complete Boolean algebra

A Boolean algebraMathworldPlanetmath A is a complete Boolean algebra if for every subset C of A, the arbitrary join and arbitrary meet of C exist.

By de Morgan’s laws, it is easy to see that a Boolean algebra is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath iff the arbitrary join of any subset exists iff the arbitrary meet of any subset exists. For a proof of this, see this link (

For an example of a complete Boolean algebra, let S be any set. Then the powerset P(S) with the usual set theoretic operationsMathworldPlanetmath is a complete Boolean algebra.

In a complete Boolean algebra, the infinite distributive and infiniteMathworldPlanetmath deMorgan’s laws hold:

  • xA=(xA) and xA=(xA)

  • (A)*=A* and (A)*=A*, where A*:={a*aA}.

In the category of complete Boolean algebras, a morphism between two objects is a Boolean algebra homomorphism that preserves arbitrary joins (equivalently, arbitrary meets), and is called a complete Boolean algebra homomorphism.

Remark There are infinitely many algebras between Boolean algebras and complete Boolean algebras. Let κ be a cardinal. A Boolean algebra A is said to be κ-complete if for every subset C of A with |C|κ, C (and equivalently C) exists. A κ-complete Boolean algebra is usually called a κ-algebra. If κ=0, the first aleph number, then it is called a countably complete Boolean algebra.

Any complete Boolean algebra is κ-complete, and any κ-complete is λ-complete for any λκ. An example of a κ-complete algebra that is not complete, take a set S with κ<|S|, then the collectionMathworldPlanetmath AP(S) consisting of any subset T such that either |T|κ or |S-T|κ is κ-complete but not complete.

A Boolean algebra homomorphism f between two κ-algebras A,B is said to be κ-complete if


for any CA with |C|κ.

Title complete Boolean algebra
Canonical name CompleteBooleanAlgebra
Date of creation 2013-03-22 18:01:09
Last modified on 2013-03-22 18:01:09
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 06E10
Related topic CompleteLattice
Defines κ-complete Boolean algebra
Defines countably complete Boolean algebra