Skip to main content Accessibility help
×
Home
Hostname: page-component-56f9d74cfd-l4dq5 Total loading time: 0.445 Render date: 2022-06-25T12:17:23.173Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

Regular subalgebras of complete Boolean algebras

Published online by Cambridge University Press:  12 March 2014

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

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
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.)

References

[1]Balcar, B. and Franek, F., Independent families in complete boolean algebras, Transactions of the American Mathematical Society, vol. 274 (1982), pp. 607618.CrossRefGoogle Scholar
[2]Baumgartner, J., Ultrafilters on ω, this Journal, vol. 60 (1995), pp. 624639.Google Scholar
[3]Blass, A., Selective ultrafilters and homogeneity, Annals of Pure and Applied Logic, vol. 38 (1988), pp. 215255.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. 745748.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. 219238.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. 305325.Google Scholar
6
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Regular subalgebras of complete Boolean algebras
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Regular subalgebras of complete Boolean algebras
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Regular subalgebras of complete Boolean algebras
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *