Galois Connection

A pair of monotone maps between posets (f : P → Q, g : Q → P) such that for all p in P and q in Q, f(p) ≤_Q q iff p ≤_P g(q). Equivalently, f is left adjoint to g when posets are viewed as categories. It yields a residual correspondence tying approximation and closure operators.

A pair of monotone maps between partially ordered sets, one left adjoint and one right adjoint, such that one map composed with the other yields order inequalities in both directions (f(a) ≤ b iff a ≤ g(b)); this correspondence links closure-like operations and kernel-like operations and organizes duality between lattices and their images.

A pair of monotone (order-preserving) maps between posets, L: P→Q and R: Q→P, such that for all p in P and q in Q, L(p) ≤ q if and only if p ≤ R(q); this yields an inclusion-reversing, closure-like correspondence linking the two orders.