representing a Boolean algebra by field of sets
In this entry, we show that every Boolean algebra is isomorphic to a field of sets, originally noted by Stone in 1936. The bulk of the proof has actually been carried out in this entry (http://planetmath.org/RepresentingADistributiveLatticeByRingOfSets), which we briefly state:
The embedding above preserves in the following sense:
iff iff iff iff . ∎
Every Boolean algebra is isomorphic to a field of sets.
From what has been discussed so far, is a Boolean algebra isomorphism between and , which is a ring of sets first of all, and a field of sets, because . ∎
Remark. There are at least two other ways to characterize a Boolean algebra as a field of sets: let be a Boolean algebra:
Every prime ideal is the kernel of a homomorphism into , and vice versa. So for an element to be not in a prime ideal is the same as saying that for some homomorphism . If we take to be the set of all homomorphisms from to , and define by , then it is easy to see that is an embedding of into .
If we appropriately topologize the sets , or , then we have the content of the Stone representation theorem.
|Title||representing a Boolean algebra by field of sets|
|Date of creation||2013-03-22 19:08:27|
|Last modified on||2013-03-22 19:08:27|
|Last modified by||CWoo (3771)|