Existentially closed model

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In model theory, a branch of mathematical logic, the notion of an existentially closed model of a theory generalizes the notions of algebraically closed fields (for the theory of fields), real closed fields (for the theory of ordered fields), existentially closed groups (for the class of groups), and dense linear orders without endpoints (for the class of linear orders).

1 Definition
2 Examples
3 References
4 External links

[edit] Definition

A substructure M of a structure N is said to be existentially closed in $N$ if for every quantifier-free formula φ(x,y₁,…,y_n) and all elements b₁,…,b_n of M such that φ(x,b₁,…,b_n) is realized in N, then φ(x,b₁,…,b_n) is also realized in M. In other words: If there is an element a in N such that φ(a,b₁,…,b_n) holds in N, then such an element also exists in M.

A model M of a theory T is called existentially closed in T if it is existentially closed in every superstructure N which is itself a model of T. More generally, a structure M is called existentially closed in a class K of structures (in which it is contained as an element) if M is existentially closed in every superstructure N which is itself an element of K.

[edit] Examples

Let σ = (+,×,0,1) be the signature of fields, i.e. +,× are binary relation symbols and 0,1 are constant symbols. Let K be the class of structures of signature σ which are fields. If A is a subfield of B, then A is existentially closed in B if and only if every system of polynomials over A which has a solution in B also has a solution in A. It follows that the existentially closed members of K are exactly the algebraically closed fields.

Similarly in the class of ordered fields, the existentially closed structures are the real closed fields. In the class of totally ordered structures, the existentially closed structures are those that are dense without endpoints.

[edit] References

Chang, Chen Chung & Keisler, H. Jerome (1990), Model Theory (3rd ed.), Studies in Logic and the Foundations of Mathematics, Elsevier, ISBN 978-0-444-88054-3
Hodges, Wilfrid (1997), A shorter model theory, Cambridge: Cambridge University Press, ISBN 978-0-521-58713-6