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Difference between revisions of "Galois correspondence"

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''between two partially ordered sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043110/g0431101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043110/g0431102.png" />''
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''between two partially ordered sets $M$ and $M'$''
  
A pair of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043110/g0431103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043110/g0431104.png" /> which satisfy the following conditions:
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A pair of mappings $\phi\colon M\to M'$ and $\psi\colon M'\to M$ which satisfy the following conditions:
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043110/g0431105.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043110/g0431106.png" />;
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if $a\leq b$, then $a\phi\geq b\phi$;
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043110/g0431107.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043110/g0431108.png" />;
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if $a'\leq b'$, then $a'\psi\geq b'\psi$;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043110/g0431109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043110/g04311010.png" />.
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$a\phi\psi\geq a$ and $a'\psi\phi\geq a'$.
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043110/g04311011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043110/g04311012.png" />.
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Here $a,b\in M$, $a', b' \in M'$.
  
The concept of a Galois correspondence is closely connected with the concept of closure in a partially ordered set; this means that if a Galois correspondence is established between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043110/g04311013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043110/g04311014.png" />, the equalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043110/g04311015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043110/g04311016.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043110/g04311017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043110/g04311018.png" />, define closure operations (cf. [[Closure relation|Closure relation]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043110/g04311019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043110/g04311020.png" />, respectively. The concept of a Galois correspondence originated in [[Galois theory|Galois theory]], which deals with the Galois correspondence between all intermediate subfields of an extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043110/g04311021.png" /> and the system of subgroups of the Galois group of this extension.
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The concept of a Galois correspondence is closely connected with the concept of closure in a partially ordered set; this means that if a Galois correspondence is established between $M$ and $M'$, the equalities $\overline a=a\phi\psi$, $a\in M$, and $\overline{a'}=a'\psi\phi$, $a'\in M'$, define closure operations (cf. [[Closure relation|Closure relation]]) in $M$ and $M'$, respectively. The concept of a Galois correspondence originated in [[Galois theory|Galois theory]], which deals with the Galois correspondence between all intermediate subfields of an extension $P\subseteq K$ and the system of subgroups of the Galois group of this extension.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Kurosh,  "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Kurosh,  "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR></table>

Revision as of 23:02, 6 February 2012

between two partially ordered sets $M$ and $M'$

A pair of mappings $\phi\colon M\to M'$ and $\psi\colon M'\to M$ which satisfy the following conditions:

if $a\leq b$, then $a\phi\geq b\phi$;

if $a'\leq b'$, then $a'\psi\geq b'\psi$;

$a\phi\psi\geq a$ and $a'\psi\phi\geq a'$.

Here $a,b\in M$, $a', b' \in M'$.

The concept of a Galois correspondence is closely connected with the concept of closure in a partially ordered set; this means that if a Galois correspondence is established between $M$ and $M'$, the equalities $\overline a=a\phi\psi$, $a\in M$, and $\overline{a'}=a'\psi\phi$, $a'\in M'$, define closure operations (cf. Closure relation) in $M$ and $M'$, respectively. The concept of a Galois correspondence originated in Galois theory, which deals with the Galois correspondence between all intermediate subfields of an extension $P\subseteq K$ and the system of subgroups of the Galois group of this extension.

References

[1] P.M. Cohn, "Universal algebra" , Reidel (1981)
[2] A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)
How to Cite This Entry:
Galois correspondence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Galois_correspondence&oldid=20865
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article