Property ((h)over-bar) and cellularity of complete Boolean algebras

A complete Boolean algebra B satisfies property ((h) over bar) iff each sequence x in 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. Herewe determine the position of property ((h) over bar) with respect to the hierarchy of conditions of the form kappa-cc. So, answering a question from Kurilic and Pavlovic (Ann Pure Appl Logic 148(1-3): 49-62, 2007), we show that "h-cc double right arrow ((h) over bar)" is not a theorem of ZFC and that there is no cardinal l, definable in ZFC, such that "l-cc double left right arrow ((h) over bar)" is a theorem of ZFC. Also, we show that the set {kappa : each kappa-cc c.B.a. has ((h) over bar)} is equal to [0, h) or [0, h] and that both values are consistent, which, with the known equality {kappa : each c.B.a. having ((h) over bar) has the kappa-cc} = [s, infinity) completes the picture.