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\begin{document}
\begin{center} {\bf Problem Set VII - The Midterm}\\
Math 609  \end{center}

Answer the following questions as clearly and completely as you can.
In solving a given exercise, you are free to assume (and encouraged to
cite) any of the previous exercises.

\begin{description}
\item[1.] Let $\La = (E)$ be a first-order language where
$E$ is a binary relation symbol. What is an $\La$-structure?

\item[2.] Let $\La = (E)$ be as above and let $\A = (A, E^{\A})$ and 
$\B = (B, E^{\B})$ be $\La$-structures. What is an $\La$-homomorphism 
$\gre : \A \to \B?$

\item[3.] Express the axioms for an equivalence relation in the language
$\La = (E).$ Denote the theory of an equivalence relation, that is, the
set of consequences of these axioms, by $T_{\eq}.$

\item[4.] Calculate $I(T_{\eq},n)$ for $n = 1,$ $2,$ $3$ and $4.$

\item[5.]  Let $\A = (A, E^{\A})$ be an $\La$-structure such that $\A \models T_{\eq}.$ 
For $a \in A,$ denote the equivalence class of $a$ modulo $E^{\A}$ by $a/ E^{\A}$
and denote by $A/E^{\A}$ the set of equivalence classes of $A$ modulo $E^{\A}.$
Prove that if $\B = (B , E^{\B}) \models T_{\eq},$ then an $\La$-homomorphism
$\gre : \A \to \B$ induces a function $\gre/E : A/E^{\A} \to B/E^{\B}$ in
the natural way.

\item[6.] Prove that an $\La$-homomorphism $\gre : \A \to \B$ is an embedding
if and only if both $\gre$ and $\gre/E$ are injective.

\item[7.] Let $n$ be a natural number and $\A \models T_{\eq}.$ Exhibit a formula
$P_n (x)$ that defines in $\A$ the set of all $a \in A$ such that $|a/ E^{\A}| = n.$

\item[Assume from now on that $T \supseteq T_{\eq}$ is a complete theory.]


\item[8.] Show that if $T \models \exists x \, P_n (x),$ then the formula $P_n (x)$ isolates
a type in $S_1 (T),$ that is, the basic open subset $[P_n (x)]$ of $S_1 (T)$
is singleton.

\item[9.] Consider the 1-type $p_{\infty} (x) := \{ \neg P_n (x) : n < \gro \}.$
Prove that if $p_{\infty}$ is $T$-consistent, then it determines a unique point
of $S_1 (T).$ 

\item[10.] Suppose that $p_{\infty}$ is $T$-consistent. 
Prove that $p_{\infty}$ is isolated in $S_1 (T)$ if and only if there
are only finitely many natural numbers $n$ for which 
$T \models \exists x \, P_n (x).$

\item[11.] Let $\A = (A, E^{\A})$ and $\B = (B, E^{\B})$ be {\em countable} 
$\La$-structures such that $\A,$ $\B \models T_{\eq}.$ Show that $\A$ and 
$\B$ are isomorphic if and only if for every natural number $n,$
$|P_n ( \A )/ E^{\A}| = |P_n ( \B )/ E^{\B}|$ and also
$|p_{\infty} ( \A )/ E^{\A}| = |p_{\infty} ( \B )/ E^{\B}|.$

\item[12.] Prove that $T$ is $\aleph_0$-categorical if and only if it has an infinite model and
the Stone space $S_1 (T)$ is finite.

\item[13.] Prove that $T$ is $\aleph_1$-categorical if and only if the following three conditions are
satisfied:
\begin{description}
\item[a)] $S_1(T)$ is finite.
\item[b)] The definable set $p ( \A )$ is infinite for exactly one of the 1-types $p \in S_1(T).$ 
\item[c)] $|p_{\infty}(\A)/ E^{\A}| \leq 1.$
\end{description}
\end{description}
\end{document}