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Power-collapsing games

Published online by Cambridge University Press:  12 March 2014

Miloš S. Kurilić  and
Boris Šobot
Miloš S. Kurilić
Affiliation:
Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia, E-mail: milos@im.ns.ac.yu
Boris Šobot
Affiliation:
Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia, E-mail: sobot@im.ns.ac.yu
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Abstract

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 = 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 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 is undetermined.

Keywords

Boolean algebrasgamesSuslin treesforcing
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 4 , December 2008 , pp. 1433 - 1457
Copyright
Copyright © Association for Symbolic Logic 2008

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References

REFERENCES

[1]Dobrinen, N., Games and generalized distributive laws in Boolean algebras, Proceedings of the American Mathematical Society, vol. 131 (2003), no. 1, pp. 309–318.CrossRefGoogle Scholar
[2]Dobrinen, N., Errata to “Games and generalized distributive laws in Boolean algebras”, Proceedings of the American Mathematical Society, vol. 131 (2003), no. 9, pp. 2967–2968.CrossRefGoogle Scholar
[3]Jech, T., More game-theoretic properties of Boolean algebras, Annals of Pure and Applied Logic, vol. 26 (1984), pp. 11–29.CrossRefGoogle Scholar
[4]Jech, T., Set theory, 2nd corr. ed., Springer, Berlin, 1997.CrossRefGoogle Scholar
[5]Kunen, K., Set theory, An introduction to independence proofs, North-Holland, Amsterdam, 1980.Google Scholar
[6]Kurilić, M. S. and Šobot, B., A game on Boolean algebras describing the collapse of the continuum, submitted.Google Scholar
[7]Zapletal, J., More on the cut and choose game, Annals of Pure and Applied Logic, vol. 76 (1995), pp. 291–301.CrossRefGoogle Scholar