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''existentially complete''
 
''existentially complete''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e1101401.png" /> be a first-order language (cf. [[Model (in logic)|Model (in logic)]]). A substructure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e1101402.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e1101404.png" />-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e1101405.png" /> (cf. [[Structure(2)|Structure]]) is called existentially closed (or existentially complete) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e1101406.png" /> if every existential sentence with parameters from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e1101407.png" /> is true in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e1101408.png" /> if it is true in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e1101409.png" />. An existential sentence with parameters from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014010.png" /> is a [[Closed formula|closed formula]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014012.png" /> is a formula without quantifiers in the first-order language of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014013.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014014.png" /> the signature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014015.png" /> (cf. [[Model theory|Model theory]]).
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Let $L$ be a first-order language (cf. [[Model (in logic)|Model (in logic)]]). A substructure $A$ of an $L$-structure $B$ (cf. [[Structure(2)|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|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|Model theory]]).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014016.png" /> is a substructure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014018.png" /> admits an embedding, fixing the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014019.png" />, in some elementary extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014020.png" /> (cf. [[Elementary theory|Elementary theory]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014021.png" /> is existentially closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014022.png" />. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014023.png" /> is existentially closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014025.png" /> is a cardinal number greater than the cardinality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014026.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014027.png" /> admits an embedding, fixing the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014028.png" />, in every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014031.png" />-saturated extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014032.png" /> (cf. also [[Model theory|Model theory]]).
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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|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|Model theory]]).
  
A member <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014033.png" /> of a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014034.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014035.png" />-structures is called existentially closed (or existentially complete) with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014036.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014037.png" /> is existentially closed in every member <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014038.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014039.png" />, provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014040.png" /> is a substructure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014041.png" />.
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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|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014042.png" /> is existentially closed in an extension field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014043.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014044.png" /> is (relatively) algebraically closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014045.png" /> (cf. [[Algebraically closed field|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014046.png" /> depending on finitely many other polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014047.png" />, provided that there is an existentially closed member <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014048.png" /> of the class containing the coefficients of the polynomials and such that every common root of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014049.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014050.png" /> is also a root of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014051.png" />. For the class of fields, the corresponding theorem is Hilbert's Nullstellensatz (cf. [[Hilbert theorem|Hilbert theorem]]). There are corresponding theorems for formally real fields (see [[Real closed field|Real closed field]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014052.png" />-valued fields (see [[P-adically closed field|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110140/e11014053.png" />-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.
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If a [[Field|field]] $K$ is existentially closed in an extension field $L$, then $K$ is (relatively) algebraically closed in $L$ (cf. [[Algebraically closed field|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|Hilbert theorem]]). There are corresponding theorems for formally real fields (see [[Real closed field|Real closed field]]), $p$-valued fields (see [[P-adically closed field|$p$-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====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Cherlin,  "Model theoretic algebra" , ''Lecture Notes in Mathematics'' , '''521''' , Springer  (1976)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Cherlin,  "Model theoretic algebra" , ''Lecture Notes in Mathematics'' , '''521''' , Springer  (1976)</TD></TR></table>

Latest revision as of 20:57, 22 December 2018

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)
How to Cite This Entry:
Existentially closed. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Existentially_closed&oldid=35477
This article was adapted from an original article by F.-V. Kuhlmann (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article