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 monotone 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 (resp. right adjoint) for the lower (resp. 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 invertible, 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
Monotone Galois connections
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 ".