Definition:Elementary Embedding

Definition
Let $\MM$ and $\NN$ be $\LL$-structures with universes $M$ and $N$ respectively.

An $\LL$-embedding $j:\MM \to \NN$ is an elementary embedding it preserves truth; 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 $a_1, \ldots, a_n \in M$.