Definition:Elementary Embedding/Partial Elementary Embedding

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

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

$j: A \to \mathcal{N}$ is a partial elementary embedding 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 $n \in \N$, all $\mathcal{L}$-formulas $\phi$ with $n$ free variables, and for all $a_1,\dots,a_n \in A$.

Note
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 $M$.

This definition is provided for by the notion of partial elementary embedding.