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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Games and general distributive laws in Boolean algebras


Author: Natasha Dobrinen
Journal: Proc. Amer. Math. Soc. 131 (2003), 309-318
MSC (2000): Primary 03G05, 06E25; Secondary 03E40
Published electronically: May 13, 2002
Erratum: Proc. Amer. Math. Soc. (recently posted)
MathSciNet review: 1929051
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Abstract: The games $\mathcal{G}_{1}^{\eta }(\kappa )$ and $\mathcal{G}_{<\lambda }^{\eta }(\kappa )$ are played by two players in $\eta ^{+}$-complete and max $(\eta ^{+},\lambda )$-complete Boolean algebras, respectively. For cardinals $\eta ,\kappa $ such that $\kappa ^{<\eta }=\eta $ or $\kappa ^{<\eta }=\kappa $, the $(\eta ,\kappa )$-distributive law holds in a Boolean algebra $\mathbf{B}$ iff Player 1 does not have a winning strategy in $\mathcal{G}_{1}^{\eta }(\kappa )$. Furthermore, for all cardinals $\kappa $, the $(\eta ,\infty )$-distributive law holds in $\mathbf{B}$ iff Player 1 does not have a winning strategy in $\mathcal{G}_{1}^{\eta }(\infty )$. More generally, for cardinals $\eta ,\lambda ,\kappa $ such that $(\kappa ^{<\lambda })^{<\eta }=\eta $, the $(\eta ,<\lambda ,\kappa )$-distributive law holds in $\mathbf{B}$ iff Player 1 does not have a winning strategy in $\mathcal{G}_{<\lambda }^{\eta }(\kappa )$. For $\eta $ regular and $\lambda \le \text{min}(\eta ,\kappa )$, $\lozenge _{\eta ^{+}}$ implies the existence of a Suslin algebra in which $\mathcal{G}_{<\lambda }^{\eta }(\kappa )$ is undetermined.


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Additional Information

Natasha Dobrinen
Affiliation: Department of Mathematics, The Pennsylvania State University, 218 McAllister Building, University Park, Pennsylvania 16802
Email: dobrinen@math.psu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06501-2
PII: S 0002-9939(02)06501-2
Keywords: Boolean algebra, distributivity, games
Received by editor(s): November 17, 2000
Received by editor(s) in revised form: August 23, 2001
Published electronically: May 13, 2002
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2002 American Mathematical Society