In
model theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...
, a branch of
mathematical logic, the notion of an existentially closed model (or existentially complete model) of a
theory generalizes the notions of
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
s (for the theory of
fields),
real closed fields (for the theory of
ordered fields),
existentially closed groups (for the theory of
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
), and
dense linear orders without endpoints (for the theory of linear orders).
Definition
A substructure ''M'' of a
structure
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
''N'' is said to be existentially closed in (or existentially complete in)
if for every
quantifier-free
formula
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...
φ(''x''
1,…,''x''
''n'',''y''
1,…,''y''
''n'') and all elements ''b''
1,…,''b''
''n'' of ''M'' such that φ(''x''
1,…,''x''
''n'',''b''
1,…,''b''
''n'') is realized in ''N'', then φ(''x''
1,…,''x''
''n'',''b''
1,…,''b''
''n'') is also realized in ''M''. In other words: If there is a tuple ''a''
1,…,''a''
''n'' in ''N'' such that φ(''a''
1,…,''a''
''n'',''b''
1,…,''b''
''n'') holds in ''N'', then such a tuple also exists in ''M''. This notion is often denoted
.
A model ''M'' of a theory ''T'' is called existentially closed in ''T'' if it is existentially closed in every superstructure ''N'' that is itself a model of ''T''. More generally, a structure ''M'' is called existentially closed in a
class
Class or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used differentl ...
''K'' of structures (in which it is contained as a member) if ''M'' is existentially closed in every superstructure ''N'' that is itself a member of ''K''.
The existential closure in ''K'' of a member ''M'' of ''K'', when it exists, is, up to
isomorphism, the least existentially closed superstructure of ''M''. More precisely, it is any extensionally closed superstructure ''M''
∗ of ''M'' such that for every existentially closed superstructure ''N'' of ''M'', ''M''
∗ is isomorphic to a substructure of ''N'' via an isomorphism that is the identity on ''M''.
Examples
Let ''σ'' = (+,×,0,1) be the
signature of fields, i.e. + and × are
binary function
In mathematics, a binary function (also called bivariate function, or function of two variables) is a function that takes two inputs.
Precisely stated, a function f is binary if there exists sets X, Y, Z such that
:\,f \colon X \times Y \rightar ...
symbols and 0 and 1 are constant symbols. Let ''K'' be the class of structures of signature ''σ'' that are fields. If ''A'' is a
subfield of ''B'', then ''A'' is existentially closed in ''B'' if and only if every system of
polynomials over ''A'' that has a solution in ''B'' also has a solution in ''A''. It follows that the existentially closed members of ''K'' are exactly the algebraically closed fields.
Similarly in the class of
ordered fields, the existentially closed structures are the
real closed fields. In the class of
linear orders, the existentially closed structures are those that are
dense without endpoints, while the existential closure of any
countable (including
empty) linear order is, up to isomorphism, the countable dense total order without endpoints, namely the
order type of the
rationals.
References
*
* {{Citation , last1=Hodges , first1=Wilfrid , author1-link=Wilfrid Hodges , title=A shorter model theory , publisher=
Cambridge University Press, location=Cambridge , isbn=978-0-521-58713-6 , year=1997
External links
Encyclopedia of Mathematics article
Model theory