representing a distributive lattice by ring of sets

representing a distributive lattice by ring of sets

In this entry, we present the proof of a fundamental fact that every

distributive lattice

is

lattice

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 (

http://planetmath.org/BirkhoffPrimeIdealTheorem

). First, a simple results from the prime ideal theorem:

Lemma 1

.

Let

L

be a distributive lattice and

a

,

b

∈

L

with

a

≠

b

. Then there is a

prime ideal

containing one or the other.

Proof.

Let

I

=

⟨

a

⟩

and

J

=

⟨

b

⟩

, the

principal ideals

generated by

a

,

b

respectively. If

I

=

J

, then

b

≤

a

and

a

≤

b

, or

a

=

b

, contradicting the

assumption

. So

I

≠

J

, which means either

a

∉

J

or

b

∉

I

. 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

:

L

→

P

⁢

(

X

)

, the

powerset

of

X

, by

F

⁢

(

a

)

:=

{

P

∣

a

∉

P

}

.

Proposition 1

.

F

is an

injection

.

Proof.

If

a

≠

b

, then by the lemma there is a prime ideal

P

containing one but not another, say

a

∈

P

and

b

∉

P

. Then

P

∉

F

⁢

(

a

)

and

P

∈

F

⁢

(

b

)

, so that

F

⁢

(

a

)

≠

F

⁢

(

b

)

. ∎

Proposition 2

.

F

is a

lattice homomorphism

.

Proof.

There are two things to show:

•

F

preserves

∧

: If

P

∈

F

⁢

(

a

∧

b

)

, then

a

∧

b

∉

P

, so that

a

∉

P

and

b

∉

P

, since

P

is a sublattice. So

P

∈

F

⁢

(

a

)

and

P

∈

F

⁢

(

b

)

as a result. On the other hand, if

P

∈

F

⁢

(

a

)

∩

F

⁢

(

b

)

, then

a

∉

P

and

b

∉

P

. Since

P

is prime,

a

∧

b

∉

P

, so that

P

∈

F

⁢

(

a

∧

b

)

. Therefore,

F

⁢

(

a

∧

b

)

=

F

⁢

(

a

)

∩

F

⁢

(

b

)

.

•

F

preserves

∨

: If

P

∈

F

⁢

(

a

∨

b

)

, then

a

∨

b

∉

P

, which implies that

a

∉

P

or

b

∉

P

, since

P

is a sublattice of

L

. So

P

∈

F

⁢

(

a

)

∪

F

⁢

(

b

)

. On the other hand, if

P

∈

F

⁢

(

a

)

∪

F

⁢

(

b

)

, then

a

∨

b

∉

P

, since

P

is a lattice ideal. Hence

F

⁢

(

a

∨

b

)

=

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.

Proof.

Let

L

,

X

,

F

be as above. Since

F

:

L

→

P

⁢

(

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 algebra

, 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

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