A substructure M of a structure N (in the same signature) that preserves the truth of every first-order formula with parameters from M; formally, for every first-order formula φ(x1,..,xn) and every tuple a from M, N ⊨ φ(a) if and only if M ⊨ φ(a). Often denoted M ≺ N.

Principle

The organizing rule is truth preservation for first-order properties with parameters taken from the smaller domain: no new first-order facts about elements of M become true in N or false in M when viewed from N.

Demonstration

Example: In the language of ordered fields, the field of rational functions with real coefficients is not an elementary substructure of the real field because a formula expressing 'there exists x with x^2=2' holds in the real field but fails in the rationals; a simpler demonstration is that any structure is an elementary substructure of itself (M ≺ M).

Misapplication

Treating any substructure (closed under the signature operations) as elementary; confusing elementary substructure with mere substructure, isomorphic copy, or with existentially closed substructure without checking full first-order preservation.

Consequence

When M ≺ N, any first-order sentence with parameters from M that holds in N already held in M; this permits transferring definability, types realized in M, and many model-theoretic arguments (back-and-forth, elementarity tests) between the two.

Reversal

The inverse notion is a substructure that is not elementary: a subset closed under the operations that nevertheless fails to preserve some first-order formula with parameters from the subset.

Boundary

Elementarity is a first-order notion relative to a fixed signature and allows parameters only from the substructure; it does not control higher-order statements, second-order properties, or meta-mathematical features like cardinality unless expressible in first-order form.

Semantic Tension

Tension arises between elementarity and weaker relations such as elementary equivalence, existential embedding, or mere isomorphism: elementary substructure demands full agreement on formulas with parameters, while those others require less.

Synthesis

An elementary substructure M ≺ N is precisely a substructure that is indistinguishable from its ambient structure N by any first-order sentence that mentions elements of M; it is the model-theoretic notion capturing 'no new first-order facts about M appear in N.'