A Theory that has as Models exactly the Reducts of another Theory

Question

I am working through the book "Model Theory" by Chang and Keisler at the moment. Now I am stuck at exercise 2.2.18 which states:

"Let $\mathcal{L}'$ be an expansion of $\mathcal{L}$ and let $T'$ be a theory in $\mathcal{L}'$. Suppose that each model $\mathfrak{A}$ for $\mathcal{L}$ has at most one expansion to a model $T'$. Prove that there is a theory $T$ in $\mathcal{L}$ such that the models of $T$ are exactly the reducts of the models of $T'$ to $\mathcal{L}$."

By a model $\mathfrak{A}$ for $\mathcal{L}$ they mean an $\mathcal{L}$-structure $\mathfrak{A}$.

It is clear to me that, if some theory $T$ does it, then it must be the theory $T$ of all consequences of $T'$ which are equivalent to an $\mathcal{L}$-sentence. Every reduct $\mathfrak{A}$ to $\mathcal{L}$ of a model of $T'$ clearly satisfies this $T$. Moreover it is somehow the biggest theory with this property, because if we add some $\mathcal{L}$-sentence $\varphi$ to $T$, it will not be a consequence of $T'$, so there will be some model $\mathfrak{A}'$ of $T'$ not satisfying $\varphi$. In particular the reduct $\mathfrak{A}$ of $\mathfrak{A}'$ to $\mathcal{L}$ will not satisfy $\varphi$.

Now my next step would be taking an $\mathcal{L}$-structure $\mathfrak{B}$ not satisfying $T$ and showing that it cannot be a reduct of a model of $T'$. But I have no idea how it shall help me that $\mathfrak{B}$ can have at most one expansion to a model of $T'$.

Thank you in advance for any help!

Answer 1

There might be another way to prove this statement : try to show that the class $\mathcal{R}$ consisting of all the $\mathcal{L}$-reducts of models of $T'$ is an elementary class. For this, we need to show that this class is closed under $\rlap{\mathcal{L}}{\rule[2.5pt]{5pt}{0.5pt}}$-isomorphisms elementary substructure and under ultraproducts.

$\mathcal{R}$ is closed under ultraproducts.

Let $(\mathfrak{A}_i)_{i\in I}$ be a familly of elements of $\mathcal{R}$, and $\mathcal{U}$ be an ultrafilter of parts of $I$. For each $i \in I$, let $\mathfrak{A}'_i$ be the unique expansion of $\mathfrak{A}_i$ to a model of $T'$. The ultraproduct $\prod_\mathcal{U} \mathfrak{A}_i$ is a reduct of $\prod_\mathcal{U} \mathfrak{A}'_i$.