complete Boolean algebra
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).
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|
|Date of creation||2013-03-22 18:01:09|
|Last modified on||2013-03-22 18:01:09|
|Last modified by||CWoo (3771)|
|Defines||-complete Boolean algebra|
|Defines||countably complete Boolean algebra|