Skip to main content Accessibility help
×
Home

Power-collapsing games

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

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 ∈ 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.

Copyright

References

Hide All
[1]Dobrinen, N., Games and generalized distributive laws in Boolean algebras, Proceedings of the American Mathematical Society, vol. 131 (2003), no. 1, pp. 309318.
[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. 29672968.
[3]Jech, T., More game-theoretic properties of Boolean algebras, Annals of Pure and Applied Logic, vol. 26 (1984), pp. 1129.
[4]Jech, T., Set theory, 2nd corr. ed., Springer, Berlin, 1997.
[5]Kunen, K., Set theory, An introduction to independence proofs, North-Holland, Amsterdam, 1980.
[6]Kurilić, M. S. and Šobot, B., A game on Boolean algebras describing the collapse of the continuum, submitted.
[7]Zapletal, J., More on the cut and choose game, Annals of Pure and Applied Logic, vol. 76 (1995), pp. 291301.

Keywords

Related content

Powered by UNSILO

Power-collapsing games

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

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.