Two structures *M* and *N* (with the same signature) are said to be **elementarily equivalent**, denoted *M* ≡ *N*, if every sentence true in *M* is true in *N* and vice versa: for every sentence ,

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Equivalently, the complete theory of *M* is the same as the complete theory of *N*.

Elementary equivalence is an equivalence relation, strictly weaker than isomorphism. For example, the field of complex numbers is elementarily equivalent to the field of algebraic numbers

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This follows from the completeness of ACF_{0}. In general if *T* is a complete theory, then any two models of *T* are elementarily equivalent. When *T* is not complete, the elementarily equivalence classes of models of *T* correspond exactly to the completions of *T*. ::: ::: :::