Difference between revisions of "Galois correspondence"
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| − | + | {{MSC|08A|12F10}} | |
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| − | A pair of | + | |
| + | 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 $a\phi\geq b\phi$; | ||
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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==== | ||
| − | + | {| | |
| + | |- | ||
| + | |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 |
Galois correspondence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Galois_correspondence&oldid=20865