complete Boolean algebraCompleteBooleanAlgebra2008-04-22 14:15:132008-04-29 10:47:31DefinitionBoolean lattice$\kappa$-complete Boolean algebracountably complete Boolean algebra\usepackage{amssymb,amscd}
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\newcommand{\mpdiff}[3]{\frac{\partial^#1 #2}{\partial #3^#1}}A Boolean algebra $A$ is a \emph{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 \PMlinkname{this link}{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 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 \emph{complete Boolean algebra homomorphism}.
\textbf{Remark} Between a Boolean algebra and a complete Boolean algebra, there are many intermediate concepts. 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 $\kappa=\aleph_0$, the first aleph number, then it is called a \emph{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 $\kappa < |S|$, then the collection $A\subseteq P(S)$ consisting of any subset $T$ such that either $|T|\le \kappa$ or $|S-T|\le \kappa$ is $\kappa$-complete but not complete.
A Boolean algebra homomorphism $f$ between two $\kappa$-algebras $A,B$ is said to be $\kappa$-complete if $$f(\bigvee \lbrace a \mid a\in C\rbrace)= \bigvee \lbrace f(a)\mid a\in C\rbrace $$ for any $C\subseteq A$ with $|C|\le \kappa$.