Table of contents |

2 Examples 3 Properties 4 Category theoretic approach 5 References |

Suppose (*A*, ≤) and (*B*, <=) are two partially ordered sets. A *Galois connection* between these posets consists of two 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*G*(*b*) ≤*a*.

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 ⊆. 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 ⊆. 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* `|->` Gal(*L*/*E*) and *G* `|->` Fix(*G*) form a Galois connection.

In algebraic geometry, the relation between sets of polynomials and their zero sets is a 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*[*X*_{1},...,*X*_{n}] and let *B* be the set of all subsets of *K*^{n}. Both *A* and *B* are naturally ordered by inclusion ⊆. If *S* is a set of polynomials, define *F*(*S*) = {**x** in *K*^{n} : *f*(**x**) = 0 for all *f* in *S*}, the set of common zeros of the polynomials in *S*. If *T* is a subset of *K*^{n}, define *G*(*T*) = {*f* in *K*[*X*_{1},...,*X*_{n}] : *f*(**x**) = 0 for all **x** in *T*}. Then *F* and *G* form a Galois connection.

Finally, 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 a Galois connection between the power sets of *X* and *Y*, if both of them are ordered by inclusion ⊆.

Furthermore, we have *G*(*F*(*a*)) ≤ *a* and *F*(*G*(*b*)) <= *b* for all *a* in *A* and *b* in *B*.

Every partially ordered set can be viewed as a category in a natural way: there's a unique morphism from *x* to *y* iff *x* ≤ *y*. A Galois connection is then nothing but a pair of adjoint functors between two categories, where the first category arises from a partially ordered set and the second category is the dual of one that arises from a partially ordered set.

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