representing a complete atomic Boolean algebra by power set
It is a known fact that every Boolean algebra is isomorphic to a field of sets (of some set) (proof here (http://planetmath.org/RepresentingABooleanLatticeByFieldOfSets)). In this entry, we show that, furthermore, if a Boolean algebra is atomic and complete, then it is isomorphic to the field of sets of some set, in other words, the powerset of some set, viewed as a Boolean algebra via the usual set-theoretic operations of union, intersection, and complement.
The proof is based on the following function, defined for any atomic Boolean algebra:
Definition. Let be an atomic Boolean algebra, and the set of its atoms. Define by
It is easy to see that iff is an atom of .
and are complement of one another in .
For any , , so that or , or or . This shows that . If , then and , so that , which is impossible, since is an atom, and by definition, must be greater than . ∎
First, by the last proposition.
Next, since is an atom. But the right hand side equals , we see that preserves .
Finally, since any atom must be greater than .
Hence, is a Boolean algebra homomorphism. ∎
is an injection.
Suppose . If , then there must be some atom such that . But this implies that , a contradiction. Hence and is injective. ∎
is conditionally complete, in the sense that if is defined for any , then
Suppose and . Let . We want to show that . If , then , or for some , since is an atom. So . Conversely, if , then , or for some . This means that , and therefore . ∎
If is complete, so is . Moreover, is surjective.
Rewording the above proposition, we have
Any complete atomic Boolean algebra is isomorphic (as complete Boolean algebras) to the powerset of some set, namely, the set of all of its atoms.
A useful application of this representation theorem is the following:
The cardinality of a finite Boolean algebra is a power of .
Every finite Boolean algebra is complete and atomic, and hence isomorphic to the powerset of a set, which is also finite, and the result follows. ∎
Remark. The proof does not depend on the representation of a Boolean algebra by a field of sets.
|Title||representing a complete atomic Boolean algebra by power set|
|Date of creation||2013-03-22 19:08:30|
|Last modified on||2013-03-22 19:08:30|
|Last modified by||CWoo (3771)|