Forcing minimal extensions of Boolean algebras

Abstract

We employ a forcing approach to extending Boolean algebras. A link between some forcings and some cardinal functions on Boolean algebras is found and exploited. We find the following applications: 1) We make Fedorchuk’s method more flexible, obtaining, for every cardinal λ \lambda of uncountable cofinality, a consistent example of a Boolean algebra A λ A_{\lambda } whose every infinite homomorphic image is of cardinality λ \lambda and has a countable dense subalgebra (i.e., its Stone space is a compact S-space whose every infinite closed subspace has weight λ \lambda ). In particular this construction shows that it is consistent that the minimal character of a nonprincipal ultrafilter in a homomorphic image of an algebra A A can be strictly less than the minimal size of a homomorphic image of A A , answering a question of J. D. Monk. 2) We prove that for every cardinal of uncountable cofinality it is consistent that 2 ω = λ 2^{\omega }=\lambda and both A λ A_{\lambda } and A ω 1 A_{\omega _{1}} exist. 3) By combining these algebras we obtain many examples that answer questions of J.D. Monk. 4) We prove the consistency of MA + ¬ \neg CH + there is a countably tight compact space without a point of countable character, complementing results of A. Dow, V. Malykhin, and I. Juhasz. Although the algebra of clopen sets of the above space has no ultrafilter which is countably generated, it is a subalgebra of an algebra all of whose ultrafilters are countably generated. This proves, answering a question of Arhangel ′ ’ skii, that it is consistent that there is a first countable compact space which has a continuous image without a point of countable character. 5) We prove that for any cardinal λ \lambda of uncountable cofinality it is consistent that there is a countably tight Boolean algebra A A with a distinguished ultrafilter ∞ \infty such that for every a ∌ ∞ a\not \ni \infty the algebra A | a A|a is countable and ∞ \infty has hereditary character λ \lambda .