Definition:Elementary Embedding

Definition
Let $\mathcal{M}$ and $\mathcal{N}$ be $\mathcal{L}$-structures with universes  $M$ and $N$ respectively.

An $\mathcal{L}$-embedding $j:\mathcal{M}\to \mathcal{N}$ is an elementary embedding if it preserves truth; that is:
 * $\mathcal{M}\models\phi(a_1,\dots, a_n) \iff \mathcal{N}\models \phi(j(a_1),\dots, j(a_n))$

holds for all $n\in\mathbb N$, all $\mathcal{L}$-formulas $\phi$ with $n$ free variables, and for all $a_1,\dots,a_n \in M$.

Partial Elementary Embedding
A common method of constructing isomorphisms and elementary embeddings in proofs is to recursively define them a finite number of elements at a time. For this purpose, it is useful to have a definition of elementary embeddings for functions which are only defined on a subset of $M$:

Let $A \subseteq M$ be a subsets of $M$.

$j:A\to \mathcal{N}$ is a partial elementary embedding if it is a partial $\mathcal{L}$-embedding which preserves truth for elements of $A$; that is:
 * $\mathcal{M}\models\phi(a_1,\dots, a_n) \iff \mathcal{N}\models \phi(j(a_1),\dots, j(a_n))$

holds for all $n\in\mathbb N$, all $\mathcal{L}$-formulas $\phi$ with $n$ free  variables, and for all $a_1,\dots,a_n \in A$.