Must complete atomless Boolean algebras of the same cardinality be isomorphic?

Question

More generally: must complete Boolean algebras of the same cardinality and with the same cardinality of atoms be isomorphic?

Answer 1

No. The quotient of the Boolean algebra $B$ of Borel subsets of $[0,1]$ by the ideal of meager (= first Baire category) sets and the quotient of the same $B$ by the ideal of sets of Lebesgue measure zero are both atomless, and they both have the cardinality of the continuum, but they are not isomorphic. Specifically, the latter satisfies a weak distributive law that fails in the former.