Galois connection
A Galois connection is the abstract pattern behind every ‘back-and-forth’ duality in mathematics: two ordered worlds linked by a pair of maps going opposite ways, such that going across and coming back never makes things worse, only ‘closes them up’. The original example is field theory — subfields versus subgroups under taking fixed fields and automorphism groups — but the same skeleton governs ideals versus varieties in algebraic geometry, and far more.
Formally, an (antitone) Galois connection between posets (P, ≤) and (Q, ≤) is a pair of order-reversing maps f: P -> Q and g: Q -> P satisfying the adjunction condition: for all p in P and q in Q, q ≤ f(p) if and only if p ≤ g(q). Equivalently, f and g are each order-reversing and both composites are inflationary: p ≤ g(f(p)) and q ≤ f(g(q)) for all p, q. (There is also a covariant, monotone version, where the maps preserve order and the connection is an adjunction between the posets viewed as categories.)
From the axioms two key facts follow automatically: the composite closure operators c = g∘f and c' = f∘g are idempotent, monotone and inflationary, and f, g restrict to mutually inverse, order-reversing bijections between the closed elements c(P) and c'(Q). This single abstraction explains why the fundamental theorem of Galois theory pairs intermediate fields with subgroups inclusion-reversingly, and why the Nullstellensatz pairs radical ideals with affine varieties — both are instances of one order-theoretic mechanism.
For a Galois extension L/K, the maps H -> L^H (fixed field) and M -> Gal(L/M) (automorphism group) form a Galois connection; the closed elements are all intermediate fields and all subgroups, in inclusion-reversing bijection.
The fundamental theorem of Galois theory is the prototypical Galois connection.