Galois connection

Template:Short description 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 Template:Math and Template:Math be two partially ordered sets. A monotone Galois connection between these posets consists of two monotone<ref>Monotonicity follows from the following condition. See the discussion of the properties. It is only explicit in the definition to distinguish it from the alternative antitone definition. One can also define Galois connections as a pair of monotone functions that satisfy the laxer condition that for all Template:Mvar in Template:Mvar, Template:Math and for all Template:Mvar in Template:Mvar, Template:Math.</ref> functions, Template:Math and Template:Math, such that for all Template:Mvar in Template:Mvar and Template:Mvar in Template:Mvar, we have

Template:Math if and only if Template:Math.

In this situation, Template:Mvar is called the lower adjoint of Template:Mvar and Template:Mvar is called the upper adjoint of F. Mnemonically, the upper/lower terminology refers to where the function application appears relative to ≤.Template:Sfn 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:

Template:Math is the least element Template:Math with Template:Math, and

Template:Math is the largest element Template:Mvar with Template:Math.

A consequence of this is that if Template:Mvar or Template:Mvar is bijective then each is the inverse of the other, i.e. Template:Math.

Given a Galois connection with lower adjoint Template:Mvar and upper adjoint Template:Mvar, we can consider the compositions Template:Math, known as the associated closure operator, and Template:Math, known as the associated kernel operator. Both are monotone and idempotent, and we have Template:Math for all Template:Mvar in Template:Mvar and Template:Math for all Template:Mvar in Template:Mvar.

A Galois insertion of Template:Mvar into Template:Mvar is a Galois connection in which the kernel operator Template:Mvar is the identity on Template:Mvar, and hence Template:Mvar is an order isomorphism of Template:Mvar onto the set of closed elements Template:MvarTemplate:Hairsp[[[:Template:Mvar]]] of Template:Mvar.<ref>Template:Cite book</ref>

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 Template:Math and Template:Math between two posets Template:Mvar and Template:Mvar, such that

Template:Math if and only if Template:Math.

The symmetry of Template:Mvar and Template:Mvar in this version erases the distinction between upper and lower, and the two functions are then called polarities rather than adjoints.Template:Sfn Each polarity uniquely determines the other, since

Template:Math is the largest element Template:Mvar with Template:Math, and

Template:Math is the largest element Template:Mvar with Template:Math.

The compositions Template:Math and Template:Math are the associated closure operators; they are monotone idempotent maps with the property Template:Math for all Template:Mvar in Template:Mvar and Template:Math for all Template:Mvar in Template:Mvar.

The implications of the two definitions of Galois connections are very similar, since an antitone Galois connection between Template:Mvar and Template:Mvar is just a monotone Galois connection between Template:Mvar and the order dual Template:Math of Template:Mvar. 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 <math>f:X\to Y</math> and <math>g:Y\to X,</math> 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 <math>(X,=)</math> and <math>(Y,=)</math> partially ordered sets. Since <math>f(x)=y</math> if and only if <math>x=g(y),</math> we have a Galois connection.

Monotone Galois connections

Floor; ceiling

A monotone Galois connection between <math>\Z,</math> the set of integers and <math>\R,</math> 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 <math>F:\Z\to\R</math> denote the embedding function, with <math>F(n)=n\in\R,</math> while <math>G:\R\to\Z</math> denotes the floor function, so <math>G(x)=\lfloor x\rfloor.</math> The equivalence <math>F(n)\leq x ~\Leftrightarrow~ n\leq G(x)</math> then translates to

<math>n\leq x ~\Leftrightarrow~ n\leq\lfloor x\rfloor.</math>

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

The dual orderings give another monotone Galois connection, now with the ceiling function:

<math>x\leq n ~\Leftrightarrow~ \lceil x\rceil\leq n.</math>

Power set; implication and conjunction

For an order-theoretic example, let Template:Mvar be some set, and let Template:Mvar and Template:Mvar both be the power set of Template:Mvar, ordered by inclusion. Pick a fixed subset Template:Mvar of Template:Mvar. Then the maps Template:Mvar and Template:Mvar, where Template:Math, and Template:Math, form a monotone Galois connection, with Template:Mvar 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 Template:Math and Template:Math. In logical terms: "implication from Template:Mvar" is the upper adjoint of "conjunction with Template:Mvar".

Lattices

Further interesting examples for Galois connections are described in the article on completeness properties. Roughly speaking, the usual functions ∨ and ∧ are lower and upper adjoints to the diagonal map Template:Math. The least and greatest elements of a partial order are given by lower and upper adjoints to the unique function Template:Math 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 Template:Mvar act transitively on Template:Mvar and pick some point Template:Mvar in Template:Mvar. Consider

<math>\mathcal{B} = \{B \subseteq X : x \in B; \forall g \in G, gB = B \ \mathrm{or} \ gB \cap B = \emptyset\},</math>

the set of blocks containing Template:Mvar. Further, let <math>\mathcal{G}</math> consist of the subgroups of Template:Mvar containing the stabilizer of Template:Mvar.

Then, the correspondence <math>\mathcal{B} \to \mathcal{G}</math>:

<math> B \mapsto H_B = \{g \in G : gx \in B\}</math>

is a monotone, one-to-one Galois connection.<ref>See Alperin, Bell, Groups and Representations (GTM 162), p. 32</ref> As a corollary, one can establish that doubly transitive actions have no blocks other than the trivial ones (singletons or the whole of Template:Mvar): this follows from the stabilizers being maximal in Template:Mvar in that case. See Doubly transitive group for further discussion.

Image and inverse image

If Template:Math is a function, then for any subset Template:Mvar of Template:Mvar we can form the image Template:Math and for any subset Template:Mvar of Template:Mvar we can form the inverse image Template:Math Then Template:Mvar and Template:Mvar form a monotone Galois connection between the power set of Template:Mvar and the power set of Template:Mvar, both ordered by inclusion ⊆. There is a further adjoint pair in this situation: for a subset Template:Mvar of Template:Mvar, define Template:Math Then Template:Mvar and Template:Mvar form a monotone Galois connection between the power set of Template:Mvar and the power set of Template:Mvar. In the first Galois connection, Template:Mvar 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 Template:Mvar connect to subgroups of Template:Math, and the closure operator on subgroups of Template:Mvar is given by Template:Math.

Span and closure

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

Syntax and semantics

A very general comment of William Lawvere<ref>William Lawvere, Adjointness in foundations, Dialectica, 1969, available here. The notation is different nowadays; an easier introduction by Peter Smith in these lecture notes, which also attribute the concept to the article cited.</ref> is that syntax and semantics are adjoint: take Template:Mvar to be the set of all logical theories (axiomatizations) reverse ordered by strength, and Template:Mvar the power set of the set of all mathematical structures. For a theory Template:Math, let Template:Math be the set of all structures that satisfy the axioms Template:MvarTemplate:Hairsp; for a set of mathematical structures Template:Math, let Template:Math be the minimum of the axiomatizations that approximate Template:Mvar (in first-order logic, this is the set of sentences that are true in all structures in Template:Mvar). We can then say that Template:Mvar is a subset of Template:Math if and only if Template:Math logically entails Template:Mvar: the "semantics functor" Template:Math and the "syntax functor" Template:Math form a monotone Galois connection, with semantics being the upper adjoint.

Antitone Galois connections

Galois theory

The motivating example comes from Galois theory: suppose Template:Math is a field extension. Let Template:Mvar be the set of all subfields of Template:Mvar that contain Template:Mvar, ordered by inclusion ⊆. If Template:Mvar is such a subfield, write Template:Math for the group of field automorphisms of Template:Mvar that hold Template:Mvar fixed. Let Template:Mvar be the set of subgroups of Template:Math, ordered by inclusion ⊆. For such a subgroup Template:Mvar, define Template:Math to be the field consisting of all elements of Template:Mvar that are held fixed by all elements of Template:Mvar. Then the maps Template:Math and Template:Math form an antitone Galois connection.

Algebraic topology: covering spaces

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

Linear algebra: annihilators and orthogonal complements

Given an inner product space Template:Mvar, we can form the orthogonal complement Template:Math of any subspace Template:Mvar of Template:Mvar. This yields an antitone Galois connection between the set of subspaces of Template:Mvar and itself, ordered by inclusion; both polarities are equal to Template:Mvar.

Given a vector space Template:Mvar and a subset Template:Mvar of Template:Mvar we can define its annihilator Template:Math, consisting of all elements of the dual space Template:Math of Template:Mvar that vanish on Template:Mvar. Similarly, given a subset Template:Mvar of Template:Math, we define its annihilator Template:Math This gives an antitone Galois connection between the subsets of Template:Mvar and the subsets of Template:Math.

Algebraic geometry

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

Fix a natural number Template:Mvar and a field Template:Mvar and let Template:Mvar be the set of all subsets of the polynomial ring Template:Math ordered by inclusion ⊆, and let Template:Mvar be the set of all subsets of Template:Math ordered by inclusion ⊆. If Template:Mvar is a set of polynomials, define the 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 zeros of the polynomials in Template:Mvar. If Template:Mvar is a subset of Template:Math, define Template:Math as the ideal of polynomials vanishing on Template:Mvar, that is

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

Then Template:Mvar and I form an antitone Galois connection.

The closure on Template:Math is the closure in the Zariski topology, and if the field Template:Mvar is algebraically closed, then the closure on the polynomial ring is the radical of ideal generated by Template:Mvar.

More generally, given a commutative ring Template:Mvar (not necessarily a polynomial ring), there is an antitone Galois connection between radical ideals in the ring and Zariski closed subsets of the affine variety Template:Math.

More generally, there is an antitone Galois connection between ideals in the ring and subschemes of the corresponding affine variety.

Connections on power sets arising from binary relations

Suppose Template:Mvar and Template:Mvar are arbitrary sets and a binary relation Template:Mvar over Template:Mvar and Template:Mvar is given. For any subset Template:Mvar of Template:Mvar, we define Template:Math Similarly, for any subset Template:Mvar of Template:Mvar, define Template:Math Then Template:Mvar and Template:Mvar yield an antitone Galois connection between the power sets of Template:Mvar and Template:Mvar, both ordered by inclusion ⊆.Template:Sfn

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), Template:ISBN</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), Template:ISBN</ref>

The 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">Template:Cite journal</ref>

Properties

In the following, we consider a (monotone) Galois connection Template:Math, where Template:Math is the lower adjoint as introduced above. Some helpful and instructive basic properties can be obtained immediately. By the defining property of Galois connections, Template:Math is equivalent to Template:Math, for all Template:Mvar in Template:Mvar. By a similar reasoning (or just by applying the duality principle for order theory), one finds that Template:Math, for all Template:Mvar in Template:Mvar. These properties can be described by saying the composite Template:Math is deflationary, while Template:Math is inflationary (or extensive).

Now consider Template:Math such that Template:Math. Then using the above one obtains Template:Math. Applying the basic property of Galois connections, one can now conclude that Template:Math. But this just shows that Template:Math preserves the order of any two elements, i.e. it is monotone. Again, a similar reasoning yields monotonicity of Template:Math. Thus monotonicity does not have to be included in the definition explicitly. However, mentioning monotonicity helps to avoid confusion about the two alternative notions of Galois connections.

Another basic property of Galois connections is the fact that Template:Math, for all Template:Mvar in Template:Mvar. Clearly we find that

Template:Math.

because Template:Math is inflationary as shown above. On the other hand, since Template:Math is deflationary, while Template:Math is monotonic, one finds that

Template:Math.

This shows the desired equality. Furthermore, we can use this property to conclude that

Template:Math

and

Template:Math

i.e., Template:Math and Template:Math are idempotent.

It can be shown (see Blyth or Erné for proofs) that a function Template:Math is a lower (respectively upper) adjoint if and only if Template:Math is a residuated mapping (respectively residual mapping). Therefore, the notion of residuated mapping and monotone Galois connection are essentially the same.

Closure operators and Galois connections

The above findings can be summarized as follows: for a Galois connection, the composite Template:Math is monotone (being the composite of monotone functions), inflationary, and idempotent. This states that Template:Math is in fact a closure operator on Template:Mvar. Dually, Template:Math is monotone, deflationary, and idempotent. Such mappings are sometimes called kernel operators. In the context of frames and locales, the composite Template:Math is called the nucleus induced by Template:Math. Nuclei induce frame homomorphisms; a subset of a locale is called a sublocale if it is given by a nucleus.

Conversely, any closure operator Template:Mvar on some poset Template:Mvar gives rise to the Galois connection with lower adjoint Template:Math being just the corestriction of Template:Mvar to the image of Template:Mvar (i.e. as a surjective mapping the closure system Template:Math). The upper adjoint Template:Math is then given by the inclusion of Template:Math into Template:Mvar, that maps each closed element to itself, considered as an element of Template:Mvar. In this way, closure operators and Galois connections are seen to be closely related, each specifying an instance of the other. Similar conclusions hold true for kernel operators.

The above considerations also show that closed elements of Template:Mvar (elements Template:Mvar with Template:Math) are mapped to elements within the range of the kernel operator Template:Math, and vice versa.

Existence and uniqueness of Galois connections

Another important property of Galois connections is that lower adjoints preserve all suprema that exist within their domain. Dually, upper adjoints preserve all existing infima. From these properties, one can also conclude monotonicity of the adjoints immediately. The adjoint functor theorem for order theory states that the converse implication is also valid in certain cases: especially, any mapping between complete lattices that preserves all suprema is the lower adjoint of a Galois connection.

In this situation, an important feature of Galois connections is that one adjoint uniquely determines the other. Hence one can strengthen the above statement to guarantee that any supremum-preserving map between complete lattices is the lower adjoint of a unique Galois connection. The main property to derive this uniqueness is the following: For every Template:Mvar in Template:Mvar, Template:Math is the least element Template:Mvar of Template:Mvar such that Template:Math. Dually, for every Template:Mvar in Template:Mvar, Template:Math is the greatest Template:Mvar in Template:Mvar such that Template:Math. The existence of a certain Galois connection now implies the existence of the respective least or greatest elements, no matter whether the corresponding posets satisfy any completeness properties. Thus, when one upper adjoint of a Galois connection is given, the other upper adjoint can be defined via this same property.

On the other hand, some monotone function Template:Math is a lower adjoint if and only if each set of the form Template:Math for Template:Mvar in Template:Mvar, contains a greatest element. Again, this can be dualized for the upper adjoint.

Galois connections as morphisms

Galois connections also provide an interesting class of mappings between posets which can be used to obtain categories of posets. Especially, it is possible to compose Galois connections: given Galois connections Template:Math between posets Template:Mvar and Template:Mvar and Template:Math between Template:Mvar and Template:Mvar, the composite Template:Math is also a Galois connection. When considering categories of complete lattices, this can be simplified to considering just mappings preserving all suprema (or, alternatively, infima). Mapping complete lattices to their duals, these categories display auto duality, that are quite fundamental for obtaining other duality theorems. More special kinds of morphisms that induce adjoint mappings in the other direction are the morphisms usually considered for frames (or locales).

Connection to category theory

Every partially ordered set can be viewed as a category in a natural way: there is a unique morphism from x to y if and only if Template:Math. A monotone Galois connection is then nothing but a pair of adjoint functors between two categories that arise from partially ordered sets. In this context, the upper adjoint is the right adjoint while the lower adjoint is the left adjoint. However, this terminology is avoided for Galois connections, since there was a time when posets were transformed into categories in a dual fashion, i.e. with morphisms pointing in the opposite direction. This led to a complementary notation concerning left and right adjoints, which today is ambiguous.

Applications in the theory of programming

Galois connections may be used to describe many forms of abstraction in the theory of abstract interpretation of programming languages.<ref>Template:Cite book
For a counterexample for the false theorem in Sect.7 (p.243 top right), see: Template:Cite tech report (However the original article only considers complete lattices)</ref><ref>Template:Cite book</ref>

References

The following books and survey articles include Galois connections using the monotone definition:

• Brian A. Davey and Hilary A. Priestley: Introduction to Lattices and Order, Cambridge University Press, 2002.
• Template:Cite book
• Marcel Erné, Jürgen Koslowski, Austin Melton, George E. Strecker, A primer on Galois connections, in: Proceedings of the 1991 Summer Conference on General Topology and Applications in Honor of Mary Ellen Rudin and Her Work, Annals of the New York Academy of Sciences, Vol. 704, 1993, pp. 103–125. (Freely available online in various file formats PS.GZ PS, it presents many examples and results, as well as notes on the different notations and definitions that arose in this area.)

Some publications using the original (antitone) definition:

• Template:Cite book
• Thomas Scott Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005, Template:Isbn.
• Template:Cite book
• Template:Cite book
• Template:Citation