complete Boolean algebra
A Boolean algebra^{} $A$ 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 complete^{} iff the arbitrary join of any subset exists iff the arbitrary meet of any subset exists. For a proof of this, see this link (http://planetmath.org/PropertiesOfArbitraryJoinsAndMeets).
For an example of a complete Boolean algebra, let $S$ be any set. Then the powerset $P(S)$ with the usual set theoretic operations^{} is a complete Boolean algebra.
In a complete Boolean algebra, the infinite distributive and infinite^{} deMorgan’s laws hold:

•
$x\wedge \bigvee A=\bigvee (x\wedge A)$ and $x\vee \bigwedge A=\bigwedge (x\vee A)$

•
${(\bigvee A)}^{*}=\bigwedge {A}^{*}$ and ${(\bigwedge A)}^{*}=\bigvee {A}^{*}$, where ${A}^{*}:=\{{a}^{*}\mid a\in A\}$.
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 $\kappa $ be a cardinal. A Boolean algebra $A$ is said to be $\kappa $complete if for every subset $C$ of $A$ with $C\le \kappa $, $\bigvee C$ (and equivalently $\bigwedge C$) exists. A $\kappa $complete Boolean algebra is usually called a $\kappa $algebra. If $\kappa ={\mathrm{\aleph}}_{0}$, the first aleph number, then it is called a countably complete Boolean algebra.
Any complete Boolean algebra is $\kappa $complete, and any $\kappa $complete is $\lambda $complete for any $\lambda \le \kappa $. An example of a $\kappa $complete algebra that is not complete, take a set $S$ with $$, then the collection^{} $A\subseteq P(S)$ consisting of any subset $T$ such that either $T\le \kappa $ or $ST\le \kappa $ is $\kappa $complete but not complete.
A Boolean algebra homomorphism $f$ between two $\kappa $algebras $A,B$ is said to be $\kappa $complete if
$$f(\bigvee \{a\mid a\in C\})=\bigvee \{f(a)\mid a\in C\}$$ 
for any $C\subseteq A$ with $C\le \kappa $.
Title  complete Boolean algebra 

Canonical name  CompleteBooleanAlgebra 
Date of creation  20130322 18:01:09 
Last modified on  20130322 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  $\kappa $complete Boolean algebra 
Defines  countably complete Boolean algebra 