Let be a first-order language (cf. Model (in logic)). A substructure of an -structure (cf. Structure) is called existentially closed (or existentially complete) in if every existential sentence with parameters from is true in if it is true in . An existential sentence with parameters from is a closed formula , where is a formula without quantifiers in the first-order language of signature , with the signature of (cf. Model theory).
If is a substructure of and admits an embedding, fixing the elements of , in some elementary extension of (cf. Elementary theory), then is existentially closed in . Conversely, if is existentially closed in and is a cardinal number greater than the cardinality of , then admits an embedding, fixing the elements of , in every -saturated extension of (cf. also Model theory).
A member of a class of -structures is called existentially closed (or existentially complete) with respect to if is existentially closed in every member of , provided that is a substructure of .
If a field is existentially closed in an extension field , then is (relatively) algebraically closed in (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 depending on finitely many other polynomials , provided that there is an existentially closed member of the class containing the coefficients of the polynomials and such that every common root of the in is also a root of . 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), -valued fields (see -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.
|[a1]||G. Cherlin, "Model theoretic algebra" , Lecture Notes in Mathematics , 521 , Springer (1976)|
Existentially closed. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Existentially_closed&oldid=14384