Galois correspondence

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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 Template:ZBL
[2]	A.G. Kurosh, "Lectures on general algebra", Chelsea (1963) (Translated from Russian) MR0158000 Template:ZBL