\begin{definition}[Definition:Elementary Embedding/Partial Elementary Embedding]
Let $\MM$ and $\NN$ be $\LL$-structures with universes $M$ and $N$ respectively.
Let $A \subseteq M$ be a subsets of $M$.
$j: A \to \NN$ is a '''partial elementary embedding''' {{iff}} it is a partial $\LL$-embedding which preserves truth for elements of $A$; that is:
:$\MM \models \map \phi {a_1, \ldots, a_n} \iff \NN \models \map \phi {\map j {a_1}, \ldots, \map j {a_n} }$
holds for all $n \in \N$, all $\LL$-formulas $\phi$ with $n$ free variables, and for all $a_1, \ldots, a_n \in A$.
\end{definition}