representing a distributive lattice by ring of sets

In this entry, we present the proof of a fundamental fact that every distributive latticeMathworldPlanetmath is latticeMathworldPlanetmath isomorphic to a ring of sets, originally proved by Birkhoff and Stone in the 1930’s. The proof uses the prime ideal theorem of Birkhoff ( First, a simple results from the prime ideal theorem:

Lemma 1.

Let L be a distributive lattice and a,bL with ab. Then there is a prime idealMathworldPlanetmathPlanetmathPlanetmath containing one or the other.


Let I=a and J=b, the principal idealsMathworldPlanetmathPlanetmath generated by a,b respectively. If I=J, then ba and ab, or a=b, contradicting the assumptionPlanetmathPlanetmath. So IJ, which means either aJ or bI. In either case, apply the prime ideal theorem to obtain a prime ideal containing I (or J) not containing b (or a). ∎

Before proving the theorem, we have one more concept to introduce:

Definition. Let L be a distributive lattice, and X the set of all prime ideals of L. Define F:LP(X), the powerset of X, by

Proposition 1.

F is an injection.


If ab, then by the lemma there is a prime ideal P containing one but not another, say aP and bP. Then PF(a) and PF(b), so that F(a)F(b). ∎

Proposition 2.

There are two things to show:

  • F preserves : If PF(ab), then abP, so that aP and bP, since P is a sublattice. So PF(a) and PF(b) as a result. On the other hand, if PF(a)F(b), then aP and bP. Since P is prime, abP, so that PF(ab). Therefore, F(ab)=F(a)F(b).

  • F preserves : If PF(ab), then abP, which implies that aP or bP, since P is a sublattice of L. So PF(a)F(b). On the other hand, if PF(a)F(b), then abP, since P is a lattice ideal. Hence F(ab)=F(a)F(b).

Therefore, F is a lattice homomorphism. ∎

The function F is called the canonical embedding of L into P(X).

Theorem 1.

Every distributive lattice is isomorphic to a ring of sets.


Let L,X,F be as above. Since F:LP(X) is an embedding, L is lattice isomorphic to F(L), which is a ring of sets. ∎

Remark. Using the result above, one can show that if L is a Boolean algebraMathworldPlanetmath, then L is isomorphic to a field of sets. See link below for more detail.

Title representing a distributive lattice by ring of sets
Canonical name RepresentingADistributiveLatticeByRingOfSets
Date of creation 2013-03-22 19:08:24
Last modified on 2013-03-22 19:08:24
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Theorem
Classification msc 06D99
Classification msc 06D05
Related topic RingOfSets
Related topic RepresentingABooleanLatticeByFieldOfSets