Definition:Embedding (Model Theory)

Definition
Let $\mathcal{M}$ and $\mathcal{N}$ be $\mathcal{L}$-structures with universes $M$ and $N$ respectively.

$j:\mathcal{M}\to\mathcal{N}$ is an $\mathcal{L}$-embedding if it is an injective map $M\to N$ which preserves interpretations of all symbols in $\mathcal{L}$; that is, such that:
 * $j(f^\mathcal{M} (a_1, \dots, a_{n_f})) = f^\mathcal{N} (j(a_1), \dots, j(a_{n_f})$ for all function symbols $f$ in $\mathcal{L}$ and $a_1, \dots, a_{n_f}$ in $M$;
 * $(a_1, \dots, a_{n_R})\in R^\mathcal{M} \iff (j(a_1), \dots, j(a_{n_R}))\in R^\mathcal{N}$ for all relation symbols $R$ in $\mathcal{L}$ and $a_1, \dots, a_{n_R}$ in $M$;
 * $j(c^\mathcal{M}) = c^\mathcal{N}$ for all constant symbols $c$ in $\mathcal{L}$.

Partial Embedding
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 embeddings using functions which are only defined on a subset of $M$:

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

$j:A\to\mathcal{N}$ is a partial $\mathcal{L}$-embedding if it is an injective map $A\to N$ which preserves interpretations of all symbols in $\mathcal{L}$ applied to elements of $A$; that is, such that:
 * $j(f^\mathcal{M} (a_1, \dots, a_{n_f})) = f^\mathcal{N} (j(a_1), \dots, j(a_{n_f})$ for  all function symbols $f$ in $\mathcal{L}$ and $a_1, \dots, a_{n_f}$ in  $A$;
 * $(a_1, \dots, a_{n_R})\in R^\mathcal{M} \iff (j(a_1), \dots, j(a_{n_R}))\in R^\mathcal{N}$ for all relation symbols $R$ in  $\mathcal{L}$ and $a_1, \dots, a_{n_R}$ in $A$;
 * $j(c^\mathcal{M}) = c^\mathcal{N}$ for all constant symbols $c$ in $\mathcal{L}$.

Isomorphism
$j:\mathcal{M}\to\mathcal{N}$ is an $\mathcal{L}$-isomorphism if it is a bijective $\mathcal{L}$-embedding.

Automorphism
$j:\mathcal{M}\to\mathcal{N}$ is an $\mathcal{L}$-automorphism if it is an $\mathcal{L}$-isomorphism and $\mathcal{M} = \mathcal{N}$.

It is often useful to talk about automorphisms which are constant on subsets of $M$. So, there is a definition and a notation for doing so:

Let $A \subseteq M$ be a subset of $M$, and let $b \in M$.

An $\mathcal{L}$-automorphism $j$ is an $A$-automorphism if $j(a)=a$ for all $a\in A$.

An $\mathcal{L}$-automorphism $j$ is an $A,b$-automorphism if it is an $(A\cup\{b\})$-automorphism; that is: $j(a)=a$ for all $a\in A$ and also $j(b)=b$.