An injective homomorphism f: A → B between structures in the same first-order language that preserves the truth of every first-order formula with parameters: for every formula φ(x1,...,xn) and tuple a from A, A ⊨ φ(a) iff B ⊨ φ(f(a)).

Principle

An elementary embedding transfers the entire first-order theory of A into B along the map, not merely the interpretations of symbols; it is a syntactic preservation condition stronger than homomorphism or elementary equivalence of images.

Demonstration

The diagonal map from A into an ultrapower A^I/U is an elementary embedding by Łoś's theorem; similarly, any isomorphism is an elementary embedding, and inclusion of an elementary substructure is the canonical elementary embedding.

Misapplication

Assuming any injective homomorphism is elementary; for instance, an injective ring homomorphism need not respect negations of atomic formulas or preserve satisfaction of arbitrary formulas, so it may fail to be elementary.

Consequence

Elementary embeddings allow transfer of types, preservation of formulas with parameters, and are used to compare models, build chains of elementary extensions, and analyze internal structure via images; they underpin many model-theoretic constructions such as elementary chains and direct limits taken elementarily.

Reversal

The reverse notion is a non-elementary embedding: an injective homomorphism that does not preserve truth of some first-order formula with parameters. Recognizing such reversals helps isolate the precise logical content lost by weaker morphisms.

Boundary

Defined within first-order languages and requires injectivity plus formula preservation; it excludes arbitrary homomorphisms, partial maps, and maps between structures in different languages unless a language embedding is fixed.

Semantic Tension

Competes with 'elementary equivalence' and with weaker notions like homomorphism and elementary substructure inclusion; an elementary embedding implies an elementary equivalence between image and domain but not conversely without surjectivity or additional structure.

Synthesis

An elementary embedding is an injective structure map that preserves satisfaction of every first-order formula with parameters, effecting a faithful logical copy of the source inside the target and serving as the canonical morphism for elementarity-preserving constructions.