example of Boolean algebras

example of Boolean algebras

Below is a list of

examples of Boolean algebras

. Note that the phrase “usual set-theoretic

operations

” refers to the operations of union

∪

,

intersection

∩

, and set

complement

′

.

1.

Let

A

be a set. The

power set

P

⁢

(

A

)

of

A

, or the

collection

of all the subsets of

A

, together with the operations of union, intersection, and set complement, the

empty set

∅

and

A

, is a

Boolean algebra

. This is the canonical example of a Boolean algebra.

2.

In

P

⁢

(

A

)

, let

F

⁢

(

A

)

be the collection of all finite subsets of

A

, and

c

⁢

F

⁢

(

A

)

the collection of all

cofinite

subsets of

A

. Then

F

⁢

(

A

)

∪

c

⁢

F

⁢

(

A

)

is a Boolean algebra.

3.

More generally, any

field of sets

is a Boolean algebra. In particular, any sigma algebra

σ

in a set is a Boolean algebra.

4.

(

product

of

algebras

) Let

A

and

B

be Boolean algebras. Then

A

×

B

is a Boolean algebra, where

(

a

,

b

)

∨

(

c

,

d

)

:=

(

a

∨

c

,

b

∨

d

)

,

(1)

(

a

,

b

)

∧

(

c

,

d

)

:=

(

a

∧

c

,

b

∧

d

)

,

(2)

(

a

,

b

)

′

:=

(

a

′

,

b

′

)

.

(3)

5.

More generally, if we have a collection of Boolean algebras

A

i

,

indexed by

a set

I

, then

∏

i

∈

I

A

i

is a Boolean algebra, where the Boolean operations are defined componentwise.

6.

In particular, if

A

is a Boolean algebra, then set of

functions

from some non-empty set

I

to

A

is also a Boolean algebra, since

A

I

=

∏

i

∈

I

A

.

7.

(

subalgebras

) Let

A

be a Boolean algebra, any subset

B

⊆

A

such that

0

∈

B

,

a

′

∈

B

whenever

a

∈

B

, and

a

∨

b

∈

B

whenever

a

,

b

∈

B

is a Boolean algebra. It is called a

Boolean subalgebra

of

A

. In particular, the

homomorphic image

of a

Boolean algebra homomorphism

is a Boolean algebra.

8.

(

quotient algebras

) Let

A

be a Boolean algebra and

I

a

Boolean ideal

in

A

. View

A

as a

Boolean ring

and

I

an ideal in

A

. Then the quotient ring

A

/

I

is

Boolean

, and hence a Boolean algebra.

9.

Let

A

be a set, and

R

n

⁢

(

A

)

be the set of all

n

-ary relations on

A

. Then

R

n

⁢

(

A

)

is a Boolean algebra under the usual set-theoretic operations. The easiest way to see this is to realize that

R

n

⁢

(

A

)

=

P

⁢

(

A

n

)

, the powerset of the

n

-fold power of

A

.

10.

The set of all

clopen sets

in a

topological space

is a Boolean algebra.

11.

Let

X

be a topological space and

A

be the collection of all

regularly open

sets in

X

. Then

A

has a Boolean

algebraic structure

. The meet and the

constant

operations follow the usual set-theoretic ones:

U

∧

V

=

U

∩

V

,

0

=

∅

and

1

=

X

. However, the join

∧

and the

complementation

′

on

A

are different. Instead, they are given by

U

′

:=

X

-

U

¯

,

(4)

U

∨

V

:=

(

U

∪

V

)

′′

.

(5)

Title

example of Boolean algebras

Canonical name

ExampleOfBooleanAlgebras

Date of creation

2013-03-22 17:52:33

Last modified on

2013-03-22 17:52:33

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Example

Classification

msc 06B20

Classification

msc 03G05

Classification

msc 06E05

Classification

msc 03G10

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