Existentially closed
existentially complete
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.
References
[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=43538