Definition:Elementary Embedding

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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 if and only if 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 $a_1, \ldots, a_n \in M$.


Partial Elementary Embedding

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

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


$j: A \to \NN$ is a partial elementary embedding if and only if it is a partial $\LL$-embedding which preserves truth for elements of $A$; 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 $a_1, \ldots, a_n \in A$.