  We compare the forcing related properties of a complete Boolean algebra B
with the properties of the convergences $\lambda_s$ (the algebraic convergence)
and $\lambda_{ls}$ on B generalizing the convergence on the Cantor and
Aleksandrov cube respectively. In particular we show that $\lambda_{ls}$ is a
topological convergence iff forcing by B does not produce new reals and that
$\lambda_{ls}$ is weakly topological if B satisfies condition $(\hbar)$
(implied by the ${\mathfrak t}$-cc). On the other hand, if $\lambda_{ls}$ is a
weakly topological convergence, then B is a $2^{\mathfrak h}$-cc algebra or in
some generic extension the distributivity number of the ground model is greater
than or equal to the tower number of the extension. So, the statement "The
convergence $\lambda_{ls}$ on the collapsing algebra $B=\ro
((\omega_2)^{<\omega})$ is weakly topological" is independent of ZFC.
