Model theory:
As in the post
Using first order sentences, axiomize the $\mathcal{L}$-theory of an equivalence relation with infinitely many infinite classes.
consider the language containing only $E$ a relation, and $T$ the theory of equivalence relations
together with
\begin{align*}
&\phi_n = \exists x_1 \ldots \exists x_n ~ \bigwedge_{i < j \le n} \neg E(x_i,x_j),
&\psi_n = \forall x \exists x_1 \ldots \exists x_n ~ \bigwedge_{i < j \le n} x_i \neq x_j \wedge \bigwedge_{i=1}^n E(x,x_j)
\end{align*}
for each $n$.
This is a theory for an infinite number of equivalence classes each infinitely large.
Let $M$ a non-empty model of $T$.
I wish to classify all the maximal types over $M$ (in one variable), i.e. the elements of
the Stone space $S_1(M)$.
I know that any maximal type over $M$ is of the form
$\mathrm{tp}_{N,1}(a)$,
where $N$ is an $\omega$-saturated elementary extension
of $M$ and $a \in N$.
For each formula of one variable $\phi$ let an element of the clopen basis of $S_1(M)$ be written as
$[\phi]$.
I wish to show that the element $a$ is in the image of $M$ under the extension if and only if
$\mathrm{tp}(a)$ is an isolated point in $S_1(M)$.
The forward direction is okay since we can take the formula $x = a$ and show that any maximal type over $M$ containing $x=a$ is $\mathrm{tp}(a)$. I am struggling with the backwards direction. If it is isolated then $\{\mathrm{tp}(a)\}$ is open, it is closed as well as it is the intersection of all the $[\phi]$ such that $\phi \in \mathrm{tp}(a)$ and so it can be written as $[\psi]$ for some formula. One thought I had was I could perhaps show that this implies $\psi$ is an atom of the assosiated Boolean algebra, and also show that atoms are precisely of the form $x = c$, but had success with neither of these claims.
I have heard that there is a correspondence between isolated points and atoms in the Boolean algebra,
but have had no success in finding anything on this.
Am I missing an assumption somewhere? I know that I have cut a corner with this example in avoiding the use of the monster model in favour of an $\omega$-saturated model.
Best Answer
First, note that $T$ is complete. This can be proved either by quantifier elimination or by Vaught's test using $\aleph_0$-categoricity.
Now suppose you have an isolated maximal type over $M$, say $[\psi(x)]$ (where I'm making explicit your convention of using $x$ as the variable in $1$-types). Since $\psi(x)$ is consistent with $T$ and $T$ is complete, $\exists x\,\psi(x)$ is provable in $T$ and therefore true in $M$. Fix some $a\in M$ such that $\psi(a)$ holds in $M$. Then the type over $M$ realized by $a$ contains $\psi(x)$ and therefore includes the type generated by $[\psi(x)]$. Since the latter is a maximal type, it coincides with the type of $a$.