An injective homomorphism f: M → N between structures in the same language that preserves the truth of every first-order formula with parameters from M: for all formulas φ(x, a) and tuples a from M, M ⊨ φ(x, a) if and only if N ⊨ φ(f(x), f(a)).

Principle

Elementary embeddings are the morphisms of model theory that respect full first-order satisfaction: they transfer all first-order properties and types from the source to the target, not merely atomic or quantifier-free facts.

Demonstration

If M is an elementary substructure of N, written M ≺ N, the inclusion map i: M ↪ N is an elementary embedding; thus any formula with parameters in M holds in M exactly when it holds in N under inclusion, exemplifying preservation of first-order truth.

Misapplication

Treating every injective homomorphism or embedding between algebraic structures as elementary; many natural embeddings (e.g., Q ↪ R as ordered fields) fail to be elementary because they do not preserve certain quantified properties like completeness or existence of limits.

Consequence

Elementary embeddings preserve complete types over the source and guarantee that model-theoretic constructions (elementary extensions, elementary chains) maintain first-order properties; they are central to back-and-forth arguments and constructions of saturated models.

Reversal

The opposite is a mere embedding that is not elementary: it injects structure but fails to preserve some first-order formulas, highlighting the distinction between syntactic preservation (elementary) and purely algebraic or categorical embedding.

Boundary

Requires same signature and first-order semantics; does not require surjectivity (surjective elementary embeddings are elementary isomorphisms) and excludes weaker notions such as elementary equivalence without a witness map or preservation only of quantifier-free formulas.

Semantic Tension

Tension exists between elementary embedding and isomorphism: an elementary embedding need not be onto, so it can relate a small model to a proper elementary submodel of a larger one; at the same time, it is strictly stronger than mere elementary equivalence which lacks an explicit embedding.

Synthesis

An elementary embedding is an injective map between models that exactly preserves first-order truth with parameters; it is the correct notion of structural inclusion in model theory for transferring formulas, types, and model-theoretic properties.