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Games played on Boolean algebras

Published online by Cambridge University Press:  12 March 2014

Matthew Foreman
Matthew Foreman*
Affiliation:
University of California, Los Angeles, California 90024
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Extract

In this paper we consider the special case of the Banach-Mazur game played on a topological space when the space also has an underlying Boolean Algebra structure. This case was first studied by Jech [2]. The version of the Banach-Mazur game we will play is the following game played on the Boolean algebra:

Players I and II alternate moves playing a descending sequence of elements of a Boolean algebra ℬ.

Player II wins the game iff Πi∈ωbi ≠ 0. Jech first considered these games and showed:

Theorem (Jech [2]). ℬ is (ω1, ∞)-distributive iff player I does not have a winning strategy in the game played on ℬ.

If ℬ has a dense ω-closed subset then it is easy to see that player II has a winning strategy in this game. This paper establishes a partial converse to this, namely it gives cardinality conditions on ℬ under which II having a winning strategy implies ω-closure.

In the course of proving the converse, we consider games of length > ω and generalize Jech's theorem to these games. Finally we present an example due to C. Gray that stands in counterpoint to the theorems in this paper.

In this section we give a few basis definitions and explain our notation. These definitions are all standard.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 48 , Issue 3 , September 1983 , pp. 714 - 723
Copyright
Copyright © Association for Symbolic Logic 1983

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References

BIBLIOGRAPHY

[1]Gray, C., Iterated forcing from the strategic point of view, Ph.D. Thesis, University of California, Berkeley, 1980.Google Scholar
[2]Jech, T., A game theoretic property of Boolean algebras, Logic Colloquium 77 (Macintyre, A.et al., Editors), North-Holland, Amsterdam, 1978, pp. 135–144.CrossRefGoogle Scholar
[3]Jech, T., More game theoretic properties of Boolean algebras, Department of Mathematics Research Report, Penn State University.CrossRefGoogle Scholar