---
tag: SA22SEC75K
kind: statement
role: theorem
name: Priestley duality
msc: [06D50, 06D05, 18F70]
uses: [0HM0HQCK6N, ZGBV8V3823, Z15FXM9WM1, ETCE287E2R]
about: [0HM0HQCK6N, ZGBV8V3823]
sources:
  - url: https://en.wikipedia.org/wiki/Duality_theory_for_distributive_lattices
---

## Statement

Let $L$ be a bounded [[distributive lattice|0HM0HQCK6N]].
A *prime filter* of $L$ is a [[filter|Z15FXM9WM1]] $F$ with $0 \notin F$ such that $a \vee b \in F$ implies $a \in F$ or $b \in F$.
Let $X_L$ be the set of prime filters of $L$, ordered by inclusion.
Give $X_L$ the topology generated by the sets $\set{F : a \in F}$ and $\set{F : a \notin F}$ for $a \in L$.
For a [[Priestley space|ZGBV8V3823]] $X$, let $\mathrm{ClopUp}(X)$ be the lattice of subsets of $X$ that are clopen [[upper sets|ETCE287E2R]].
Its meet and join are intersection and union.

1. $X_L$ is a Priestley space, and the map $a \mapsto \set{F \in X_L : a \in F}$ is an isomorphism of bounded lattices from $L$ onto $\mathrm{ClopUp}(X_L)$.
2. For every Priestley space $X$, the map $x \mapsto \set{U \in \mathrm{ClopUp}(X) : x \in U}$ is a homeomorphism and order isomorphism from $X$ onto $X_{\mathrm{ClopUp}(X)}$.
3. These constructions extend to a dual equivalence of categories.
On one side are the bounded distributive lattices with bounded lattice homomorphisms; on the other, the Priestley spaces with continuous order-preserving maps.
A homomorphism $h \colon L \to M$ corresponds to the map $X_M \to X_L$, $G \mapsto h^{-1}(G)$.

## Remarks

The key input is the [[prime ideal theorem for distributive lattices|ME8GM0VTWP]]: if $a \not\le b$, some prime filter contains $a$ and not $b$.
This gives the injectivity in part 1 and the separation axiom of $X_L$.
Compactness of $X_L$ comes from viewing $X_L$ as a closed subspace of the product space $\set{0, 1}^L$.

Example: if $L$ is finite, the prime filters are the sets ${\uparrow} j$ for $j$ join-irreducible, and $X_L$ is a finite poset with the discrete topology.
Then part 1 becomes [[Birkhoff's representation theorem|6RDM3069KC]] up to reversing the order.

When $L$ is a [[Boolean algebra|0XADJKM699]], every prime filter is an ultrafilter and the order on $X_L$ is trivial.
Then the duality restricts to Stone duality between Boolean algebras and [[Stone spaces|ZV38HA5PAQ]].

## Explanations

### L1 human

Every bounded distributive lattice can be drawn as a lattice of sets: the clopen upper sets of a special ordered space.
Conversely, from such a space one gets back the lattice, and from the lattice the space.
So questions about distributive lattices can be translated into questions about ordered compact spaces, and back.

### L2 human

Priestley duality is a widely used form of duality for distributive lattices.
Homomorphisms turn into continuous order-preserving maps in the opposite direction, so algebraic questions become geometric ones.
It extends Stone duality for Boolean algebras.
Under it, Heyting algebras correspond to [[Esakia spaces|996BWM6ZQ7]], a special class of Priestley spaces.
An equivalent version, due to Stone, uses [[spectral spaces|G3W2WXY2VN]] instead of Priestley spaces.