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 $F:L\to P(X)$ defined by $F(a)=\{P\mid a\notin P\}$ is an embedding^{}.
Now, if $L$ is a Boolean lattice, then every element $a\in L$ has a complement^{} ${a}^{\prime}\in L$. ${a}^{\prime}$ is in fact uniquely determined by $a$.
Proposition 1.
The embedding $F$ above preserves ${}^{\mathrm{\prime}}$ in the following sense:
$$F({a}^{\prime})=XF(a).$$ 
Proof.
$P\in F({a}^{\prime})$ iff ${a}^{\prime}\notin P$ iff $a\in P$ iff $P\notin F(a)$ iff $P\in XF(a)$. ∎
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 $XF(a)=F({a}^{\prime})$. ∎
Remark. There are at least two other ways to characterize a Boolean algebra as a field of sets: let $L$ be a Boolean algebra:

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Every prime ideal is the kernel of a homomorphism^{} into $\mathrm{\U0001d7d0}:=\{0,1\}$, and vice versa. So for an element $a$ to be not in a prime ideal $P$ is the same as saying that $\varphi (a)=1$ for some homomorphism $\varphi :L\to \mathrm{\U0001d7d0}$. If we take $Y$ to be the set of all homomorphisms from $L$ to $\mathrm{\U0001d7d0}$, and define $G:L\to P(Y)$ by $G(a)=\{\varphi \mid \varphi (a)=1\}$, then it is easy to see that $G$ is an embedding of $L$ into $P(Y)$.

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Every prime ideal is a maximal ideal^{}, and vice versa. Furthermore, $P$ is maximal iff ${P}^{\prime}$ is an ultrafilter^{}. So if we define $Z$ to be the set of all ultrafilters of $L$, and set $H:L\to P(Z)$ by $H(a)=\{U\mid a\in U\}$, then it is easy to see that $H$ is an embedding of $L$ into $P(Z)$.
If we appropriately topologize the sets $X,Y$, or $Z$, 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  20130322 19:08:27 
Last modified on  20130322 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 