What is the stone space $S_n(T)$ for a theory with infinitely many equivalence classes, each class infinite?

Question

Let $L$ be the (first-order) language with one binary relation symbol $E$, and $T$ be the $L$-theory asserting that $E$ is an equivalence relation with infinitely many classes, each of which is infinite. Can you describe or otherwise characterize the stone spaces $S_n(T)$, for all $n$?

Answer 1

First, of course, note that the theory is complete and has quantifier elimination (in the language with one binary relation symbol for the equivalence). The entire analysis to follow essentially uses quantifier elimination to say that a type that completely describes equality and equivalence among its variables is a complete type.

To start, see that there is only one 1-type in $S_1(T)$, since there are no interesting unary relations.

Next, consider 2-types. A 2-type $p(x,y)$ gets to say whether $x$ and $y$ are equal, and whether they are equivalent. There are three possibilities: equal (and therefore equivalent), not equal but still equivalent, and not equivalent (and therefore not equal). Therefore, $S_2(T)$ consists of just these three points.

For higher $n$, a type specifies equality and equivalence for each pair of variables. There are, however, some dependencies between the different choices: for example, if $x$ is equivalent (or equal!) to $y$ and $y$ is equivalent to $z$, $x$ must be equivalent to $z$. Therefore $S_n(T)$ will consist of all the consistent ways of making these choices for all pairs of variables, so $S_n(T)$ is finite. Exactly how many points it contains is a not too difficult (but not too enlightening) combinatorics problem.