View source for Galois correspondence

''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|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====
<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>