complete Boolean algebra
A Boolean algebra is a complete Boolean algebra if for every subset of , the arbitrary join and arbitrary meet of 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 be any set. Then the powerset 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:
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and
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and , where .
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 be a cardinal. A Boolean algebra is said to be -complete if for every subset of with , (and equivalently ) exists. A -complete Boolean algebra is usually called a -algebra. If , the first aleph number, then it is called a countably complete Boolean algebra.
Any complete Boolean algebra is -complete, and any -complete is -complete for any . An example of a -complete algebra that is not complete, take a set with , then the collection consisting of any subset such that either or is -complete but not complete.
A Boolean algebra homomorphism between two -algebras is said to be -complete if
for any with .
Title | complete Boolean algebra |
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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 | -complete Boolean algebra |
Defines | countably complete Boolean algebra |