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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Galois connection canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11099163 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Galois connection Context triple: [Évariste Galois, conceptNamedAfter, Galois connection]
-
A.
Curry–Howard correspondence
The Curry–Howard correspondence is a foundational principle in logic and computer science that establishes a deep analogy between proofs and programs, and between logical propositions and types in programming languages.
-
B.
Lattice Theory
Lattice Theory is a foundational mathematical text that systematically develops the theory of lattices and ordered structures, profoundly influencing modern algebra and order theory.
-
C.
Continuous Lattices
Continuous Lattices is a foundational work in domain theory and lattice theory that introduced a mathematical framework for modeling computation and denotational semantics.
-
D.
Birkhoff’s representation theorem for finite distributive lattices
Birkhoff’s representation theorem for finite distributive lattices is a fundamental result in lattice theory that characterizes every finite distributive lattice as isomorphic to the lattice of lower (order) ideals of a finite poset.
-
E.
Tarski’s fixed point theorem
Tarski’s fixed point theorem is a fundamental result in order theory and lattice theory that guarantees the existence of fixed points for monotone functions on complete lattices, with wide applications in logic, computer science, and economics.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Galois connection Target entity description: 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.
-
A.
Curry–Howard correspondence
The Curry–Howard correspondence is a foundational principle in logic and computer science that establishes a deep analogy between proofs and programs, and between logical propositions and types in programming languages.
-
B.
Lattice Theory
Lattice Theory is a foundational mathematical text that systematically develops the theory of lattices and ordered structures, profoundly influencing modern algebra and order theory.
-
C.
Continuous Lattices
Continuous Lattices is a foundational work in domain theory and lattice theory that introduced a mathematical framework for modeling computation and denotational semantics.
-
D.
Birkhoff’s representation theorem for finite distributive lattices
Birkhoff’s representation theorem for finite distributive lattices is a fundamental result in lattice theory that characterizes every finite distributive lattice as isomorphic to the lattice of lower (order) ideals of a finite poset.
-
E.
Tarski’s fixed point theorem
Tarski’s fixed point theorem is a fundamental result in order theory and lattice theory that guarantees the existence of fixed points for monotone functions on complete lattices, with wide applications in logic, computer science, and economics.
- F. None of above. chosen
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
ⓘ
linked to:
Lattice Theory
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Galois connection Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Galois