Property $${(\hbar)}$$ and cellularity of complete Boolean algebras

Abstract

A complete Boolean algebra $${\mathbb{B}}$$ satisfies property $${(\hbar)}$$ iff each sequence x in $${\mathbb{B}}$$ has a subsequence y such that the equality lim sup z n = lim sup y n holds for each subsequence z of y. This property, providing an explicit definition of the a posteriori convergence in complete Boolean algebras with the sequential topology and a characterization of sequential compactness of such spaces, is closely related to the cellularity of Boolean algebras. Here we determine the position of property $${(\hbar)}$$ with respect to the hierarchy of conditions of the form κ-cc. So, answering a question from Kurilic and Pavlovic (Ann Pure Appl Logic 148(1–3):49–62, 2007), we show that $${``\mathfrak{h}{\rm -cc}\Rightarrow (\hbar)"}$$ is not a theorem of ZFC and that there is no cardinal $${\mathfrak{k}}$$ , definable in ZFC, such that $${``\mathfrak{k} {\rm -cc} \Leftrightarrow (\hbar)"}$$ is a theorem of ZFC. Also, we show that the set $${\{ \kappa : {\rm each}\, \kappa{\rm -cc\, c.B.a.\, has}\, (\hbar ) \}}$$ is equal to $${[0, \mathfrak{h})}$$ or $${[0, {\mathfrak h}]}$$ and that both values are consistent, which, with the known equality $${{\{\kappa : {\rm each\, c.B.a.\, having }\, (\hbar )\, {\rm has\, the}\, \kappa {\rm -cc } \} =[{\mathfrak s}, \infty )}}$$ completes the picture.