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| In [[model theory]], a branch of [[mathematical logic]], the notion of an '''existentially closed model''' (or '''existentially complete model''') of a [[theory (mathematical logic)|theory]] generalizes the notions of [[algebraically closed field]]s (for the theory of [[field (mathematics)|field]]s), [[real closed field]]s (for the theory of ordered fields), [[divisible group|existentially closed group]]s (for the class of groups), and dense linear orders without endpoints (for the class of linear orders).
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| ==Definition==
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| A substructure ''M'' of a [[structure (mathematical logic)|structure]] ''N'' is said to be '''existentially closed in''' (or '''existentially complete in''') <math>N</math> if for every quantifier-free formula φ(''x'',''y''<sub>1</sub>,…,''y''<sub>n</sub>) and all elements ''b''<sub>1</sub>,…,''b''<sub>n</sub> of ''M'' such that φ(''x'',''b''<sub>1</sub>,…,''b''<sub>n</sub>) is realized in ''N'', then φ(''x'',''b''<sub>1</sub>,…,''b''<sub>n</sub>) is also realized in ''M''. In other words: If there is an element ''a'' in ''N'' such that φ(''a'',''b''<sub>1</sub>,…,''b''<sub>n</sub>) holds in ''N'', then such an element also exists in ''M''.
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| A model ''M'' of a theory ''T'' is called existentially closed in ''T'' if it is existentially closed in every superstructure ''N'' which is itself a model of ''T''. More generally, a structure ''M'' is called existentially closed in a class ''K'' of structures (in which it is contained as a member) if ''M'' is existentially closed in every superstructure ''N'' which is itself a member of ''K''.
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| 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''<sup>∗</sup> of ''M'' such that for every existentially closed superstructure ''N'' of ''M'', ''M''<sup>∗</sup> is isomorphic to a substructure of ''N'' via an isomorphism that is the identity on ''M''.
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| ==Examples==
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| Let σ = (+,×,0,1) be the [[signature (logic)|signature]] of fields, i.e. +,× are binary relation symbols and 0,1 are constant symbols. Let ''K'' be the class of structures of signature σ which are fields.
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| If ''A'' is a subfield of ''B'', then ''A'' is existentially closed in ''B'' if and only if every system of [[polynomial]]s over ''A'' which 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.
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| Similarly in the class of [[ordered field]]s, the existentially closed structures are the [[real closed field]]s. In the class of [[total order|totally ordered structures]], the existentially closed structures are those that are [[dense order|dense]] without endpoints, while the existential closure of any countable (including empty) total order is, up to isomorphism, the countable dense total order without endpoints, namely the [[order type]] of the [[rationals]].
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| ==References==
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| * {{Citation | last1=Chang | first1=Chen Chung | author1-link=Chen Chung Chang | last2=Keisler | first2=H. Jerome | author2-link=Howard Jerome Keisler | title=Model Theory | origyear=1973 | publisher=Elsevier | edition=3rd | series=Studies in Logic and the Foundations of Mathematics | isbn=978-0-444-88054-3 | year=1990}}
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| * {{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}}
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| ==External links==
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| *[http://eom.springer.de/e/e110140.htm EoM article]
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| [[Category:Model theory]]
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