A structure B is an elementary extension of a structure A (same first-order language) if A is an elementary substructure of B; equivalently, every first-order formula with parameters from A holds in A exactly when it holds in B. Denoted B ≻ A.

Principle

An elementary extension adds elements (or interpretations) without changing the truth of first-order formulas referencing the original structure's elements; it extends the domain while preserving all parametrized properties of the smaller model.

Demonstration

Ultrapowers provide canonical examples: for any structure A and nonprincipal ultrafilter U, the ultrapower A^I/U is an elementary extension of A via the diagonal map. Thus new elements appear in the ultrapower but all formulas with parameters from A keep the same truth value.

Misapplication

Confusing mere extension (A ⊆ B) with elementary extension; for example, adding new constants or elements arbitrarily to produce B does not guarantee elementarity—existential statements might become true in B that were false in A.

Consequence

Elementary extensions permit saturation arguments, realization of types absent in A, and controlled transfer of theories; they are central to constructing models with desired cardinality or saturation properties while retaining the original model's parameterized theory.

Reversal

The inverse is an elementary substructure of a given model; reversing perspective emphasizes that an elementary extension may add realizations of types not present earlier, whereas a non-elementary extension changes truth values for some formulas with original parameters.

Boundary

Limited to first-order settings and to structures in the same language; existence of elementary extensions with specific features depends on compactness, Löwenheim–Skolem, and ultraproduct constructions; not every desired property can be achieved without additional set-theoretic assumptions.

Semantic Tension

Tension arises with 'conservative extension' of theories versus elementary extension of models: a conservative theory extension preserves theorems in a language, while an elementary extension preserves truth of formulas with parameters in a specific model; the two notions are related but distinct.

Synthesis

An elementary extension is a superstructure that enlarges a model's domain while preserving the truth of all first-order formulas with parameters from the original model, enabling the controlled addition of elements without altering parameterized first-order properties.