Regular subalgebras of complete Boolean algebras

Aleksander Błaszczyk; Saharon Shelah

doi:10.2307/2695044

Regular subalgebras of complete Boolean algebras

Aleksander Błaszczyk^*
, Saharon Shelah

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

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.

Original language	English
Pages (from-to)	792-800
Number of pages	9
Journal	Journal of Symbolic Logic
Volume	66
Issue number	2
DOIs	https://doi.org/10.2307/2695044
State	Published - Jun 2001

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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.",

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AB - 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.

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Regular subalgebras of complete Boolean algebras

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