Boolean algebras as unions of chains of subalgebras (Q1239181)

For every infinite Boolean algebra \(B\), \(\operatorname{cf} B\) (the cofinality of \(B\)) is defined to be the least infinite cardinal \(\kappa\) such that \(B\) is the union of a strictly ascending chain of subalgebras \(B_\alpha\), \(\alpha<\kappa\). It is proved that \(\operatorname{cf} B\leq 2^{\aleph_0}\) and \(\operatorname{cf} B=\aleph_1\) if \(B\) is \(\sigma\)-complete. Moreover, some classes of Boolean algebras are described such that \(\operatorname{cf} B=\aleph_0\), for each member of the classes. It seems to be unknown whether there is some \(B\) such that \(\operatorname{cf}B\geq \aleph_2\).