# Definition:Elementary Embedding/Partial Elementary Embedding

## Definition

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 $a_1, \ldots, a_n \in A$.

## 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**.

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