---
tag: VH12PN9V13
kind: concept
name: Stone-type duality for varieties
msc: ["08C20", "06D50", "06E15"]
sources:
  - url: https://en.wikipedia.org/wiki/Stone_duality
---

## Summary

A Stone-type duality for a variety or quasivariety of algebras is a dual equivalence of categories.
It matches the algebras with [[Stone spaces|ZV38HA5PAQ]] that carry additional structure.
Under it, homomorphisms of algebras correspond to continuous structure-preserving maps in the opposite direction.
The prototype is [[Stone duality|B9T33D5MJK]] between Boolean algebras and Stone spaces; [[Priestley duality|SA22SEC75K]] and Esakia duality for Heyting algebras are further examples.
Such dualities turn algebraic questions into topological ones, and the theory of [[natural dualities|W5VG6HS8KW]] gives a uniform method to build many of them.

## Explanations

### L1 human

A Boolean algebra can be pictured as the clopen subsets of a compact, totally disconnected space, and this picture loses nothing.
A Stone-type duality is a dictionary of this kind for another class of algebras.
Algebras become spaces, subalgebras become quotient spaces, quotients become subspaces, and products of finitely many algebras become disjoint unions of spaces.
For example, a finite distributive lattice corresponds to the finite ordered set of its join-irreducible elements.