Examples of Stone algebras which are not Boolean algebras

Question

Grätzer, in his Lattice Theory: Foundation, describes a Stone algebra as a distributive lattice with pseudocomplementation $L$ which satisfies the Stone identity: for every $a \in L$, $\neg a \vee \neg\neg a = 1$, where "$\neg$" is the pseudocomplementation operation. Unfortunately, he doesn't delve too much into examples of this type of structure; specifically, I'm interested in examples of structures which are not Boolean algebras as well. I tried to find some topological examples, but couldn't come up with any (but then again, I know very little of topology). Any ideas?

Answer 1

Natural examples of distributive lattices with pseudocomplementation come from lattices of open sets in topological spaces. Topologically, if $a$ is an open set, $\lnot a$ is the interior of the complement of $a$, and $\lnot\lnot a$ is the interior of the closure of $a$.

Now the Stone identity says that the space $X$ is the union of the sets $\lnot a$ and $\lnot\lnot a$. This means that $X$ is disconnected and $\lnot\lnot a$ is both open and closed, which happens if and only if $\lnot\lnot a = \text{cl}(a)$, i.e. the closure of $a$ is open. A space in which the closure of any open set is open is called an "extremally disconnected space". These are exactly the projective objects in the category of compact Hausdorff spaces.

---

Edit: You can find a proof here that the Stone space of any complete Boolean algebra is extremally disconnected. So, for an explicit example, take $S(\mathcal{P}(\mathbb{N}))$. The clopen sets form a Boolean algebra, but the lattice of open sets form a Stone algebra. As an example of an open set which is not clopen, take $\bigcup_{n\in\mathbb{N}} [\{n\}]$, the set of all principal ultrafilters. Its closure is the whole space!

(Edit 2: Why? We want to show that the set of principal ultrafilters is dense in the Stone space $S(\mathcal{P}(\mathbb{N}))$. A basic open set in the Stone space is $[A] = \{U\text{ an ultrafilter}\mid A\in U\}$ for $A\in\mathcal{P}(\mathbb{N})$. If $A = \emptyset$, then $[A]$ is empty, since no ultrafilter contains $\emptyset$. If not, pick $n\in A$. The principal ultrafilter $U_n$ generated by $n$ is $\{X\in\mathcal{P}(\mathbb{N})\mid n\in X\}$. Since $n\in A$, $A\in U_n$, so $U_n\in [A]$.)

You ask about regular open sets in the comments. Note that in an extramlly disconnected space, an open set is regular if and only if it is clopen.

My favorite source for all these sorts of things is Johnstone's Stone Spaces.

Answer 2

I know you requested topological examples but here is another example of a Stone algebra that is not Boolean:

The chain $\mathbf{3}=\{0,e,1\}$ where $0<e<1$, $0=\neg e=\neg 1$, and $1=\neg 0$.

In some sense, this is the main example since every Stone algebra is isomorphic to a subalgebra of a direct product of $\mathbf{2}$'s and $\mathbf{3}$'s (where $\mathbf{2}$ is the 2-element Boolean algebra).