nLab
existentially closed model

Contents

Idea

Existential closedness is a property of first-order structures (in the same way that model completeness is a property of first-order theories) meant to generalize the properties of the theory ACF of algebraically closed fields.

As discussed in section 8.1 of (Hodges93):

Take a first-order language \mathcal{L} without relation symbols and a class K\mathbf{K} of \mathcal{L}-structures. For example, \mathcal{L} might be the language of rings and K\mathbf{K} the class of fields, or \mathcal{L} might be the language of groups and K\mathbf{K} the class of groups.

We say that a structure AA in K\mathbf{K} is existentially closed in K\mathbf{K} if whenever EE is a finite set of equations and inequations with parameters from AA and EE has a simultaneous solution in some extension BB of AA with BB in K\mathbf{K}, then EE has a solution already in AA.

(c.f. algebraically closed field).

Definition

Definition
  1. An existential formula is a formula of the form yφ(x,y)\exists y \varphi(x,y) where φ(x,y)\varphi(x,y) is quantifier-free.

  2. A universal formula is a formula of the form yφ(x,y)\forall y \varphi(x,y) where φ(x,y)\varphi(x,y) is quantifier-free.

Definition

We say that a model MM of a first-order theory TT is existentially closed if for every NN also modelling TT, every embedding MNM \hookrightarrow N, every tuple aMa \in M, and every existential formula yφ(x,y)\exists y \varphi(x,y), then whenever Nyφ(a,y)N \models \exists y \varphi(a, y), we also have that Myφ(a,y)M \models \exists y \varphi(a,y).

(“The existence of witnesses in a superstructure to existential statements about points in the existentially closed model is reflected to the existentially closed model.”)

In the terminology of the passage from (Hodges93), we say that MTM \models T is an existentially closed model of TT if it is existentially closed in the class K=dfMod(T)\mathbf{K} \overset{\operatorname{df}}{=} \mathbf{Mod}(T).

We say that a theory TT is an “existentially closed theory” (i.e., model complete) if all of its models are existentially closed.

Examples

The theory ACF of algebraically closed fields is existentially closed. An existential formula yφ(x,y)\exists y \varphi(x,y) in ACF\mathsf{ACF} says “for a choice of coefficients xx, there exists a point yy in affine space (of fixed dimension) which is in these finitely many affine varieties depending on xx but not these finitely many affine varieties depending on xx”.

Checking that ACF\mathsf{ACF} is existentially closed therefore means checking that if EKE \hookrightarrow K is an embedding of algebraically closed fields, and xx is a choice of coefficients from EE, then if such a point yy exists in affine KK-space, then we can find a similar yy in affine EE-space.

However, we can do so very cheaply after we know that ACF\mathsf{ACF} eliminates quantifiers (and therefore so does the theory of algebraically closed fields containing EE): yφ(a,y)\exists y \varphi(a,y) is equivalent to some quantifier-free sentence involving aa, and quantifier-free sentences are always transferred by embeddings of structures, so we are done.

Remarks

  • Theories which eliminate quantifiers are model complete: all of their models are existentially closed (again, because quantifier-free sentences are always transferred by embeddings of structures).

References

  • Wilfrid Hodges, Model theory, Cambridge Univ. Press 1993

Last revised on May 22, 2017 at 23:45:34. See the history of this page for a list of all contributions to it.