---
tag: DA619M38C3
kind: concept
name: Quantale
msc: ["06F07"]
specializes: [D7A0ZGZT5S]
sources:
  - url: https://en.wikipedia.org/wiki/Quantale
same_as:
  wikidata: Q7268888
  wikipedia: Quantale
---

## Summary

A quantale is a [[complete lattice|D7A0ZGZT5S]] with an associative multiplication that distributes over arbitrary joins on both sides.
Lattices of ideals of a ring, frames of open sets, and power sets of monoids are quantales.
Quantales generalise frames and locales to a non-commutative or non-idempotent setting.
They appear in noncommutative topology and linear logic.
The multiplication of a quantale always has left and right residuals, which links quantales with [[residuated lattices|E337A6WVH3]].

## Explanations

### L1 human

Take the ideals of a commutative ring.
You can join any family of ideals by taking the ideal they generate, and you can multiply two ideals.
Multiplying an ideal by the join of a family gives the join of the products.
A quantale is any complete lattice with a multiplication with this property.