PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (197)
Orphanage
Unclass'd
Unproven (498)
Corrections (22)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: Very high
[parent] complete Boolean algebra (Definition)

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.

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 complete Boolean algebra homomorphism.

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 $ \vert C\vert\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 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 < \vert S\vert$, then the collection $ A\subseteq P(S)$ consisting of any subset $ T$ such that either $ \vert T\vert\le \kappa$ or $ \vert S-T\vert\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

$\displaystyle f(\bigvee \lbrace a \mid a\in C\rbrace)= \bigvee \lbrace f(a)\mid a\in C\rbrace $
for any $ C\subseteq A$ with $ \vert C\vert\le \kappa$.



"complete Boolean algebra" is owned by CWoo.
(view preamble)

View style:

See Also: complete lattice

Also defines:  $\kappa$-complete Boolean algebra, countably complete Boolean algebra

This object's parent.

Attachments:
Sikorski's extension theorem (Theorem) by CWoo
Log in to rate this entry.
(view current ratings)

Cross-references: collection, algebra, aleph number, cardinal, complete Boolean algebra homomorphism, preserves, Boolean algebra homomorphism, objects, morphism, category, operations, powerset, proof, iff, complete, easy to see, de Morgan's laws, arbitrary meet, arbitrary join, subset, Boolean algebra
There are 5 references to this entry.

This is version 4 of complete Boolean algebra, born on 2008-04-22, modified 2008-04-29.
Object id is 10535, canonical name is CompleteBooleanAlgebra.
Accessed 535 times total.

Classification:
AMS MSC: 06E10 (Order, lattices, ordered algebraic structures :: Boolean algebras :: Chain conditions, complete algebras)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)