Definition:Galois Connection
Definition
Let $\struct {S, \preceq}$ and $\struct {T, \precsim}$ be ordered sets.
Let $g: S \to T$, $d: T \to S$ be mappings.
Then $\tuple {g, d}$ is a Galois connection if and only if:
- $g$ and $d$ are increasing mappings and
- $\forall s \in S, t \in T: t \precsim \map g s \iff \map d t \preceq s$
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