Residuated mapping

Summary

A residuated mapping is a monotone map $f \colon P \to Q$ between partially ordered sets with a special property. For each $y$, the inequality $f(x) \le y$ has a largest solution $x$. Equivalently, $f$ is the lower adjoint of a Galois connection; its upper adjoint is called the residual of $f$. Residuated maps are the basis of residuated lattices, which model division in ordered algebra and implication in logic.