Galois connection
E904575
A Galois connection is a pair of order-reversing (or order-preserving) maps between partially ordered sets that form an adjoint relationship, linking their structures in a way that generalizes many dualities in mathematics.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
adjunction
ⓘ
mathematical concept ⓘ order-theoretic notion ⓘ |
| alsoKnownAs | Galois correspondence ⓘ |
| appearsIn |
lattice-theoretic treatments of algebra
ⓘ
theory of complete lattices ⓘ |
| captures |
correspondence between closure systems
ⓘ
duality between substructures ⓘ |
| characterizedBy |
equivalence of certain order relations
ⓘ
inequalities involving compositions of the maps ⓘ |
| definedOn | partially ordered sets ⓘ |
| field |
category theory
ⓘ
lattice theory ⓘ order theory ⓘ |
| formalizedAs | adjoint functor pair between posets viewed as categories ⓘ |
| generalizes |
adjunctions in category theory
ⓘ
classical Galois correspondence in field theory ⓘ closure operators ⓘ duality between subgroups and intermediate fields ⓘ kernel-image correspondences ⓘ polarity in formal concept analysis ⓘ |
| hasComponent |
lower adjoint
ⓘ
upper adjoint ⓘ |
| hasCondition | for all a and b, f(a) ≤ b iff a ≤ g(b) ⓘ |
| hasDirection |
order-preserving
ⓘ
order-reversing ⓘ |
| hasDual | dual Galois connection obtained by order reversal ⓘ |
| hasHistoricalOrigin | Évariste Galois's work on field extensions ⓘ |
| implies |
existence of closure operator on one poset
ⓘ
existence of kernel operator on the other poset ⓘ |
| involves |
adjoint pair of functions
ⓘ
pair of monotone maps ⓘ |
| isSpecialCaseOf | adjunction in a 2-category of posets ⓘ |
| property |
adjoint relationship between posets
ⓘ
order-reversing or order-preserving structure ⓘ |
| relates |
structure of one poset to another
ⓘ
two partially ordered sets ⓘ |
| studiedIn |
Birkhoff's lattice theory
NERFINISHED
ⓘ
Tarski's work on closure operators ⓘ |
| usedIn |
abstract algebra
ⓘ
abstract interpretation ⓘ computer science ⓘ formal concept analysis ⓘ logic ⓘ program analysis ⓘ topology ⓘ |
| yields |
closure operators on posets
ⓘ
interior operators on posets ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Évariste Galois