Definition:Galois Connection
(Redirected from Definition: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.
Then $\struct {g, d}$ is a Galois connection if and only if:
- $g$ and $d$ are increasing mappings and
- $\forall s \in S, t \in T: t \precsim \map g s \iff \map d t \preceq s$
$g$ is upper adjoint and $d$ is lower adjoint of a Galois connection.
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Source of Name
This entry was named for Évariste Galois.
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