I am stuck in following exercise:
Let $T$ be a theory with following property: for all models $M$, $N$, and $P$ of $T$ if $M \subset P$, $N \subset P$, and $M \cap N \neq \emptyset$, then $M \cap N \models T$. Prove that $T$ is $\forall\exists$-axiomatizable.
I know that according to Chang-Los-Suszko's theorem it suffice to show that $T$ is stable with respect to union of chains of models, but I can't see how given condition helps.
Any hints? Thanks!
Suppose $M_0\subseteq M_1\subseteq M_2\subseteq \dots$ is a chain of models of $T$, and let $M$ be the union. As you noted in the question, it suffices to show that $M$ is a model of $T$.
First, use compactness to show that $T\cup \text{Diag}(M)$ is consistent, so that $M$ is a substructure of a model $N_1\models T$.
Second, show that there are models $N_2\models T$ and $N_3\models T$ such that $N_1\subseteq N_3$ and $N_2\subseteq N_3$, and $M = N_1\cap N_2$.
Hint: Start with the $L(N_1)$-theory $T\cup\text{Diag}(N_1)$. Add a unary relation $P$ to the language, and add axioms saying that $P$ picks out a substructure $N_2$ which is a model of $T$, and $N_1\cap N_2 = M$. Finish by compactness.