---
tag: W5VG6HS8KW
kind: concept
name: Natural duality
msc: ["08C20", "06D50"]
sources:
  - url: https://en.wikipedia.org/wiki/Stone_duality
---

## Summary

A natural duality represents each algebra in the [[quasivariety|WRXEKT4NQS]] generated by a finite algebra $M$.
The algebra becomes an algebra of continuous structure-preserving maps on a topological structure.
The dual of an algebra $A$ is the set of homomorphisms from $A$ to $M$.
It is viewed inside a power of a topological structure $\underset{\sim}{M}$ on the same set as $M$.
[[Stone duality|B9T33D5MJK]] for Boolean algebras and [[Priestley duality|SA22SEC75K]] for bounded distributive lattices are the model cases.
The theory gives general conditions under which such a duality exists.
It also tells when the duality is a dual equivalence of categories, a [[Stone-type duality|VH12PN9V13]].

## Explanations

### L1 human

Stone showed that a Boolean algebra can be rebuilt from the set of its homomorphisms into the two-element Boolean algebra, once that set carries the right topology.
Natural duality asks the same question for other finite "building block" algebras $M$.
Each algebra $A$ built from copies of $M$ is turned into a space of homomorphisms $A \to M$, with extra relations or operations remembered from $M$.
The goal is that $A$ can be recovered as the continuous structure-preserving maps from that space back to $M$.