Elementary Equivalence - Elementary Embeddings

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, …, x_n) and all elements a₁, …, a_n of N,

N φ(a₁, …, a_n) implies M φ(h(a₁), …, h(a_n)).

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).