# Definition:Galois Connection

## Definition

Let $\struct {S, \preceq_S}$ and $\struct {T, \preceq_T}$ be ordered sets.

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

Then $\tuple {g, d}$ is a **Galois connection** if and only if:

- $g$ and $d$ are increasing mappings
- $\forall s \in S, t \in T: t \preceq_T \map g s \iff \map d t \preceq_S s$

### Upper Adjoint

- $g$ is called the
**upper adjoint**of the Galois connection.

### Lower Adjoint

- $d$ is called the
**lower adjoint**of the Galois connection.

## Also known as

In some sources the **upper adjoint** is called the **right adjoint** and the **lower adjoint** is called the **left adjoint**.

The terminology **right adjoint** and **left adjoint** is also used to denote the functors of an adjuntion in the context of category theory.

So the use of the terms **upper adjoint** and **lower adjoint** for a **Galois connection** serves to differentiate the context of order theory from the context of category theory.

The mappings are named for the position that each mapping has with respect to the orderings involved in the defining condition of **Galois connection**.

## Notation

A **Galois connection** is often denoted as $f = \struct{\upperadjoint f, \loweradjoint f}$ where $\upperadjoint f : S \to T$ denotes the **upper adjoint** and $\loweradjoint f : T \to S$ denotes the **lower adjoint**.

When $g : S \to T$ is known to be an **upper adjoint** of a **Galois connection**, the **lower adjoint** can be denoted as $\loweradjoint g : T \to S$.

Similarly, when $d : T \to S$ is known to be a **lower adjoint** of a **Galois connection**, the **upper adjoint** can be denoted as $\upperadjoint d : S \to T$.

## Also see

- Results about
**Galois connections**can be found**here**.

## Source of Name

This entry was named for Évariste Galois.

## Technical Notes for Lower Adjoint and Upper Adjoint

The $\LaTeX$ code for \(\loweradjoint f\) is `\loweradjoint f`

.

The $\LaTeX$ code for \(\upperadjoint f\) is `\upperadjoint f`

.

## Sources

- 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott:
*A Compendium of Continuous Lattices*

- Mizar article WAYBEL_1:def 10