A Galois connection can also be defined on preordered sets or classes; this article presents the common case of posets.
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 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 ≤.
[3] The above definition is common in many applications today, and prominent in lattice and domain theory.
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 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.
Because the equality relation is reflexive, transitive and antisymmetric, it is, trivially, a partial order, making
The embedding of integers is customarily done implicitly, but to show the Galois connection we make it explicit.
A similar Galois connection whose lower adjoint is given by the meet (infimum) operation can be found in any Heyting algebra.
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}.
These considerations give some impression of the ubiquity of Galois connections in order theory.
[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.
Pick some mathematical object X that has an underlying set, for instance a group, ring, vector space, etc.
Now F and G form a monotone Galois connection between subsets of X and subobjects of X, if both are 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.
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.
More generally, there is an antitone Galois connection between ideals in the ring and subschemes of the corresponding affine variety.
Then F and G yield an antitone Galois connection between the power sets of X and Y, both ordered by inclusion ⊆.
[8] Theory and applications of Galois connections arising from binary relations are studied in formal concept analysis.
By a similar reasoning (or just by applying the duality principle for order theory), one finds that f ∗( f∗(y)) ≤ y, for all y in B.
However, mentioning monotonicity helps to avoid confusion about the two alternative notions of Galois connections.
This states that f∗∘ f ∗ is in fact a closure operator on A. Dually, f ∗∘ f∗ is monotone, deflationary, and idempotent.
Nuclei induce frame homomorphisms; a subset of a locale is called a sublocale if it is given by a nucleus.
Another important property of Galois connections is that lower adjoints preserve all suprema that exist within their domain.
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.
On the other hand, some monotone function f is a lower adjoint if and only if each set of the form { x ∈ A | f (x) ≤ b }, for b in B, contains a greatest element.
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 x ≤ y.
A monotone Galois connection is then nothing but a pair of adjoint functors between two categories that arise from partially ordered sets.
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.