In this note we give a characterization of complete atomic Boolean algebras by means of complete atomic lattices. We find that unicity of the representation of the maximum as union of atoms and Lambda-infinite distributivity law are necessary and sufficient conditions for the lattice to be a complete atomic Boolean algebra.
We compare the forcing-related properties of a complete Boolean algebra with the properties of the convergences (the algebraic convergence) and on generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that is a topological convergence iff forcing by does not produce new reals and that is weakly topological if satisfies condition (implied by the -cc). On the other hand, if is a weakly topological convergence, then is a -cc algebra...
Using ♢ , we construct a rigid atomless Boolean algebra that has no uncountable antichain and that admits the elimination of the Malitz quantifier .