Show there is a substructure $N_0\subseteq M_0$ but there is no $L$-Structure $N$ so that $N\equiv M$

Question

Give example of a Language $L$ and $L_0\subset L$ and $L$-Structure $M$ so that $M_0$ the reduction of $M$ to $L_0$ there is a substructure $N_0\subseteq M_0$ but there is no $L$-Structure $N$ so that $N\equiv M$

So I need a substructure of the reduced structure $M_0$ that doesn't extend to a structure which is equivalent to $M$. I'm not even sure how to start this problem.

Answer 1

Just take $N_0$ not elementary in $M_0$. Then there's a $L_0$-sentence $\varphi$ that is satisfied by $N_0$ but not by $M_0$. $\varphi$ is also a $L$-sentence. However you interpret $L \setminus L_0$, you can't have $N\equiv_L M$ since $N$ satisfies $\varphi$ and $M$ doesn't.