Elementary Embeddings
An elementary embedding of a structure N into a structure M of the same signature σ is a map h: N → M such that for every first-order σ-formula φ(x1, …, xn) and all elements a1, …, an of N,
- N φ(a1, …, an) implies M φ(h(a1), …, h(an)).
Every elementary embedding is a 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).
Read more about this topic: Elementary Equivalence
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