Exercise 20, § 6

A lattice $X$ is said to be a Boolean algebra if it is distributive, has a least element $o$, and a greatest element $i$ and if, for each $x \in X$, there is an element $x' \in X$ such that $\inf (x, x') = o$ and $\sup (x, x') = i$; such an element is said to be a complement of $x$ in $X$.

a) Let $x \to A_x$ be the injection of a Boolean algebra $X$ into the set of subsets of the set $\Omega$ of prime prefilters on $X$, defined in Exercise 18 d). Show that if $x'$ is a complement of $x$ in $X$, then $A_{x'} = \mathbf{C}A_x$ in $\Omega$; deduce that an element $x$ has only one complement $x'$, that the complement of $x'$ is $x$, and that the complement of $\inf (x, y)$ is $\sup (x', y')$; give direct proofs of these statements. The ideals of the Boolean ring $A$ corresponding to $X$, other than $A$, are the coprefilters on $X$.

b) Show that, in a Boolean algebra $X$, every prime prefilter is maximal (remark that, for each $x \in X$, every prime prefilter must contain either $x$ or its complement).

c) Conversely, let $X$ be a distributive lattice which has a least element $o$ and a greatest element $i$, and is such that every prime prefilter on $X$ is maximal. Show that $X$ is a Boolean algebra. (For an element $x \in X$, consider the coprefilter $C$ (Exercise 17) consisting of those $y \in X$ such that $\inf (x, y) = o$; if $x$ has no complement, there is a maximal coprefilter $M$ containing $x$ and $C$; remark that the complement $U$ of $M$ in $X$ is a prime prefilter [Exercise 17 b) and 18 b)] and use Exercise 17 f)).

d) Show that if $X$ is a topological space then the set $\mathfrak{B}$ of subsets of $X$ which are both open and closed is a Boolean algebra if ordered by inclusion. If we take $X$ to be the topological space associated with the Fréchet filter (no. 5), show that the Boolean algebra $\mathfrak{B}$ thus obtained is countably infinite and therefore cannot be isomorphic to the ordered set $\mathfrak{P}(A)$ of all subsets of a set $A$.