In order theory the term Galois connection can mean both: “adjunction between posets” and “dual adjunction between posets”; the former notion is sometimes called “monotone Galois connection” and the latter “antitone Galois connection”. In this article the term “Galois connection” shall mean “dual adjunction between posets”.
More explicitly, given posets and , a Galois connection between and is a pair of order-reversing maps and such that and for all , .
A Galois correspondence is a Galois connection which is an adjoint equivalence (so and for all , ).
Any Galois connection , induces a Galois correspondence between and , given by the composites and .
For any of the form , we have and also where the inequality follows from and antitonicity of . Hence for all . Similarly for all .
Frequently Galois connections between collections of subsets arise where is “the set of all standing in some relation to every ” and dually is “the set of all standing in some relation to every .” See orthogonality for one example.
The Galois theory normally taught in graduate-level algebra courses (and based on the work of Évariste Galois) involves a Galois connection between the intermediate fields of a Galois extension and the subgroups of the corresponding Galois group.
Every Galois connection between full power sets,
is of the form in the first example: there is some binary relation from to such that
Indeed, define by stipulating that is true if and only if . Because is a left adjoint, it takes colimits in (in this case, unions) to colimits in , which are intersections in . Since every in is a union of singletons , this gives
which is another way of writing the formula for given above. We observe that
if and only if
(now viewing extensionally in terms of subsets). This last symmetrical expression in and means
which means we have a Galois connection between and under this definition; since is uniquely determined by the presence of a Galois connection with , we conclude that all Galois connections between power sets arise in this way, via a relation between and .
Wojciech Dzik, Jouni Järvinen, Michiro Kondo, Characterising intermediate tense logics in terms of Galois connections (arXiv:1401.7646).
M. Erné, E. Klossoswki, A. Melton, G. E. Strecker, A primer in Galois connections , Annals New York Academy of Sciences 704 (1993) pp.103-125. (draft)
O. Ore, Galois connexions , Trans. AMS 55 (1944) pp.493-513. (pdf)