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:
if L is a distributive lattice
, and X the set of all prime ideals
of L, then the map F:L→P(X) defined by F(a)={P∣a∉P} is an embedding
.
Now, if L is a Boolean lattice, then every element a∈L has a complement a′∈L. a′ is in fact uniquely determined by a.
Proposition 1.
The embedding F above preserves ′ in the following sense:
F(a′)=X-F(a). |
Proof.
P∈F(a′) iff a′∉P iff a∈P iff P∉F(a) iff P∈X-F(a). ∎
Theorem 1.
Every Boolean algebra is isomorphic to a field of sets.
Proof.
From what has been discussed so far, F is a Boolean algebra isomorphism between L and F(L), which is a ring of sets first of all, and a field of sets, because X-F(a)=F(a′). ∎
Remark. There are at least two other ways to characterize a Boolean algebra as a field of sets: let L be a Boolean algebra:
-
•
Every prime ideal is the kernel of a homomorphism
into 𝟐:=, 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 .
-
•
Every prime ideal is a maximal ideal
, and vice versa. Furthermore, is maximal iff is an ultrafilter
. So if we define to be the set of all ultrafilters of , and set 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 |
Canonical name | RepresentingABooleanAlgebraByFieldOfSets |
Date of creation | 2013-03-22 19:08:27 |
Last modified on | 2013-03-22 19:08:27 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 13 |
Author | CWoo (3771) |
Entry type | Theorem |
Classification | msc 06E20 |
Classification | msc 06E05 |
Classification | msc 03G05 |
Classification | msc 06B20 |
Classification | msc 03G10 |
Related topic | FieldOfSets |
Related topic | RepresentingADistributiveLatticeByRingOfSets |
Related topic | LatticeHomomorphism |
Related topic | RepresentingACompleteAtomicBooleanAlgebraByPowerSet |
Related topic | StoneRepresentationTheorem |
Related topic | MHStonesRepresentationTheorem |