---
tag: V76Q9B6VS8
kind: concept
name: Residuated mapping
msc: [06A15]
sources:
  - url: https://en.wikipedia.org/wiki/Residuated_mapping
same_as:
  wikidata: Q7315523
  wikipedia: Residuated mapping
---

## Summary

A residuated mapping is a monotone map $f \colon P \to Q$ between [[partially ordered sets|011C29Q1S1]] 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|MBVJ4JQXC5]]; 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.

## Explanations

### L1 human

For a positive number $a$, the inequality $a x \le y$ in the reals has the largest solution $x = y / a$.
The map $x \mapsto a x$ is residuated, and its residual is division by $a$.
In general, a residuated map is one for which "how large can $x$ be so that $f(x)$ stays below $y$?" always has an answer.