Hostname: page-component-f554764f5-44mx8 Total loading time: 0 Render date: 2025-04-18T05:18:26.033Z Has data issue: false hasContentIssue false
  • English
  • Français

Regular subalgebras of complete Boolean algebras

Published online by Cambridge University Press:  12 March 2014

Aleksander Błaszczyk  and
Saharon Shelah
Aleksander Błaszczyk
Affiliation:
Institute of Mathematics, Silesian University, Bankowa 14. 40-007 Katowice, Poland, E-mail: ablasz.cz@ux2.math.us.edu.ps
Saharon Shelah
Affiliation:
Department of Mathematics, Hebrew University, Givat Ram. 91904 Jerusalem, Israel Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA, E-mail: shelah@math.huji.ac.il
Get access Rights & Permissions [Opens in a new window]

Abstract

It is proved that the following conditions are equivalent:

(a) there exists a complete, atomless, σ–centered Boolean algebra, which does not contain any regular, atomless, countable subalgebra.

(b) there exists a nowhere dense ultrafilter on ω.

Therefore, the existence of such algebras is undecidable in ZFC. In “forcing language” condition (a) says that there exists a non–trivial σ–centered forcing not adding Cohen reals.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 2 , June 2001 , pp. 792 - 800
Copyright
Copyright © Association for Symbolic Logic 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Balcar, B. and Franek, F., Independent families in complete boolean algebras, Transactions of the American Mathematical Society, vol. 274 (1982), pp. 607–618.CrossRefGoogle Scholar
[2]Baumgartner, J., Ultrafilters on ω, this Journal, vol. 60 (1995), pp. 624–639.Google Scholar
[3]Blass, A., Selective ultrafilters and homogeneity, Annals of Pure and Applied Logic, vol. 38 (1988), pp. 215–255.CrossRefGoogle Scholar
[4]Dow, A., Gubbi, A. V., and Szymanski, A., Rigid Stone spaces within ZFC, Proceedings of the American Mathematical Society, vol. 102, 1988, pp. 745–748.Google Scholar
[5]Gitik, M. and Shelah, S., More of simple forcing notions and forcing with ideals, Annals of Pure and Applied Logic, vol. 59 (1993), pp. 219–238.CrossRefGoogle Scholar
[6]Heindorf, L. and Shapiro, L. B., Nearly projective Boolean algebras, Lecture Notes in Mathematics, vol. 1596, Springer-Verlag, 1994.CrossRefGoogle Scholar
[7]Shelah, S., There may be no nowhere dense ultrafilter, Proceedings of the Logic Colloquium Haifa'95, Lecture Notes in Mathematical Logic, vol. 11, Springer-Verlag, 1998, pp. 305–325.Google Scholar