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ISSN 1088-6826(e) ISSN 0002-9939(p)

Directive trees and games on posets

Author(s): Tetsuya Ishiu; Yasuo Yoshinobu
Journal: Proc. Amer. Math. Soc. 130 (2002), 1477-1485.
MSC (2000): Primary 03E40; Secondary 03E65
Posted: October 12, 2001
MathSciNet review: 1879973
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Abstract | References | Similar articles | Additional information

Abstract: We show that for any infinite cardinal $\kappa$ , every $(\kappa +1)$ -strategically closed poset is $\kappa ^{+}$ -strategically closed if and only if $\square _{\kappa }$ holds. This extends previous results of Velleman, et.al.

References:

[F]: M. Foreman, Games played on boolean algebras, J. of Symbolic Logic 48 (1983), 714-723. MR 85h:03064
[G]: C. Gray, Iterated forcing from the strategic point of view, Ph.D. Thesis, University of California, Berkeley, 1980.
[I]: T. Ishiu, On Axiom A and games on boolean algebras (Japanese), Master Thesis, Waseda University, 1997.
[J1]: T. Jech, A game theoretic property of Boolean algebras, Logic Colloquium '77 (A. Macintyre, L. Pacholski and J. Paris, eds.), North-Holland, Amsterdam, 1978, pp. 135-144. MR 80c:90184
[J2]: T. Jech, More game-theoretic properties of Boolean algebras, Ann. Pure Appl. Logic 26 (1984), 11-29. MR 85j:03110
[Vc]: B. Velickovic, Jensen's $\square$ principles and the Novák number of partially ordered sets, J. of Symbolic Logic 51 (1986), 47-58. MR 87h:03074
[Vm]: D. Velleman, On a generalization of Jensen's $\square _{\kappa }$ , and strategic closure of partial orders, J. of Symbolic Logic 48 (1983), 1046-1052. MR 85g:03076

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Additional Information:

Tetsuya Ishiu
Affiliation: Department of Mathematics, University of California, Irvine, California 92697
Email: tishiu@math.uci.edu

Yasuo Yoshinobu
Affiliation: Graduate School of Human Informatics, Nagoya University, Nagoya 464-8601, Japan
Email: yosinobu@math.nagoya-u.ac.jp

DOI: 10.1090/S0002-9939-01-06235-9
PII: S 0002-9939(01)06235-9
Keywords: Directive trees, strategic closure
Received by editor(s): November 2, 2000
Posted: October 12, 2001
Additional Notes: The second author was partially supported by Grant-in-Aid for Scientific Research (No.11640112), Ministry of Education, Science and Culture, Japan
Communicated by: Carl G. Jockusch, Jr.
Copyright of article: Copyright 2001, American Mathematical Society

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