Complete Boolean Algebra - The Completion of A Boolean Algebra

The Completion of A Boolean Algebra

The completion of a Boolean algebra can be defined in several equivalent ways:

• The completion of A is (up to isomorphism) the unique complete Boolean algebra B containing A such that A is dense in B; this means that for every nonzero element of B there is a smaller non-zero element of A.
• The completion of A is (up to isomorphism) the unique complete Boolean algebra B containing A such that every element of B is the supremum of some subset of A.

The completion of a Boolean algebra A can be constructed in several ways:

• The completion is the Boolean algebra of regular open sets in the Stone space of prime ideals of A. Each element x of A corresponds to the open set of prime ideals not containing x (which open and closed, and therefore regular).
• The completion is the Boolean algebra of regular cuts of A. Here a cut is a subset U of A+ (the non-zero elements of A) such that if q is in U and p≤q then p is in U, and is called regular if whenever p is not in U there is some r ≤ p such that U has no elements ≤r. Each element p of A corresponds to the cut of elements ≤p.

If A is a metric space and B its completion then any isometry from A to a complete metric space C can be extended to a unique isometry from B to C. The analogous statement for complete Boolean algebras is not true: a homomorphism from a Boolean algebra A to a complete Boolean algebra C cannot necessarily be extended to a (supremum preserving) homomorphism of complete Boolean algebras from the completion B of A to C. (By Sikorski's extension theorem it can be extended to a homomorphism of Boolean algebras from B to C, but this will not in general be a homomorphism of complete Boolean algebras; in other words, it need not preserve suprema.)