Elementary Embedding

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.

An injective homomorphism f: M → N between structures in the same language that preserves the truth of every first-order formula with parameters from M: for all formulas φ(x, a) and tuples a from M, M ⊨ φ(x, a) if and only if N ⊨ φ(f(x), f(a)).

An injective homomorphism f: A → B between structures in the same first-order language that preserves the truth of every first-order formula with parameters: for every formula φ(x1,...,xn) and tuple a from A, A ⊨ φ(a) iff B ⊨ φ(f(a)).