Existentially closed

existentially complete

Let $L$ be a first-order language (cf. Model (in logic)). A substructure $A$ of an $L$-structure $B$ (cf. Structure) is called existentially closed (or existentially complete) in $B$ if every existential sentence with parameters from $A$ is true in $(A,|A|)$ if it is true in $(B,|A|)$. An existential sentence with parameters from $A$ is a closed formula $\exists x_1\dots\exists x_n\ \Phi(x_1,\dots,x_n)$, where $\Phi$ is a formula without quantifiers in the first-order language of signature $\langle\Omega,|A|\rangle$, with $\Omega$ the signature of $L$ (cf. Model theory).

If $A$ is a substructure of $B$ and $B$ admits an embedding, fixing the elements of $A$, in some elementary extension of $A$ (cf. Elementary theory), then $A$ is existentially closed in $B$. Conversely, if $A$ is existentially closed in $B$ and $\alpha$ is a cardinal number greater than the cardinality of $B$, then $B$ admits an embedding, fixing the elements of $A$, in every $\alpha$-saturated extension of $A$ (cf. also Model theory).

A member $A$ of a class $K$ of $L$-structures is called existentially closed (or existentially complete) with respect to $K$ if $A$ is existentially closed in every member $B$ of $K$, provided that $A$ is a substructure of $B$.

If a field $K$ is existentially closed in an extension field $L$, then $K$ is (relatively) algebraically closed in $L$ (cf. Algebraically closed field). Hence, a field that is existentially closed with respect to all fields must be algebraically closed, and a formally real field that is existentially closed with respect to all formally real fields must be a real closed field. Existentially closed fields or rings (with respect to suitable classes) give rise to a corresponding Nullstellensatz. This is a theorem describing the form of a polynomial $g$ depending on finitely many other polynomials $f_1,\dots,f_m$, provided that there is an existentially closed member $A$ of the class containing the coefficients of the polynomials and such that every common root of the $f_i$ in $A$ is also a root of $g$. For the class of fields, the corresponding theorem is Hilbert's Nullstellensatz (cf. Hilbert theorem). There are corresponding theorems for formally real fields (see Real closed field), $p$-valued fields (see $p$-adically closed field), differential fields, division rings, commutative rings, and commutative regular rings. The general model-theoretic framework was considered by V. Weispfenning in 1977.

References

[a1]	G. Cherlin, "Model theoretic algebra" , Lecture Notes in Mathematics , 521 , Springer (1976)