Directive trees and games on posets

Abstract

We show that for any infinite cardinal κ, every (κ+1)-strategically closed poset is κ +-strategically closed if and only if κ holds. This extends previous results of Velleman, et.al.

Approachability and Games on Posets

“…Generalized directive trees. Here we introduce the generalized notion of directive trees, which was first introduced in [5]. Let us review some basic terminology on trees.…”

“…A sector of a tree T is a union of several components of T. For a tree T and a € T, the history of a in 7" is the unique branch of T whose top element is a. Note that for an infinite cardinal K and a limit ordinal A > K, being (A, < £)directive is equivalent for £ e (K, min(A, K + ) ] and is also equivalent to being (A, K-directive in the sense of [5]. PROOF.…”

“…As for G n -type games, Velleman [11] has shown that for any infinite cardinal K, every cr-closed, (K + 1)-strategically closed poset is K + -strategically closed if and only if D K holds. Ishiu and Yoshinobu [5] have shown that the condition 'a-closed' can be dropped in Velleman's result.…”

“…In this paper we continue to study such prolongation of strategic closure properties and also of strong strategic closure properties. [5] is frequently referred to, although most of this paper is self-contained. In Section 2 of this paper, several versions of square principles we will use are introduced.…”

“…In Section 4, we prove a theorem (Theorem 4.1) which describes when a prolongation of strategic closure properties happens. Most results in Section 3 and Section 4 are generalizations of those of [5], and the basic idea of their proofs are already given there. But we write down the full proofs for the sake of completeness.…”