representing a complete atomic Boolean algebra by power set

representing a complete atomic Boolean algebra by power set

It is a known fact that every

Boolean algebra

is

isomorphic

to a

field of sets

(of some set) (proof here (

http://planetmath.org/RepresentingABooleanLatticeByFieldOfSets

)). In this entry, we show that, furthermore, if a Boolean algebra is atomic and

complete

, then it is isomorphic to

the

field of sets of some set, in other words, the

powerset

of some set, viewed as a Boolean algebra via the usual set-theoretic

operations

of union,

intersection

, and

complement

.

The proof is based on the following

function

, defined for any atomic Boolean algebra:

Definition

. Let

B

be an atomic Boolean algebra, and

X

the

set of its atoms. Define

f

:

B

→

P

⁢

(

X

)

by

f

⁢

(

x

)

:=

{

a

∣

a

≤

x

}

.

It is easy to see that

f

⁢

(

x

)

=

{

x

}

iff

x

is an atom of

B

.

Proposition 1

.

f

⁢

(

x

)

and

f

⁢

(

x

′

)

are complement of one another in

X

.

Proof.

For any

a

∈

X

,

a

≤

1

=

x

∨

x

′

, so that

a

≤

x

or

a

≤

x

′

, or

a

∈

f

⁢

(

x

)

or

a

∈

f

⁢

(

x

′

)

. This shows that

f

⁢

(

x

)

∪

f

⁢

(

x

′

)

=

X

. If

a

∈

f

⁢

(

x

)

∩

f

⁢

(

x

′

)

, then

a

≤

x

and

a

≤

x

′

, so that

a

≤

x

∧

x

′

=

0

, which is impossible, since

a

is an atom, and by definition, must be greater than

0

.
∎

Proposition 2

.

f

is a

Boolean algebra homomorphism

.

Proof.

First,

f

⁢

(

x

′

)

=

X

-

f

⁢

(

x

)

by the last

proposition

.

Next,

f

⁢

(

x

∨

y

)

=

{

a

∣

a

≤

x

∨

y

}

=

{

a

∣

a

≤

x

⁢

or

⁢

a

≤

y

}

since

a

is an atom. But the

right hand side

equals

{

a

∣

a

≤

x

}

∪

{

a

∣

a

≤

y

}

=

f

⁢

(

x

)

∪

f

⁢

(

y

)

, we see that

f

preserves

∨

.

Finally,

f

⁢

(

0

)

=

{

a

∣

a

≤

0

}

=

∅

since any atom must be greater than

0

.

Hence,

f

is a Boolean algebra homomorphism.
∎

Proposition 3

.

f

is an

injection

.

Proof.

Suppose

f

⁢

(

x

)

=

∅

. If

x

≠

0

, then there must be some atom

a

such that

a

≤

x

. But this implies that

f

⁢

(

x

)

≠

∅

, a

contradiction

. Hence

x

=

0

and

f

is injective. ∎

Proposition 4

.

f

is conditionally complete, in the sense that if

⋁

A

is defined for any

A

⊆

B

, then

f

⁢

(

⋁

A

)

=

⋃

{

f

⁢

(

x

)

∣

x

∈

A

}

.

Proof.

Suppose

y

=

⋁

A

and

Y

=

f

⁢

(

y

)

. Let

Z

=

⋃

{

f

⁢

(

x

)

∣

x

∈

A

}

. We want to show that

Y

=

Z

. If

a

∈

Y

, then

a

≤

y

, or

a

≤

x

for some

x

∈

A

, since

a

is an atom. So

a

∈

f

⁢

(

x

)

⊆

Z

.

Conversely

, if

a

∈

Z

, then

a

∈

f

⁢

(

x

)

, or

a

≤

x

for some

x

∈

A

. This means that

a

≤

x

≤

⋁

A

=

y

, and therefore

a

∈

f

⁢

(

y

)

=

Y

.
∎

Proposition 5

.

If

B

is complete, so is

f

. Moreover,

f

is

surjective

.

Proof.

The first

sentence

is a direct

consequence

of the previous proposition. For the second setnence, let

Y

∈

P

⁢

(

X

)

. Let

y

=

⋁

Y

.

x

exists because

B

is complete. So

f

⁢

(

y

)

=

f

⁢

(

⋁

Y

)

=

⋃

{

f

⁢

(

x

)

∣

x

∈

Y

}

=

⋃

{

{

x

}

∣

x

∈

Y

}

=

Y

, since each

x

∈

Y

is an atom. ∎

Rewording the above proposition, we have

Theorem 1

.

Any complete atomic Boolean algebra is isomorphic (as

complete Boolean algebras

) to the powerset of some set, namely, the set of all of its atoms.

A useful application of this representation theorem is the following:

Corollary 1

.

The

cardinality

of a finite Boolean algebra is a power of

2

.

Proof.

Every finite Boolean algebra is complete and atomic, and hence isomorphic to the powerset of a set, which is also finite, and the result follows. ∎

Remark

. The proof does not depend on the

representation

of a Boolean algebra by a field of sets.

Title

representing a complete atomic Boolean algebra by

power set

Canonical name

RepresentingACompleteAtomicBooleanAlgebraByPowerSet

Date of creation

2013-03-22 19:08:30

Last modified on

2013-03-22 19:08:30

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06E10

Related topic

RepresentingABooleanLatticeByFieldOfSets

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