Games and general distributive laws in Boolean algebras
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- by Natasha Dobrinen PDF
- Proc. Amer. Math. Soc. 131 (2003), 309-318 Request permission
Erratum: Proc. Amer. Math. Soc. 131 (2003), 2967-2968.
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.References
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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
- 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.
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 309-318
- MSC (2000): Primary 03G05, 06E25; Secondary 03E40
- DOI: https://doi.org/10.1090/S0002-9939-02-06501-2
- MathSciNet review: 1929051