Existentially closed model

In

Definition

A substructure M of a structure N is said to be existentially closed in (or existentially complete in) ${\displaystyle N}$ if for every quantifier-free formula φ(x₁,…,x_n,y₁,…,y_n) and all elements b₁,…,b_n of M such that φ(x₁,…,x_n,b₁,…,b_n) is realized in N, then φ(x₁,…,x_n,b₁,…,b_n) is also realized in M. In other words: If there is a tuple a₁,…,a_n in N such that φ(a₁,…,a_n,b₁,…,b_n) holds in N, then such a tuple also exists in M. This notion is often denoted ${\displaystyle M\prec _{1}N}$.

A model M of a theory T is called existentially closed in T if it is existentially closed in every superstructure N that 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 a member) if M is existentially closed in every superstructure N that is itself a member of K.

The existential closure in K of a member M of K, when it exists, is, up to isomorphism, the least existentially closed superstructure of M. More precisely, it is any extensionally closed superstructure M^∗ of M such that for every existentially closed superstructure N of M, M^∗ is isomorphic to a substructure of N via an isomorphism that is the identity on M.

Examples

Let σ = (+,×,0,1) be the signature of fields, i.e. + and × are binary function symbols and 0 and 1 are constant symbols. Let K be the class of structures of signature σ that 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 that 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

• Tarski-Vaught test

References

• ISBN 978-0-444-88054-3
• ISBN 978-0-521-58713-6

• Encyclopedia of Mathematics article