 This is Sikorski’s extension criterion. Every BA \(A\) can be embedded in a complete BA \(B\) in such a way that every element of \(B\) is the least upper bound of a set of elements of \(A\). \(B\) is unique up to \(A\)-isomorphism, and is called the completion of \(A\). If \(f\) is a homomorphism from a BA \(A\) into a complete BA \(B\), and if \(A\) is a subalgebra of \(C\), then \(f\) can be extended to a homomorphism of \(C\) into \(B\). This is Sikorski’s extension theorem. Another general algebraic notion which applies to Boolean algebras is the notion of a free algebra. This can be concretely constructed for BAs. Namely, the free BA on \(\kappa\) is the BA of closed-open subsets of the two element discrete space raised to the \(\kappa\) power.