In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fundamental theorem of Galois theory about the correspondence between subgroups and subfields, discovered by the French mathematician Évariste Galois.

A Galois connection can also be defined on preordered sets or classes; this article presents the common case of posets. The literature contains two closely related notions of "Galois connection". In this article, we will refer to them as (monotone) Galois connections and antitone Galois connections.

A Galois connection is rather weak compared to an order isomorphism between the involved posets, but every Galois connection gives rise to an isomorphism of certain sub-posets, as will be explained below. The term Galois correspondence is sometimes used to mean a bijective Galois connection; this is simply an order isomorphism (or dual order isomorphism, depending on whether we take monotone or antitone Galois connections).

Definitions

(Monotone) Galois connection

Let (A, ≤) and (B, ≤) be two partially ordered sets. A monotone Galois connection between these posets consists of two monotone^[1] functions: F : A → B and G : B → A, such that for all a in A and b in B, we have

F(a) ≤ b if and only if a ≤ G(b).

In this situation, F is called the lower adjoint of G and G is called the upper adjoint of F. Mnemonically, the upper/lower terminology refers to where the function application appears relative to ≤.^[2] The term "adjoint" refers to the fact that monotone Galois connections are special cases of pairs of adjoint functors in category theory as discussed further below. Other terminology encountered here is left adjoint (respectively right adjoint) for the lower (respectively upper) adjoint.

An essential property of a Galois connection is that an upper/lower adjoint of a Galois connection uniquely determines the other:

F(a) is the least element ~b with a ≤ G(~b), and

G(b) is the largest element ~a with F(~a) ≤ b.

A consequence of this is that if F or G is bijective then each is the inverse of the other, i.e. F = G ⁻¹.

Given a Galois connection with lower adjoint F and upper adjoint G, we can consider the compositions GF : A → A, known as the associated closure operator, and FG : B → B, known as the associated kernel operator. Both are monotone and idempotent, and we have a ≤ GF(a) for all a in A and FG(b) ≤ b for all b in B.

A Galois insertion of B into A is a Galois connection in which the kernel operator FG is the identity on B, and hence G is an order isomorphism of B onto the set of closed elements GF  of A.^[3]

Antitone Galois connection

The above definition is common in many applications today, and prominent in lattice and domain theory. However the original notion in Galois theory is slightly different. In this alternative definition, a Galois connection is a pair of antitone, i.e. order-reversing, functions F : A → B and G : B → A between two posets A and B, such that

b ≤ F(a) if and only if a ≤ G(b).

The symmetry of F and G in this version erases the distinction between upper and lower, and the two functions are then called polarities rather than adjoints.^[4] Each polarity uniquely determines the other, since

F(a) is the largest element b with a ≤ G(b), and

G(b) is the largest element a with b ≤ F(a).

The compositions GF : A → A and FG : B → B are the associated closure operators; they are monotone idempotent maps with the property a ≤ GF(a) for all a in A and b ≤ FG(b) for all b in B.

The implications of the two definitions of Galois connections are very similar, since an antitone Galois connection between A and B is just a monotone Galois connection between A and the order dual B^op of B. All of the below statements on Galois connections can thus easily be converted into statements about antitone Galois connections.

Examples

Bijections

The bijection of a pair of functions ${\displaystyle f:X\to Y}$ and ${\displaystyle g:Y\to X,}$ each other's inverse, forms a (trivial) Galois connection, as follows. Because the equality relation is reflexive, transitive and antisymmetric, it is, trivially, a partial order, making ${\displaystyle (X,=)}$ and ${\displaystyle (Y,=)}$ partially ordered sets. Since ${\displaystyle f(x)=y}$ if and only if ${\displaystyle x=g(y),}$ we have a Galois connection.

Monotone Galois connections

Floor; ceiling

A monotone Galois connection between ${\displaystyle \mathbb {Z} ,}$ the set of integers and ${\displaystyle \mathbb {R} ,}$ the set of real numbers, each with its usual ordering, is given by the usual embedding function of the integers into the reals and the floor function truncating a real number to the greatest integer less than or equal to it. The embedding of integers is customarily done implicitly, but to show the Galois connection we make it explicit. So let ${\displaystyle F:\mathbb {Z} \to \mathbb {R} }$ denote the embedding function, with ${\displaystyle F(n)=n\in \mathbb {R} ,}$ while ${\displaystyle G:\mathbb {R} \to \mathbb {Z} }$ denotes the floor function, so ${\displaystyle G(x)=\lfloor x\rfloor .}$ The equivalence ${\displaystyle F(n)\leq x~\Leftrightarrow ~n\leq G(x)}$ then translates to

${\displaystyle n\leq x~\Leftrightarrow ~n\leq \lfloor x\rfloor .}$

This is valid because the variable ${\displaystyle n}$ is restricted to the integers. The well-known properties of the floor function, such as ${\displaystyle \lfloor x+n\rfloor =\lfloor x\rfloor +n,}$ can be derived by elementary reasoning from this Galois connection.

The dual orderings give an antitone Galois connection, now with the ceiling function:

${\displaystyle x\leq n~\Leftrightarrow ~\lceil x\rceil \leq n.}$

Power set; implication and conjunction

For an order-theoretic example, let U be some set, and let A and B both be the power set of U, ordered by inclusion. Pick a fixed subset L of U. Then the maps F and G, where F(M ) = L ∩ M, and G(N ) = N ∪ (U \ L), form a monotone Galois connection, with F being the lower adjoint. A similar Galois connection whose lower adjoint is given by the meet (infimum) operation can be found in any Heyting algebra. Especially, it is present in any Boolean algebra, where the two mappings can be described by F(x) = (a ∧ x) and G( y) = ( y ∨ ¬a) = (a ⇒ y). In logical terms: "implication from a" is the upper adjoint of "conjunction with a".

Lattices

Further interesting examples for Galois connections are described in the article on completeness properties. Roughly speaking, it turns out that the usual functions ∨ and ∧ are lower and upper adjoints to the diagonal map X → X × X. The least and greatest elements of a partial order are given by lower and upper adjoints to the unique function X → {1}. Going further, even complete lattices can be characterized by the existence of suitable adjoints. These considerations give some impression of the ubiquity of Galois connections in order theory.

Transitive group actions

Let G act transitively on X and pick some point x in X. Consider

${\displaystyle {\mathcal {B}}=\{B\subseteq X:x\in B;\forall g\in G,gB=B\ \mathrm {or} \ gB\cap B=\emptyset \},}$

the set of blocks containing x. Further, let ${\displaystyle {\mathcal {G}}}$ consist of the subgroups of G containing the stabilizer of x.

Then, the correspondence ${\displaystyle {\mathcal {B}}\to {\mathcal {G}}}$:

${\displaystyle B\mapsto H_{B}=\{g\in G:gx\in B\}}$

is a monotone, one-to-one Galois connection.^[5] As a corollary, one can establish that doubly transitive actions have no blocks other than the trivial ones (singletons or the whole of X): this follows from the stabilizers being maximal in G in that case. See Doubly transitive group for further discussion.

Image and inverse image

If  f : X → Y is a function, then for any subset M of X we can form the image F(M ) =  f M = { f (m) | m ∈ M} and for any subset N of Y we can form the inverse image G(N ) =  f ⁻¹N = {x ∈ X |  f (x) ∈ N}. Then F and G form a monotone Galois connection between the power set of X and the power set of Y, both ordered by inclusion ⊆. There is a further adjoint pair in this situation: for a subset M of X, define H(M) = {y ∈ Y |  f ⁻¹{y} ⊆ M}. Then G and H form a monotone Galois connection between the power set of Y and the power set of X. In the first Galois connection, G is the upper adjoint, while in the second Galois connection it serves as the lower adjoint.

In the case of a quotient map between algebraic objects (such as groups), this connection is called the lattice theorem: subgroups of G connect to subgroups of G/N, and the closure operator on subgroups of G is given by H = HN.

Span and closure

Pick some mathematical object X that has an underlying set, for instance a group, ring, vector space, etc. For any subset S of X, let F(S ) be the smallest subobject of X that contains S, i.e. the subgroup, subring or subspace generated by S. For any subobject U of X, let G(U ) be the underlying set of U. (We can even take X to be a topological space, let F(S ) the closure of S, and take as "subobjects of X " the closed subsets of X.) Now F and G form a monotone Galois connection between subsets of X and subobjects of X, if both are ordered by inclusion. F is the lower adjoint.

Syntax and semantics

A very general comment of William Lawvere^[6] is that syntax and semantics are adjoint: take A to be the set of all logical theories (axiomatizations) reverse ordered by strength, and B the power set of the set of all mathematical structures. For a theory T ∈ A, let Mod(T ) be the set of all structures that satisfy the axioms T ; for a set of mathematical structures S ∈ B, let Th(S ) be the minimum of the axiomatizations that approximate S (in first-order logic, this is the set of sentences that are true in all structures in S). We can then say that S is a subset of Mod(T ) if and only if Th(S ) logically entails T: the "semantics functor" Mod and the "syntax functor" Th form a monotone Galois connection, with semantics being the upper adjoint.

Antitone Galois connections

Galois theory

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

Algebraic topology: covering spaces

Analogously, given a path-connected topological space X, there is an antitone Galois connection between subgroups of the fundamental group π₁(X) and path-connected covering spaces of X. In particular, if X is semi-locally simply connected, then for every subgroup G of π₁(X), there is a covering space with G as its fundamental group.

Linear algebra: annihilators and orthogonal complements

Given an inner product space V, we can form the orthogonal complement F(X ) of any subspace X of V. This yields an antitone Galois connection between the set of subspaces of V and itself, ordered by inclusion; both polarities are equal to F.

Given a vector space V and a subset X of V we can define its annihilator F(X ), consisting of all elements of the dual space V ^∗ of V that vanish on X. Similarly, given a subset Y of V ^∗, we define its annihilator G(Y ) = { x ∈ V | φ(x) = 0 ∀φ ∈ Y }. This gives an antitone Galois connection between the subsets of V and the subsets of V ^∗.

Algebraic geometry

In algebraic geometry, the relation between sets of polynomials and their zero sets is an antitone Galois connection.

Fix a natural number n and a field K and let A be the set of all subsets of the polynomial ring K ordered by inclusion ⊆, and let B be the set of all subsets of Kⁿ ordered by inclusion ⊆. If S is a set of polynomials, define the variety of zeros as

${\displaystyle V(S)=\{x\in K^{n}:f(x)=0{\mbox{ for all }}f\in S\},}$

the set of common zeros of the polynomials in S. If U is a subset of Kⁿ, define I(U ) as the ideal of polynomials vanishing on U, that is

${\displaystyle I(U)=\{f\in <!--\; \in \; -->K:f(x)=0\text{ for all }}$Zdroj:https://en.wikipedia.org?pojem=Galois_connection
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