User:Zero sharp/Maps between structures
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[edit] Maps between structures
Fix a language, L and let M and N be two L-structures. For symbols from the language, such as a constant c, let cM be the interpretation of c in M and similarly for the other classes of symbols (functions and relations).
A map j from the domain of M to the domain of N is a homomorphism if the following conditions hold:
- for every constant symbol
, we have j(cM) = cN. - for every n-ary function symbol
and
, we have
, - for every n-ary relation symbol
and
we have
,
If in addition, the map j is injective and the third condition is modified to read:
- for every n-ary relation symbol
and
we have 
then the map j is an embedding (of M into N).
Equivalent definitions of homomorphism and embedding are:
If for all atomic formulas φ and sequences of elements from M, 
where
is the image of
under j:
then j is a homomorphism. If instead:
then j is an embedding.
![M \models \phi [\bar{a}] \Rightarrow N \models \phi [\bar{b}]](../../../../math/e/f/3/ef390e8e5b0fbe0e31599e65ac2cdc32.png)

![M \models \phi [\bar{a}] \Leftrightarrow N \models \phi [\bar{b}]](../../../../math/9/d/2/9d2fd811d80ecad923ca3d17f87a2532.png)

