# Definition:Galois Connection

## Definition

Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be ordered sets.

Let $g: S \to T$, $d: T \to S$ be mappings.

Then $\left({g, d}\right)$ is Galois connection if and only if:

$g$ and $d$ are increasing mappings and
$\forall s \in S, t \in T: t \precsim g\left({s}\right) \iff d\left({t}\right) \preceq s$

$g$ is upper adjoint and $d$ is lower adjoint of a Galois connection.

## Source of Name

This entry was named for Évariste Galois.