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In model theory, a branch of mathematical logic, two structures M and N of the same signature σ are called elementarily equivalent if they satisfy the same first-order σ-sentences.
If N is a substructure of M, one often needs a stronger condition. In this case N is called an elementary substructure of M if every first-order σ-formula φ(a1, …, an) with parameters a1, …, an from N is true in N if and only if it is true in M. If N is an elementary substructure of M, then M is called an elementary extension of N. An embedding h: N → M is called an elementary embedding of N into M if h(N) is an elementary substructure of M.
A substructure N of M is elementary if and only if it passes the Tarski–Vaught test: every first-order formula φ(x, b1, …, bn) with parameters in N that has a solution in M also has a solution in N when evaluated in M. One can prove that two structures are elementarily equivalent with the Ehrenfeucht–Fraïssé games.
Elementary embeddings are used in the study of large cardinals, including rank-into-rank.