A structure N is an elementary extension of a structure M (same signature) if M is an elementary substructure of N; equivalently, every first-order formula with parameters from M has the same truth value in M and in N. Notation: M ≺ N or N ⪰ M.

Principle

The core idea is enlarging a model without altering any first-order truths about the smaller model's elements: extensions that add new elements but do not change formulas with parameters from the original domain.

Demonstration

Example: Take a countable model M of a complete theory and form a saturated or sufficiently large elementary extension N that realizes additional types over M; N contains new elements realizing types that were omitted in M while preserving all first-order truths about M's tuples.

Misapplication

Assuming any proper superstructure is elementary, or that embedding into a larger structure automatically preserves all parameterized first-order formulas; confusing elementary extension with mere extension or elementary equivalence without embedding.

Consequence

Elementary extensions allow expansion of a model to realize types, perform compactness constructions, and compare models by saturation and cardinality while guaranteeing that the original model's internal first-order properties remain unchanged.

Reversal

A non-elementary extension is a superstructure that, although containing M as a substructure, makes some first-order statements about tuples from M true or false differently than M does.

Boundary

Elementarity as extension depends on the signature and the allowed parameters; it does not assert preservation of higher-order or set-theoretic properties, nor does it force isomorphism or preservation of cardinalities unless expressible in the language.

Semantic Tension

Tension exists between elementary extensions and other enlargement notions (e.g., algebraic or topological closures): elementary extension preserves first-order theory with parameters, while other closures preserve different structural features.

Synthesis

An elementary extension N of M is a superstructure that adds new elements or witnesses without changing any first-order facts about elements of M; it is the controlled way to enlarge models while keeping M intact from the perspective of first-order logic.