Model theory exam question:Show that if $M$ is isomorphic to $N$, then there is an extention to signature $S*$ so that $M*$ isomorphic to $N*$

Question

So I had an exam today, and this question was really difficult for me:

Let $S$ be a signature with $S=\{<,f\}$, $n_f=2$. $M=(\mathbb R;<,+)$ and $N=(\mathbb R_{>0};<,\cdot)$, $+, \cdot$ the regular addition and multiplication functions, and $<$ is the regular order.

(a) Show that $M$ is isomorphic to $N$ My solution: Using the function $\alpha: M \rightarrow N$, $\alpha(x)=e^x$

(b) [The problematic part] Show there is an extension of $N$ to the signature $S^*=\{<,f,g\}$, so that $n_f=2,n_g=2$ so that $N^*=(\mathbb R;<,\cdot,g^{N*})$ is isomorphic to $M^*=(\mathbb R;<,+,\cdot)$.

My attempt to solve (probably wrong): I tried to show that structure $M,N$ are isomorphic $\iff$ there is a bijection $\alpha: M \rightarrow N$ so that for every sentence $\phi$, $M \vDash \phi (\overline b) \iff N \vDash \phi(\overline b)$.

Then my half-made claim is that since $M,N,M^*,N^*$ are elementary equivalent, and $M,N$ are isomorphic, then by above claim (which I'm not sure if I proved correctly), the same $\alpha$ holds for $M*,N*$, so they are isomorphic.

However, this is probably the wrong way of solution. What is the correct one?

Thank you!

Answer 1

This is an instance of a more general principle:

If $\mathcal{M}, \mathcal{N}$ are $L$-structures with $i:\mathcal{M}\rightarrow\mathcal{N}$ an isomorphism between them, $L'$ is an expansion of $L$, and $\mathcal{M}'$ is an expansion of $\mathcal{M}$ to $L'$, then there is a unique expansion to $L'$, $\mathcal{N}'$, of $\mathcal{N}$ such that $i$ is still an isomorphism between $\mathcal{M}'$ and $\mathcal{N}'$.

It is no harder to answer your question than it is to prove this more general result.

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HINT: Suppose $L$ is the empty language, $\mathcal{M}, \mathcal{N}$ are sets of the same cardinality, and $i$ is a bijection between them (so, an isomorphism). Let $L'$ be the language with a single unary predicate, $U$, and let $\mathcal{M}'$ be an expansion of $L$ to $L'$.

Suppose I want to interpret $U$ on $\mathcal{N}$ so that $i$ is still an isomorphism. Let $a\in\mathcal{N}$; how do I decide whether I want $U^{\mathcal{N}'}(a)$ to hold? (Look at $i^{-1}(a)$ . . .)