##  [Elementary Embedding](/elementary-embedding-2) 

 Definition

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

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

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

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

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

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

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

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

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.