Definition:Galois Connection/Upper Adjoint

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Definition

Let $\struct {S, \preceq}$ and $\struct {T, \precsim}$ be ordered sets.

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

Let $\tuple {g, d}$ be a Galois connection.


Then:

$g$ is called the upper adjoint of the 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


Technical Note

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


Sources


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