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 and be two partially ordered sets. A monotone Galois connection between these posets consists of two monotoneMonotonicity 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 in , and for all in , . functions, and , such that for all in and in , we have
if and only if .
In this situation, is called the lower adjoint of and is called the upper adjoint of F. Mnemonically, the upper/lower terminology refers to where the function application appears relative to ≤. 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:
is the least element with , and is the largest element with .
A consequence of this is that if or is bijective then each is the inverse of the other, i.e. .
Given a Galois connection with lower adjoint and upper adjoint , we can consider the compositions , known as the associated closure operator, and , known as the associated kernel operator. Both are monotone and idempotent, and we have for all in and for all in .
A Galois insertion of into is a Galois connection in which the kernel operator is the identity on , and hence is an order isomorphism of onto the set of closed elements [] of .
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 and between two posets and , such that
if and only if .
The symmetry of and in this version erases the distinction between upper and lower, and the two functions are then called polarities rather than adjoints. Each polarity uniquely determines the other, since
is the largest element with , and is the largest element with .
The compositions and are the associated closure operators; they are monotone idempotent maps with the property for all in and for all in .
The implications of the two definitions of Galois connections are very similar, since an antitone Galois connection between and is just a monotone Galois connection between and the order dual of . 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 and 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 and partially ordered sets. Since if and only if we have a Galois connection.
Monotone Galois connections
Floor; ceiling A monotone Galois connection between the set of integers, and 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 denote the embedding function, with while denotes the floor function, so The equivalence then translates to
This is valid because the variable is restricted to the integers. The well-known properties of the floor function, such as can be derived by elementary reasoning from this Galois connection.
The dual orderings give another monotone Galois connection, now with the ceiling function:
Power set; implication and conjunction For an order-theoretic example, let be some set, and let and both be the power set of , ordered by inclusion. Pick a fixed subset of . Then the maps and , where , and , form a monotone Galois connection, with 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 and . In logical terms: "implication from " is the upper adjoint of "conjunction with ".
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 . The least and greatest elements of a partial order are given by lower and upper adjoints to the unique function 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 act transitively on and pick some point in . Consider
the set of blocks containing . Further, let consist of the subgroups of containing the stabilizer of .
Then, the correspondence :
is a monotone, one-to-one Galois connection.See Alperin, Bell, Groups and Representations (GTM 162), p. 32 As a corollary, one can establish that doubly transitive actions have no blocks other than the trivial ones (singletons or the whole of ): this follows from the stabilizers being maximal in in that case. See Doubly transitive group for further discussion.
Image and inverse image If is a function, then for any subset of we can form the image and for any subset of we can form the inverse image Then and form a monotone Galois connection between the power set of and the power set of , both ordered by inclusion ⊆. There is a further adjoint pair in this situation: for a subset of , define Then and form a monotone Galois connection between the power set of and the power set of . In the first Galois connection, 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 connect to subgroups of , and the closure operator on subgroups of is given by .
Span and closure Pick some mathematical object that has an underlying set, for instance a group, ring, vector space, etc. For any subset of , let be the smallest subobject of that contains , i.e. the subgroup, subring or subspace generated by . For any subobject of , let be the underlying set of . (We can even take to be a topological space, let the closure of , and take as "subobjects of the closed subsets of .) Now and form a monotone Galois connection between subsets of and subobjects of , if both are ordered by inclusion. is the lower adjoint.
Syntax and semantics A very general comment of William LawvereWilliam 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. is that syntax and semantics are adjoint: take to be the set of all logical theories (axiomatizations) reverse ordered by strength, and the power set of the set of all mathematical structures. For a theory , let be the set of all structures that satisfy the axioms ; for a set of mathematical structures , let be the minimum of the axiomatizations that approximate (in first-order logic, this is the set of sentences that are true in all structures in ). We can then say that is a subset of if and only if logically entails : the "semantics functor" and the "syntax functor" form a monotone Galois connection, with semantics being the upper adjoint.
Antitone Galois connections
Galois theory The motivating example comes from Galois theory: suppose is a field extension. Let be the set of all subfields of that contain , ordered by inclusion ⊆. If is such a subfield, write for the group of field automorphisms of that hold fixed. Let be the set of subgroups of , ordered by inclusion ⊆. For such a subgroup , define to be the field consisting of all elements of that are held fixed by all elements of . Then the maps and form an antitone Galois connection.
Algebraic topology: covering spaces Analogously, given a path-connected topological space , there is an antitone Galois connection between subgroups of the fundamental group and path-connected covering spaces of . In particular, if is semi-locally simply connected, then for every subgroup of , there is a covering space with as its fundamental group.
Linear algebra: annihilators and orthogonal complements Given an inner product space , we can form the orthogonal complement of any subspace of . This yields an antitone Galois connection between the set of subspaces of and itself, ordered by inclusion; both polarities are equal to .
Given a vector space and a subset of we can define its annihilator , consisting of all elements of the dual space of that vanish on . Similarly, given a subset of , we define its annihilator This gives an antitone Galois connection between the subsets of and the subsets of .
Algebraic geometry In algebraic geometry, the relation between sets of polynomials and their zero sets is an antitone Galois connection.
Fix a natural number and a field and let be the set of all subsets of the polynomial ring ordered by inclusion ⊆, and let be the set of all subsets of ordered by inclusion ⊆. If is a set of polynomials, define the variety of zeros as
the set of common zeros of the polynomials in . If is a subset of , define as the ideal of polynomials vanishing on , that is
Then and I form an antitone Galois connection.
The closure on is the closure in the Zariski topology, and if the field is algebraically closed, then the closure on the polynomial ring is the radical of ideal generated by .
More generally, given a commutative ring (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 .
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 and are arbitrary sets and a binary relation over and is given. For any subset of , we define Similarly, for any subset of , define Then and yield an antitone Galois connection between the power sets of and , both ordered by inclusion ⊆.