Elementary Extension

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.

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.