PC$_\Delta$ class closed under elementary $\sigma$-substructures has an theory of signature $\sigma$

Question

Let $K$ be a PC$_\Delta$ class of signature $\sigma$, i.e. there is a class of $\sigma'\supseteq\sigma$ structures $K'$ and a $\sigma'$-theory $T'$ such that $K'$ is the class of $\sigma'$-structures that model $T'$, such that $K$ consists of the $\sigma$-reducts of structures in $K'$.

To show is that $K$ is elementary with respect to signature $\sigma$ if and only if $K$ is closed under elementary substructures of signature $\sigma$.

I can prove the $(\Rightarrow)$ direction, but am having trouble showing the $(\Leftarrow)$ direction.

Answer 1

This exercise can be solved by applying some standard strategies. Rather than giving you a full solution, I'll try to explain how these strategies (which are useful in many other situations) present themselves and lead naturally to a solution.

1. To show that a class of structures can be axiomatized by a set of sentences of a certain form, you can look at the set of all sentences of that form satisfied by the class of structures. So in this case, consider the $\sigma$-theory $$T = \{\varphi\mid \varphi\text{ is a $\sigma$-sentence, and for all }M\in K, M\models \varphi\}.$$ Clearly if $M\in K$, then $M\models T$. So it suffices to show the converse.

2. Suppose $M$ is a $\sigma$-structure which satisfies $T$. We want to show $M\in K$. Since we're assuming $K$ is closed under elementary substructures, it suffices to find an elementary extension of $M$ which is in $K$. And to do this, we can find a $\sigma'$-structure $M'$ which is a model of $T'$, and whose reduct to $\sigma$ is an elementary extension of $M$.

3. To show that a structure $M$ has an elementary extension $N$ satisfying certain properties, we use the method of diagrams: Write down the elementary diagram of $M$ (in the language $\sigma_M$ which is $\sigma$ together with a constant symbol naming every element of $M$) together with the properties that we want $N$ to satisfy. So in this case we look at the $\sigma'_M$-theory $$\text{EDiag}_\sigma(M)\cup T'.$$

4. It remains to check that the theory above is consistent, by compactness, using the fact that $M\models T$. I'll leave this to you - but let me know if you have difficulty with it.