Elementary Substructure

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.

A substructure A of a structure B (in the same first-order language) such that for every first-order formula φ(x1,...,xn) and every tuple a from A, B ⊨ φ(a) if and only if A ⊨ φ(a). Commonly denoted A ≺ B.