\begin{definition}[Definition:Embedding (Model Theory)]
Let $\MM$ and $\NN$ be $\LL$-structures with universes $M$ and $N$ respectively.
$j: \MM \to \NN$ is an '''$\LL$-embedding''' {{iff}} it is an injective map $M \to N$ which preserves interpretations of all symbols in $\LL$; that is, such that:
:$\map j {\map {f^\MM} {a_1, \dots, a_{n_f} } } = \map {f^\NN} {\map j {a_1}, \ldots, \map j {a_{n_f} } }$ for all function symbols $f$ in $\LL$ and $a_1, \dots, a_{n_f}$ in $M$
:$\tuple {a_1, \ldots, a_{n_R} } \in R^\MM \iff \tuple {\map j {a_1}, \dots, \map j {a_{n_R} } } \in R^\NN$ for all relation symbols $R$ in $\LL$ and $a_1, \dots, a_{n_R}$ in $M$
:$\map j {c^\MM} = c^\NN$ for all constant symbols $c$ in $\LL$.
\end{definition}