuum, submitted.
J. Zapletal, More on the cut and choose game, Annals of Pure and Applied Logic, vol. 76 (1995), pp. 291--301.
Mathematical Reviews (MathSciNet): MR1366514
Digital Object Identifier: doi:10.1016/0168-0072(95)00002-X
Zentralblatt MATH: 0849.03043
previous :: next