# Galois correspondence

From Encyclopedia of Mathematics

*between two partially ordered sets and *

A pair of mappings and which satisfy the following conditions:

if , then ;

if , then ;

and .

Here , .

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 and , the equalities , , and , , define closure operations (cf. Closure relation) in and , respectively. The concept of a Galois correspondence originated in Galois theory, which deals with the Galois correspondence between all intermediate subfields of an extension 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=12061

This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article