dual space of a Boolean algebra

Let $B$ be a Boolean algebra, and $B^{*}$ the set of all maximal ideals of $B$. In this entry, we will equip $B^{*}$ with a topology so it is a Boolean space.

Definition. For any $a\in B$, define $M(a):=\{M\in B^{*}\mid a\notin M\}$, and $ℬ:=\{M(a)\mid a\in B\}$.

It is know that in a Boolean algebra, maximal ideals and prime ideals coincide. From this entry (http://planetmath.org/RepresentingABooleanLatticeByFieldOfSets), we have the three following properties concerning $M(a)$:

$$M(a)\cap M(b)=M(a\wedge b),M(a)\cup M(b)=M(a\vee b),B^{*}-M(a)=M(a^{\prime }).$$

Furthermore, if $M(a)=M(b)$, then $a=b$.

From these properties, we see that $M(0)=\mathrm{\varnothing }$ and $M(1)=B^{*}$. As a result, we see that

Proposition 1.

$B^{*}$ is a topological space, whose topology $\mathrm{T}$ is generated by the basis $\mathrm{B}$.

Proof.

$\mathrm{\varnothing }$ and $B^{*}$ are both open, as they are $M(0)$ and $M(1)$ respectively. Also, the intersection of open sets $M(a)$ and $M(b)$ is again open, since it is $M(a\wedge b)$. ∎

We may in fact treat $ℬ$ as a subbasis for $𝒯$, since finite intersections of elements of $ℬ$ remain in $ℬ$.

Proposition 2.

Each member of $\mathrm{B}$ is closed, hence $\mathrm{T}$ is generated by a basis of clopen sets. In other words, $B^{\mathrm{*}}$ is zero-dimensional.

Proof.

Each $M(a)$ is open, by definition, and closed, since it is the complement of the open set $M(a^{\prime })$. ∎

Proposition 3.

$B^{*}$ is Hausdorff.

Proof.

If $M,N\in B^{*}$ such that $M\ne N$, then there is some $a\in B$ such that $a\in M$ and $a\notin N$. This means that $N\in M(a)$ and $M\notin M(a)$, which means that $M\in B^{*}-M(a)=M(a^{\prime })$. Since $M(a)$ and $M(a^{\prime })$ are open and disjoint, with $N\in M(a)$ and $M\in M(a^{\prime })$, we see that $B^{*}$ is Hausdorff. ∎

Now, based on a topological fact, every zero-dimensional Hausdorff space is totally disconnected. Hence $B^{*}$ is totally disconnected.

Proposition 4.

$B^{*}$ is compact.

Proof.

Suppose $\{U_i\mid i\in I\}$ is a collection of open sets whose union is $B^{*}$. Since each $U_i$ is a union of elements of $ℬ$, we might as well assume that $B^{*}$ is covered by elements of $ℬ$. In other words, we may assume that each $U_i$ is some $M(a_i)\in ℬ$.

Let $J$ be the ideal generated by the set $\{a_i\mid i\in I\}$. If $J\ne B$, then $J$ can be extended to a maximal ideal $M$. Since each $a_i\in M$, we see that $M\notin M(a_i)$, so that $M\notin \bigcup \{M(a_i)\mid i\in I\}=B^{*}$, which is a contradiction. Therefore, $J=B$. In particular, $1\in J$, which means that $1$ can be expressed as the join of a finite number of the $a_i$’s:

$$1=\bigvee \{a_i\mid i\in K\},$$

where $K$ is a finite subset of $J$. As a result, we have

$$\bigcup \{M(a_i)\mid i\in K\}=M(\bigvee \{a_i\mid i\in K\})=M(1)=B^{*}.$$

So $B^{*}$ has a finite subcover, and hence is compact. ∎

Collecting the last three results, we see that $B^{*}$ is a Boolean space.

Remark. It can be shown that $B$ is isomorphic to the Boolean algebra of clopen sets in $B^{*}$. This is the famous Stone representation theorem.

Title	dual space of a Boolean algebra
Canonical name	DualSpaceOfABooleanAlgebra
Date of creation	2013-03-22 19:08:35
Last modified on	2013-03-22 19:08:35
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	6
Author	CWoo (3771)
Entry type	Definition
Classification	msc 06E05
Classification	msc 03G05
Classification	msc 06B20
Classification	msc 03G10
Classification	msc 06E20
Related topic	StoneRepresentationTheorem
Related topic	MHStonesRepresentationTheorem