---
tag: E337A6WVH3
kind: concept
name: Residuated lattice
msc: ["06F05", "03G10"]
specializes: [WH4SFAACAS]
sources:
  - url: https://en.wikipedia.org/wiki/Residuated_lattice
same_as:
  wikidata: Q9462417
  wikipedia: Residuated lattice
---

## Summary

A residuated lattice is a [[lattice|WH4SFAACAS]] with a [[monoid|0N6R902FG1]] multiplication that has left and right division operations, called residuals.
The residuals are characterised by an adjunction with the multiplication and the order.
[[Heyting algebras|NG943XNQWH]], [[Boolean algebras|0XADJKM699]], lattice-ordered groups and the lattices of ideals of a ring are examples.
Residuated lattices are the algebraic semantics of substructural logics, where the multiplication is a conjunction and the residuals are implications.

## Explanations

### L1 human

In the positive real numbers, $xy \le z$ holds exactly when $y \le z/x$.
So division is the best answer to the question "how large can $y$ be so that $x$ times $y$ stays below $z$?".
A residuated lattice asks for such a best answer in an abstract ordered setting, from the left and from the right.
In logic, "$x$ and $y$ imply $z$" holds exactly when "$y$ implies ($x$ implies $z$)", and this is the same rule.