Galois connection

Summary

A Galois connection is a pair of maps $f \colon P \to Q$ and $g \colon Q \to P$ between partially ordered sets with $f(x) \le y$ exactly when $x \le g(y)$. It is a weak form of an order isomorphism, and it is the order-theoretic case of an adjunction between functors. The correspondence between subgroups and subfields in Galois theory is the classical example, which gives the notion its name. Every Galois connection produces a closure operator $g \circ f$ on $P$.