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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
Corresponding

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

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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
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