Definition:Galois Connection/Lower 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:
- $d$ is called the lower 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 \(\loweradjoint f\) is \loweradjoint 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