Editing Elementary equivalence (section)

==Elementary embeddings==

An '''elementary embedding''' of a structure ''N'' into a structure ''M'' of the same signature ''σ'' is a map ''h'':&nbsp;''N''&nbsp;→&nbsp;''M'' such that for every first-order ''σ''-formula ''φ''(''x''<sub>1</sub>,&nbsp;…,&nbsp;''x''<sub>''n''</sub>) and all elements ''a''<sub>1</sub>,&nbsp;…,&nbsp;''a''<sub>n</sub> of&nbsp;''N'', 
:''N'' <math>\models</math> ''φ''(''a''<sub>1</sub>,&nbsp;…,&nbsp;''a''<sub>''n''</sub>) if and only if ''M'' <math>\models</math> ''φ''(''h''(''a''<sub>1</sub>),&nbsp;…,&nbsp;''h''(''a''<sub>''n''</sub>)).
Every elementary embedding is a [[Structure (mathematical logic)#Homomorphisms|strong homomorphism]], and its image is an elementary substructure.

Elementary embeddings are the most important maps in model theory.  In [[set theory]], elementary embeddings whose domain is ''V'' (the universe of set theory) play an important role in the theory of [[large cardinals]] (see also [[Critical point (set theory)|Critical point]]).