The Journal of Symbolic Logic

Research Article

Power-collapsing games

Miloš S. Kurilića1 and Boris Šobota2

a1 Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia, E-mail: milos@im.ns.ac.yu

a2 Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia, E-mail: sobot@im.ns.ac.yu

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 p α ∈ (0, p) and Black responds choosing i α ∈{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 2 κ 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.

(Received February 19 2008)

Key words and phrases

  • Boolean algebras;
  • games;
  • Suslin trees;
  • forcing