Elementary equivalence

A first-order theory is complete if and only if any two of its models are elementarily equivalent.

The downward Löwenheim–Skolem theorem gives a countable elementary substructure for any infinite first-order structure in at most countable signature; the upward Löwenheim–Skolem theorem gives elementary extensions of any infinite first-order structure of arbitrarily large cardinality.

Every elementary embedding is a strong homomorphism, and its image is an elementary substructure.