Galois connection

The Galoisverbindung is named after Évariste Galois. It refers to the following facts:

Definition: A Galoisverbindung between two partially ordered sets is a pair of pictures, if and are antitone and their compositions and are extensive. That is, it must have the following properties must be fulfilled:

Properties

A Galoisverbindung between and has the following properties:

• Symmetry is a Galoisverbindung between and.
• , By symmetry as well.
• Is directed to a closure operator, and thus is a shell on operator.

Application

Often, while power sets, and about. These are partially ordered by inclusion. Under a Galoisverbindung between the quantities and then one sees a Galoisverbindung between and. Such can be obtained with the aid of relations: Be a relation between and. The illustrations

,

Then provide a Galoisverbindung and forth between.

Examples

• For every natural number integer division by form, and the multiplication, ie, A Galoisverbindung between and.

• Between one body ( with lower body ) and the Galois group of is the following relation:

• Consider a vector space and a second vector space consisting of linear functionals of, that is, a subspace of the dual space. We define the relation on by

• In algebraic geometry, there is a Galois connection, eg between the affine algebraic sets in and the ideals in the polynomial ring, with an algebraically closed field called. It assigns each algebraic quantity the ideal of all polynomials that vanish on this set and assigns to each ideal that algebraic set to, is the common zero set of all polynomials in this ideal.

• In the Universal algebra, more precisely in the theory of equations, there is a Galoisverbindung between the equations and the classes of algebras. Here are algebras and terms of a fixed type. The Galoisverbindung Galoisverbindung is called the equation of the theory and differs from the original definition from the effect that not being operated only on quantity but on classes. It is a system of equations over the set of variables and a class of algebras:

• Algebra
• Order theory