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 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 $a_1,\dots,a_n \in M$.