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'complete Boolean algebra'
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| Title of object: |
complete Boolean algebra |
| Canonical Name: |
CompleteBooleanAlgebra |
| Type: |
Definition |
| Created on: |
2008-04-22 14:15:13 |
| Modified on: |
2008-04-29 01:12:49 |
| Classification: |
msc:06E10 |
| Defines: |
$\kappa$-complete Boolean algebra, countably complete Boolean algebra |
Preamble:
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Content:
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}.
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}. 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 $|C|\le \kappa$. |
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