##  [Elementary Embedding](/elementary-embedding-0) 

 Definition

An injective homomorphism f: M → N between structures in the same first-order language such that for every first-order formula φ(x1,...,xn) and every tuple a from M, M ⊨ φ(a) if and only if N ⊨ φ(f(a)); equivalently, f preserves and reflects all first-order truths with parameters from M.

 

 

 

 

 

 





## Principle

Principle

Elementarity is characterized by preservation and reflection of first-order formulas: an elementary embedding identifies M with an elementary substructure of N, so M and its image satisfy exactly the same first-order properties with parameters from M.

 

 

 

 

 





## Demonstration

Demonstration

The diagonal map from a structure M into a sufficiently saturated ultrapower of M is elementary; conversely, the inclusion of an elementary submodel M ⊆ N is an elementary embedding, which justifies transferring definable properties between M and N.

 

 

 

 

## Misapplication

Misapplication

Treating any injective homomorphism or any embedding preserving atomic or quantifier-free formulas as elementary; such maps need not preserve existential or universal formulas and so can fail elementarity.

 

 

 

 

 





## Consequence

Consequence

Elementary embeddings allow transfer of first-order statements and constructions of elementary chains and elementary extensions; they are the morphisms of model theory preserving the full first-order structure and underpin saturation and compactness arguments.

 

 

 

 

## Reversal

Reversal

A non-elementary embedding (mere embedding) can preserve some structural facts but fail to reflect first-order truths; the reversal emphasizes the difference between syntactic preservation of atomic facts and semantic preservation of all first-order formulas.

 

 

 

 

 





## Boundary

Boundary

Elementarity is a first-order notion: it requires a common signature and concerns first-order formulas only; it does not guarantee preservation of infinitary or higher-order properties or of combinatorial invariants beyond first-order expressibility.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension between elementary embedding and elementary equivalence: two structures can be elementarily equivalent (same theory) without there being an elementary embedding between them; elementarity is stronger than mere shared theory but weaker than isomorphism.

 

 

 

 

 





## Synthesis

Synthesis

An elementary embedding is an injective map that preserves and reflects all first-order truths, identifying the source with an elementary substructure of the target and enabling coherent transfer of definable and model-theoretic properties.