In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets (posets). Galois connections generalize the correspondence between subgroups and subfields investigated in Galois theory. They find applications in various mathematical theories.

A Galois connection is rather weaker than an isomorphism between the involved posets, but every Galois connection gives rise to an isomorphism of certain sub-posets, as will be explained below.

Like Galois theory, Galois connections are named after the French mathematician Évariste Galois.

Definition

Let (A, ≤) and (B, ≤) be two partially ordered sets. A Galois connection between these posets consists of two monotone 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. This terminology relates the Galois connections to category theory discussed below. As detailed below, each part of a Galois connection uniquely determines the other mapping. Viewing two functions that form a Galois connection as two specifications of the same object, it is convenient to denote a pair of corresponding lower and upper adjoints by f ^∗ and f _∗, respectively. Note that the asterisk is placed above the function symbol to denote the lower adjoint.

Alternative definition

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) . (Note: This is a correction of an earlier definition.)

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.

Both notions of a Galois connection are still present in the literature. In this article the term (monotone) Galois connection will always refer to a Galois connection in the former sense. If the alternative definition is applied, the term antitone Galois connection or order-reversing Galois connection is used.

The implications of both definitions are in fact 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.

Note however that for an antitone Galois connection, it does not make sense to talk about the lower and upper adjoint: the situation is completely symmetrical.

Examples

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 $subseteq$. 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 $subseteq$. 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 $mapsto$ Gal(L /E) and G $mapsto$ Fix(G) form an antitone Galois connection.

Algebraic topology: covering spaces

Analogously, given a path-connected topological space $X$, there is a Galois connection between subgroups of the fundamental group $pi\_1(X)$ and path-connected covering spaces of $X$. In particular, if X is semi-locally simply connected, then for every subgroup $G$ of $pi\_1(X)$, there is a covering space with $G$ as its fundamental group.

Order theory

Power set

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) is the intersection of L and M, and G(N) is the union of N and (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 $wedge$ x) and G(y) = (y $vee$ $neg$ a) = (a $Rightarrow$ y). In logical terms: "implication" is the upper adjoint of "conjunction".

Lattices

Further interesting examples for Galois connections are described in the article on completeness properties. It turns out that the usual functions $vee$ and $wedge$ are adjoints in two suitable Galois connections. The same is true for the mappings from the one element set that point out the least and greatest elements of a partial order. 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.

Binary relations and annihilators

Suppose X and Y are arbitrary sets and a binary relation R over X and Y is given. For any subset M of X, we define F(M) = { y$in$Y : mRy for all m$in$M}. Similarly, for any subset N of Y, define G(N) = { x$in$X : xRn for all n$in$N}. Then F and G yield an antitone Galois connection between the power sets of X and Y, both ordered by inclusion $subseteq$.

An important special case in linear algebra is the annihilator, which includes the orthogonal complement as a special case.

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[X₁,...,X_n] ordered by inclusion $subseteq$, and let B be the set of all subsets of Kⁿ ordered by inclusion $subseteq$. If S is a set of polynomials, define the variety of zeros as

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

the set of common zeros of the polynomials in S.

If U is a subset of Kⁿ, define the radical ideal of polynomials vanishing on U as

$I(U)\; =\; \{\; f\; in\; K[X\_1,dots,X\_n]\; :\; f(x)\; =\; 0\; mbox\{\; for\; all\; \}\; x\; in\; U\}.$

Then V and I form an antitone Galois connection.

The closure on the polynomial ring is "radical ideal generated by U", while the closure on $K^n$ is the closure in the Zariski topology.

More generally, given a ring R (not necessarily a polynomial ring), there is an antitone Galois connection between radical ideals in the ring and subvarieties of the affine variety $operatorname\{Spec\},R$ (namely Spec of the ring).

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

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$in$M} and for any subset N of Y we can form the inverse image G(N) = f ^-1(N) = {x$in$X : f(x)$in$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 $subseteq$. There is a further adjoint pair in this situation: for a subset M of X, define H(M) = {y$in$Y : f ^-1({y}) $subseteq$ 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 $bar\; H\; =\; HN$.

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 if the sets and subobjects are ordered by inclusion. F is the lower adjoint.

Syntax and semantics

A very general comment of Martin Hyland is that syntax and semantics are adjoint: take A to be the set of all logical theories (axiomatizations), and B the power set of the set of all mathematical structures. For a theory T$in$A, let F(T) be the set of all structures that satisfy the axioms T; for a set of mathematical structures S, let G(S) be the minimal axiomatization of S. We can then say that F(T) is a subset of S if and only if T logically implies G(S): the "semantics functor" F and the "syntax functor" G form a monotone Galois connection, with semantics being the lower adjoint.

Properties

In the following, we consider a (monotone) Galois connection f = (f ^∗, f _∗), where f ^∗: A → B is the lower adjoint as introduced above. Some helpful and instructive basic properties can be obtained immediately. By the defining property of Galois connections, f ^∗(x) ≤ f ^∗(x) is equivalent to x ≤ f _∗(f ^∗(x)), for all x in A. 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. These properties can be described by saying the composite f ^∗$circ$f _∗ is deflationary, while f _∗$circ$f ^∗ is inflationary (or extensive).

Now if one considers any elements x and y of A such that x ≤ y, then one can clearly use the above findings to obtain x ≤ f _∗(f ^∗(y)). Applying the basic property of Galois connections, one can now conclude that f ^∗(x) ≤ f ^∗(y). But this just shows that f ^∗ preserves the order of any two elements, i.e. it is monotone. Again, a similar reasoning yields monotonicity of f _∗. 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 f _∗(f ^∗(f _∗(x))) = f _∗(x), for all x in B. Clearly we find that

f _∗(f ^∗(f _∗(x))) ≥ f _∗(x)

because f _∗$circ$f ^∗ is inflationary as shown above. Similarly, since f ^∗$circ$f _∗ is deflationary, one finds that

f ^∗ f _∗ f ^∗ f _∗(x) ≤ f ^∗ f _∗(x) ≤ x,

which is equivalent to

f _∗(f ^∗(f _∗(x))) ≤ f _∗(x).

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

f ^∗(f _∗(f ^∗(f _∗(x)))) = f ^∗(f _∗(x)),

i.e., f ^∗$circ$f _∗ is idempotent.

Closure operators and Galois connections

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

Conversely, any closure operator c on some poset A gives rise to the Galois connection with lower adjoint f ^∗ being just the corestriction of c to the image of c (i.e. as a surjective mapping the closure system c(A)). The upper adjoint f _∗ is then given by the inclusion of c(A) into A, that maps each closed element to itself, considered as an element of A. 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 A (elements x with f _∗(f ^∗(x)) = x) are mapped to elements within the range of the kernel operator f ^∗ $circ$ f _∗, 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 x in A, f ^∗(x) is the least element y of B such that x ≤ f _∗(y). Dually, for every y in B, f _∗(y) is the greatest x in A such that f ^∗(x) ≤ y. 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 adjoint of a Galois connection is given, the other can be defined via this property. On the other hand, some arbitrary function f is a lower adjoint if and only if each set of the form { x in A | f(x) ≤ b }, b in B, 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 (f ^∗, f _∗) between posets A and B and (g ^∗, g _∗) between B and C, the composite (g ^∗$circ$f ^∗, f _∗$circ$g _∗) 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, this 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 x ≤ y. A 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 arrows 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.

Notes

References

A freely available introduction to Galois connections, presenting many examples and results. Also includes notes on the different notations and definitions that arose in this area:

• M. Erné, J. Koslowski, A. Melton, G. 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. Available online in various file formats: PS.GZ PS

The following standard reference books also include Galois connections using modern notation and definitions:

• B. A. Davey and H. A. Priestley: Introduction to lattices and Order, Cambridge University Press, 2002.
• G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott: Continuous Lattices and Domains, Cambridge University Press, 2003.

Finally, some publications using the original (antitone) definition:

• Garrett Birkhoff: Lattice Theory, Amer. Math. Soc. Coll. Pub., Vol 25, 1940
• Øystein Ore: Galois Connexions, Transactions of the American Mathematical Society 55 (1944), pp. 493-513