Elementary Extension

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## Elementarily equivalent structures

## Elementary substructures and elementary extensions

## Tarski-Vaught test

## Elementary embeddings

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Elementary Extension

In model theory, a branch of mathematical logic, two structures *M* and *N* of the same signature *?* are called **elementarily equivalent** if they satisfy the same first-order *?*-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 *?*-formula *?*(*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 *?*(*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.

Two structures *M* and *N* of the same signature *?* are **elementarily equivalent** if every first-order sentence (formula without free variables) over *?* 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 ?o?-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** of *M* if *N* and *M* are structures of the same signature *?* such that for all first-order *?*-formulas *?*(*x*_{1}, ..., *x*_{n}) with free variables *x*_{1}, ..., *x*_{n}, and all elements *a*_{1}, ..., *a*_{n} of *N*, *?*(*a*_{1}, ..., *a*_{n}) holds in *N* if and only if it holds in *M*:

*N**?*(*a*_{1}, ...,*a*_{n}) iff*M**?*(*a*_{1}, ...,*a*_{n}).

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

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

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

- 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 o Heidelberg o Berlin: Springer Verlag, ISBN 0-387-90170-1

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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