Consider the following problem:
Suppose $\mathcal{M}$ is an $L$-structure and $A \subseteq M$ is non-empty. Prove that if every $L_A$-formula in one free variable that has a realisation in $\mathcal{M}$ actually has a realisation in $A$, then $A$ is the universe of an elementary substructure of $\mathcal{M}$.
What does "has a realisation" mean in this context?