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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*(*a*_{1}, …, *a*_{n}) with parameters *a*_{1}, …, *a*_{n} 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*, *b*_{1}, …, *b*_{n}) 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.

Two structures *M* and *N* of the same signature *σ* are **elementarily equivalent** if every first-order sentence (formula without free variables) over *σ* is true in *M* if and only if it is true in *N*, i.e. if *M* and *N* have the same complete first-order theory.
If *M* and *N* are elementarily equivalent, one writes *M* ≡ *N*.

A first-order theory is complete if and only if any two of its models are elementarily equivalent.

For example, consider the language with one binary relation symbol '<'. The model **R** of real numbers with its usual order and the model **Q** of rational numbers with its usual order are elementarily equivalent, since they both interpret '<' as an unbounded dense linear ordering. This is sufficient to ensure elementary equivalence, because the theory of unbounded dense linear orderings is complete, as can be shown by the Łoś–Vaught test.

More generally, any first-order theory with an infinite model has non-isomorphic, elementarily equivalent models, which can be obtained via the Löwenheim–Skolem theorem. Thus, for example, there are non-standard models of Peano arithmetic, which contain other objects than just the numbers 0, 1, 2, etc., and yet are elementarily equivalent to the standard model.

*N* is an **elementary substructure** or **elementary submodel** of *M* if *N* and *M* are structures of the same signature *σ* such that for all first-order *σ*-formulas *φ*(*x*_{1}, …, *x*_{n}) with free variables *x*_{1}, …, *x*_{n}, and all elements *a*_{1}, …, *a*_{n} of *N*, *φ*(*a*_{1}, …, *a*_{n}) holds in *N* if and only if it holds in *M*:

This definition first appears in Tarski, Vaught (1957).^{[1]} It follows that *N* is a substructure of *M*.

If *N* is a substructure of *M*, then both *N* and *M* can be interpreted as structures in the signature *σ*_{N} consisting of *σ* together with a new constant symbol for every element of *N*. Then *N* is an elementary substructure of *M* if and only if *N* is a substructure of *M* and *N* and *M* are elementarily equivalent as *σ*_{N}-structures.

If *N* is an elementary substructure of *M*, one writes *N* *M* and says that *M* is an **elementary extension** of *N*: *M* *N*.

The downward Löwenheim–Skolem theorem gives a countable elementary substructure for any infinite first-order structure in at most countable signature; the upward Löwenheim–Skolem theorem gives elementary extensions of any infinite first-order structure of arbitrarily large cardinality.

The **Tarski–Vaught test** (or **Tarski–Vaught criterion**) is a necessary and sufficient condition for a substructure *N* of a structure *M* to be an elementary substructure. It can be useful for constructing an elementary substructure of a large structure.

Let *M* be a structure of signature *σ* and *N* a substructure of *M*. Then *N* is an elementary substructure of *M* if and only if for every first-order formula *φ*(*x*, *y*_{1}, …, *y*_{n}) over *σ* and all elements *b*_{1}, …, *b*_{n} from *N*, if *M* *x* *φ*(*x*, *b*_{1}, …, *b*_{n}), then there is an element *a* in *N* such that *M* *φ*(*a*, *b*_{1}, …, *b*_{n}).

An **elementary embedding** of a structure *N* into a structure *M* of the same signature *σ* is a map *h*: *N* → *M* such that for every first-order *σ*-formula *φ*(*x*_{1}, …, *x*_{n}) and all elements *a*_{1}, …, *a*_{n} of *N*,

*N**φ*(*a*_{1}, …,*a*_{n}) if and only if*M**φ*(*h*(*a*_{1}), …,*h*(*a*_{n})).

Every elementary embedding is a strong homomorphism, and its image is an elementary substructure.

Elementary embeddings are the most important maps in model theory. In set theory, elementary embeddings whose domain is *V* (the universe of set theory) play an important role in the theory of large cardinals (see also Critical point).

**^**E. C. Milner, The use of elementary substructures in combinatorics (1993). Appearing in*Discrete Mathematics*, vol. 136, issues 1--3, 1994, pp.243--252.

- Chang, Chen Chung; Keisler, H. Jerome (1990) [1973],
*Model Theory*, Studies in Logic and the Foundations of Mathematics (3rd ed.), Elsevier, ISBN 978-0-444-88054-3. - Hodges, Wilfrid (1997),
*A shorter model theory*, Cambridge: Cambridge University Press, ISBN 978-0-521-58713-6. - Monk, J. Donald (1976),
*Mathematical Logic*, Graduate Texts in Mathematics, New York • Heidelberg • Berlin: Springer Verlag, ISBN 0-387-90170-1

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