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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, 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 ${}^{{\prime}}$ 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:

Every prime ideal is the kernel of a homomorphism into $\boldsymbol{2}:=\{0,1\}$, and vice versa. So for an element $a$ to be not in a prime ideal $P$ is the same as saying that $\phi(a)=1$ for some homomorphism $\phi:L\to\boldsymbol{2}$. If we take $Y$ to be the set of all homomorphisms from $L$ to $\boldsymbol{2}$, and define $G:L\to P(Y)$ by $G(a)=\{\phi\mid\phi(a)=1\}$, then it is easy to see that $G$ is an embedding of $L$ into $P(Y)$.

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.
Mathematics Subject Classification
06E20 no label found06E05 no label found03G05 no label found06B20 no label found03G10 no label found Forums
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