---
tag: ME8GM0VTWP
kind: statement
role: theorem
name: Prime ideal theorem for distributive lattices
msc: [06D05, "03E25"]
uses: [0HM0HQCK6N, 5GW2T2AM1A, Z15FXM9WM1, QKWVXJWHP6]
about: [0HM0HQCK6N, QKWVXJWHP6]
sources:
  - url: https://en.wikipedia.org/wiki/Boolean_prime_ideal_theorem
---

## Statement

Let $L$ be a [[distributive lattice|0HM0HQCK6N]].
Let $I$ be an [[ideal|5GW2T2AM1A]] of $L$ and let $F$ be a [[filter|Z15FXM9WM1]] of $L$ with $I \cap F = \emptyset$.
Then there is a [[prime ideal|QKWVXJWHP6]] $P$ of $L$ such that $I \subseteq P$ and $P \cap F = \emptyset$.

## Remarks

The usual proof applies [[Zorn's lemma|04RXTQA5ZA]] to the set of ideals that contain $I$ and are disjoint from $F$, ordered by inclusion.
Let $P$ be a maximal one, and suppose $x \wedge y \in P$ with $x \notin P$ and $y \notin P$.
By maximality, the ideals generated by $P \cup \set{x}$ and by $P \cup \set{y}$ meet $F$, so $f \le p \vee x$ and $g \le q \vee y$ for some $f, g \in F$ and $p, q \in P$.
Then $f \wedge g \le (p \vee q) \vee (x \wedge y)$ by distributivity, so $f \wedge g \in P \cap F$, a contradiction.

Example: in the lattice of divisors of $12$, with $\gcd$ as meet and $\lcm$ as join, take $I = \set{1, 2}$ and $F = \set{3, 6, 12}$.
The set $P = \set{1, 2, 4}$ of divisors not divisible by $3$ is a prime ideal with $I \subseteq P$ and $P \cap F = \emptyset$.

Distributivity is necessary.
In the [[diamond lattice M3|JQ6FJH2TP8]] with middle elements $a, b, c$, the proper ideals are $\set{0}$ and $\set{0, x}$ for a middle element $x$.
None of them is prime: for instance $a \wedge b = 0$, while neither $a$ nor $b$ lies in $\set{0, c}$.
So the ideal $\set{0}$ and the filter $\set{1}$ admit no separating prime ideal.

The [[Boolean prime ideal theorem|STPBXBKSZP]] is the special case of Boolean algebras.
Over ZF set theory, the two statements are equivalent, and both are strictly weaker than the [[axiom of choice|0HTGB13FE2]].

## Explanations

### L1 human

An ideal is a family of "small" elements and a filter is a family of "large" elements.
If they do not overlap, the theorem finds a prime ideal that keeps all the small ones and avoids all the large ones.
Prime ideals are the "points" of a distributive lattice, so the theorem says there are enough points to separate things.

### L2 human

The theorem is the key step in representing a distributive lattice as a lattice of sets: map $a$ to the set of prime filters that contain it.
Separating $a \not\le b$ by a prime ideal containing $b$ and missing $a$ shows that this map is injective.
This is the basis of Stone's representation theorem for distributive lattices and of [[Priestley duality|SA22SEC75K]].
It needs a form of choice, and over ZF it is equivalent to the [[Boolean prime ideal theorem|STPBXBKSZP]].