Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-02-06501-2
Published electronically: May 13, 2002
Erratum: Proc. Amer. Math. Soc. (recently posted)
MathSciNet review: 1929051
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. M. Foreman, Games played on Boolean algebras, J. Symbolic Logic 48 (3) (1983), 714-723. MR 85h:03064
  • 2. S. Fuchino, H. Mildenberger, S. Shelah and P. Vojtás, On absolutely divergent series, Fund. Math. 160 (3) (1999), 255-268. MR 2000g:03107
  • 3. T. Jech, A game theoretic property of Boolean algebras, Logic Colloquium '77 (1978), 135-144. MR 80c:90184
  • 4. -, More game-theoretic properties of Boolean algebras, Ann. Pure and Appl. Logic 26 (1984), 11-29. MR 85j:03110
  • 5. -, Distributive Laws, Handbook of Boolean Algebra, Vol. 2, North-Holland, Amsterdam, 1989, pp. 317-331. CMP 21:10
  • 6. A. Kamburelis, On the weak distributivity game, Ann. Pure and Appl. Logic 66 (1994), 19-26. MR 95d:04004
  • 7. S. Koppelberg, Handbook of Boolean Algebra, Vol. 1, North-Holland, Amsterdam, 1989. MR 90k:06002

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 03G05, 06E25, 03E40

Retrieve articles in all journals with MSC (2000): 03G05, 06E25, 03E40


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: https://doi.org/10.1090/S0002-9939-02-06501-2
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

American Mathematical Society