Namespaces
Variants
Actions

Difference between revisions of "Galois correspondence"

From Encyclopedia of Mathematics
Jump to: navigation, search
(TeX)
(links)
 
(2 intermediate revisions by one other user not shown)
Line 1: Line 1:
''between two partially ordered sets $M$ and $M'$''
+
{{MSC|08A|12F10}}
 +
{{TEX|done}}
  
A pair of mappings $\phi\colon M\to M'$ and $\psi\colon M'\to M$ which satisfy the following conditions:
+
 
 +
A Galois correspondence is a pair of [[antitone mapping]]s $\phi\colon M\to M'$ and $\psi\colon M'\to M$ between two [[partially ordered set]]s $M$ and $M'$, which satisfy the following conditions:
  
 
if $a\leq b$, then $a\phi\geq b\phi$;
 
if $a\leq b$, then $a\phi\geq b\phi$;
Line 11: Line 13:
 
Here $a,b\in M$, $a', b' \in M'$.
 
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|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.
+
The concept of a Galois correspondence is closely connected with the concept of [[Closure relation|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>
+
{|
 +
|-
 +
|valign="top"|[1]||valign="top"| P.M. Cohn,  "Universal algebra", Reidel  (1981) {{MR|0620952}} {{ZBL|0461.08001}}
 +
|-
 +
|valign="top"|[2]||valign="top"| A.G. Kurosh,  "Lectures on general algebra", Chelsea  (1963)  (Translated from Russian) {{MR|0158000}} {{ZBL|0121.25901}}
 +
|-
 +
|}

Latest revision as of 20:00, 23 December 2016

2020 Mathematics Subject Classification: Primary: 08A Secondary: 12F10 [MSN][ZBL]


A Galois correspondence is a pair of antitone mappings $\phi\colon M\to M'$ and $\psi\colon M'\to M$ between two partially ordered sets $M$ and $M'$, 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) MR0620952 Zbl 0461.08001
[2] A.G. Kurosh, "Lectures on general algebra", Chelsea (1963) (Translated from Russian) MR0158000 Zbl 0121.25901
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