Regular subalgebras of complete Boolean algebras

Abstract

AbstractIt 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.