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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 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 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 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.