Editing Galois connection (section)

===Antitone Galois connections===

====Galois theory====
The motivating example comes from Galois theory: suppose {{math|''L''/''K''}} is a [[field extension]]. Let {{mvar|A}} be the set of all subfields of {{mvar|L}} that contain {{mvar|K}}, ordered by inclusion ⊆. If {{mvar|E}} is  such a subfield, write {{math|Gal(''L''/''E'')}} for the group of [[field automorphism]]s of {{mvar|L}} that hold {{mvar|E}} fixed. Let {{mvar|B}} be the set of subgroups of {{math|Gal(''L''/''K'')}}, ordered by inclusion ⊆. For such a subgroup {{mvar|G}}, define {{math|Fix(''G'')}} to be the field consisting of all elements of {{mvar|L}} that are held fixed by all elements of {{mvar|G}}. Then the maps {{math|''E'' {{mapsto}} Gal(''L''/''E'')}} and {{math|''G'' {{mapsto}} Fix(''G'')}} form an antitone Galois connection.

====Algebraic topology: covering spaces====
Analogously, given a [[path-connected]] [[topological space]] {{mvar|X}}, there is an antitone Galois connection between subgroups of the [[fundamental group]] {{math|''π''<sub>1</sub>(''X'')}} and path-connected [[covering space]]s of {{mvar|X}}. In particular, if {{mvar|X}} is [[semi-locally simply connected]], then for every subgroup {{mvar|G}} of {{math|''π''<sub>1</sub>(''X'')}}, there is a covering space with {{mvar|G}} as its fundamental group.

====Linear algebra: annihilators and orthogonal complements====
Given an [[inner product space]] {{mvar|V}}, we can form the [[orthogonal complement]] {{math|''F''(''X''&hairsp;)}} of any subspace {{mvar|X}} of {{mvar|V}}. This yields an antitone Galois connection between the set of subspaces of {{mvar|V}} and itself, ordered by inclusion; both polarities are equal to {{mvar|F}}.

Given a [[vector space]] {{mvar|V}} and a subset {{mvar|X}} of {{mvar|V}} we can define its annihilator {{math|''F''(''X''&hairsp;)}}, consisting of all elements of the [[dual space]] {{math|''V''&hairsp;<sup>∗</sup>}} of {{mvar|V}} that vanish on {{mvar|X}}. Similarly, given a subset {{mvar|Y}} of {{math|''V''&hairsp;<sup>∗</sup>}}, we define its annihilator {{math|''G''(''Y''&thinsp;) {{=}} {&hairsp;''x'' ∈ ''V'' {{!}} ''φ''(''x'') {{=}} 0 ∀''φ'' ∈ ''Y''&hairsp;}.}} This gives an antitone Galois connection between the subsets of {{mvar|V}} and the subsets of {{math|''V''&hairsp;<sup>∗</sup>}}.

====Algebraic geometry====
In [[algebraic geometry]], the relation between sets of [[polynomial]]s and their zero sets is an antitone Galois connection.

Fix a [[natural number]] {{mvar|n}} and a [[field (mathematics)|field]] {{mvar|K}} and let {{mvar|A}} be the set of all subsets of the [[polynomial ring]] {{math|''K''[''X''<sub>1</sub>, ..., ''X<sub>n</sub>'']}} ordered by inclusion ⊆, and let {{mvar|B}} be the set of all subsets of {{math|''K''<sup>&hairsp;''n''</sup>}} ordered by inclusion ⊆. If {{mvar|S}} is a set of polynomials, define the [[Algebraic geometry#Affine varieties|variety]] of zeros as

:<math>V(S) = \{x \in K^n : f(x) = 0 \mbox{ for all } f \in S\},</math>

the set of common [[root of a polynomial|zeros]] of the polynomials in {{mvar|S}}. If {{mvar|U}} is a subset of {{math|''K''<sup>&hairsp;''n''</sup>}}, define {{math|''I''(''U''&hairsp;)}} as the [[ideal (ring theory)|ideal]] of polynomials vanishing on {{mvar|U}}, that is

:<math>I(U) = \{f \in K[X_1,\dots,X_n] : f(x) = 0 \mbox{ for all } x \in U\}.</math>

Then {{mvar|V}} and ''I'' form an antitone Galois connection.

The closure on {{math|''K''<sup>&hairsp;''n''</sup>}} is the closure in the [[Zariski topology]], and if the field {{mvar|K}} is [[Algebraically closed field|algebraically closed]], then the closure on the polynomial ring is the [[Radical of an ideal|radical]] of ideal generated by {{mvar|S}}.

More generally, given a [[commutative ring]] {{mvar|R}} (not necessarily a polynomial ring), there is an antitone Galois connection between radical ideals in the ring and Zariski closed subsets of the [[Algebraic geometry#Affine varieties|affine variety]] {{math|[[Spectrum of a ring|Spec]](''R'')}}.

More generally, there is an antitone Galois connection between ideals in the ring and [[subscheme]]s of the corresponding [[Algebraic geometry#Affine varieties|affine variety]].

====Connections on power sets arising from binary relations====
Suppose {{mvar|X}} and {{mvar|Y}} are arbitrary sets and a [[binary relation]] {{mvar|R}} over {{mvar|X}} and {{mvar|Y}} is given. For any subset {{mvar|M}} of {{mvar|X}}, we define {{math|''F''(''M''&hairsp;) {{=}} {&hairsp;''y'' ∈ ''Y'' {{!}} ''mRy'' ∀''m'' ∈ ''M''&hairsp;}.}} Similarly, for any subset {{mvar|N}} of {{mvar|Y}}, define {{math|''G''(''N''&hairsp;) {{=}} {&hairsp;''x'' ∈ ''X'' {{!}} ''xRn'' ∀''n'' ∈ ''N''&hairsp;}.}} Then {{mvar|F}} and {{mvar|G}} yield an antitone Galois connection between the power sets of {{mvar|X}} and {{mvar|Y}}, both ordered by inclusion ⊆.{{sfn|Birkhoff|1940|loc=§32; 3rd edition (1967): Ch. V, §7 and §8}}

Up to isomorphism ''all'' antitone Galois connections between power sets arise in this way. This follows from the "Basic Theorem on Concept Lattices".<ref>Ganter, B. and Wille, R. ''Formal Concept Analysis -- Mathematical Foundations'', Springer (1999), {{ISBN|978-3-540-627715}}</ref> Theory and applications of Galois connections arising from binary relations are studied in [[formal concept analysis]]. That field uses Galois connections for mathematical data analysis.  Many algorithms for Galois connections can be found in the respective literature, e.g., in.<ref>Ganter, B. and Obiedkov, S. ''Conceptual Exploration'', Springer (2016), {{ISBN|978-3-662-49290-1}}</ref>

The [[General Concept Lattice|general concept lattice]] in its primitive version incorporates both the monotone and antitone Galois connections to furnish its upper and lower bounds of nodes for the concept lattice, respectively.<ref name=":1">{{Cite journal |last1=Liaw |first1=Tsong-Ming |last2=Lin |first2=Simon C. |date=2020-10-12 |title=A general theory of concept lattice with tractable implication exploration |url=https://www.sciencedirect.com/science/article/pii/S0304397520302826 |url-status=live |journal=Theoretical Computer Science |language=en |volume=837 |pages=84–114 |doi=10.1016/j.tcs.2020.05.014 |issn=0304-3975 |s2cid=219514253 |archive-url=https://web.archive.org/web/20200528022615/https://www.sciencedirect.com/science/article/pii/S0304397520302826 |archive-date=2020-05-28 |access-date=2023-07-19}}</ref>