A Game on Partial Orderings

We study the determinacy of the game Gκ(A) introduced in [FuKoShe] for uncountable regular κ and several classes of partial orderings A. Among trees or Boolean algebras, we can always find an A such that Gκ(A) is undetermined. For the class of linear orders, the existence of such A depends on the size of κ. In particular we obtain a characterization of κ = κ in terms of determinacy of the game Gκ(L) for linear orders L. We consider in this paper the question whether for every partially ordered set (A,≤), the game Gκ(A) described below is determined, i.e. whether one of the players has a winning strategy. Here and in the following, except for the motivation given below, κ is always a regular uncountable cardinal. More precisely we study the question for trees, Boolean algebras and linear orderings. In fact there are trees, resp. Boolean algebras, A of size κ for which Gκ(A) is not determined (Propositions 6 and 11); for linear orders, the situation is more complex: if κ = κ, then for every linear order L, Gκ(L) is determined (Proposition 2); otherwise there is a linear order L of size κ such that Gκ(L) is not determined (Proposition 8). The motivation for this question comes from the paper [FuKoShe] which in turn was motivated by [HeSha]. A Boolean algebra A is said to have the Freese-Nation property if there exists a function f which assigns to every a ∈ A a finite subset f(a) of A such that if a, b ∈ A satisfy a ≤ b, then a ≤ x ≤ b holds for some x ∈ f(a)∩f(b). This property is closely related to projectivity; in fact, every projective Booleran algebra has the Freese-Nation property (but not conversely). Heindorf proved that the Freese-Nation property is equivalent to open-generatedness, a notion originally introduced in topology by Ščepin. In [FuKoShe], it is generalized to from ω to regular cardinals κ and from Boolean algebras to arbitrary partial orderings. This generalization is called κ-Freese-Nation property and the following equivalence was proved: a partial ordering A has the κ-Freese-Nation property iff there is a closed unbounded subset C of [A] such that C ≤κ A holds for all C ∈ C iff in the game 1991 Mathematics Subject Classification. 03E05.