nLab existentially closed model

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 $\mathbf{K}$ of $\mathcal{L}$-structures. For example, $\mathcal{L}$ might be the language of rings and $\mathbf{K}$ the class of fields, or $\mathcal{L}$ might be the language of groups and $\mathbf{K}$ the class of groups.

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

(c.f. algebraically closed field).

Definition

Definition

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

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

Definition

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

(“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 $M \models T$ is an existentially closed model of $T$ if it is existentially closed in the class $\mathbf{K} \overset{\operatorname{df}}{=} \mathbf{Mod}(T)$.

We say that a theory $T$ 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 $\exists y \varphi(x,y)$ in $\mathsf{ACF}$ says “for a choice of coefficients $x$, there exists a point $y$ in affine space (of fixed dimension) which is in these finitely many affine varieties depending on $x$ but not these finitely many affine varieties depending on $x$”.

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

However, we can do so very cheaply after we know that $\mathsf{ACF}$ eliminates quantifiers (and therefore so does the theory of algebraically closed fields containing $E$): $\exists y \varphi(a,y)$ is equivalent to some quantifier-free sentence involving $a$, 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).

Related concepts

• model complete theory

• quantifier elimination

• model companion?

References

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