Boolean algebra homomorphism

Boolean algebra homomorphism

Let

A

and

B

be

Boolean algebras

. A

function

f

:

A

→

B

is called a

Boolean algebra homomorphism

, or

homomorphism

for short, if

f

is a

{

0

,

1

}

-

lattice homomorphism

(

http://planetmath.org/LatticeHomomorphism

) such that

f

respects

′

:

f

⁢

(

a

′

)

=

f

⁢

(

a

)

′

.

Typically, to show that a function between two Boolean algebras is a Boolean algebra homomorphism, it is not necessary to check every defining condition. In fact, we have the following:

1.

if

f

respects

′

, then

f

respects

∨

iff it respects

∧

;

2.

if

f

is a lattice homomorphism, then

f

respects

0

and

1

iff it respects

′

.

The first assertion can be shown by de Morgan’s laws. For example, to see the LHS implies RHS,

f

(

a

∧

b

)

=

f

(

(

a

′

∨

b

′

)

′

)

=

f

(

a

′

∨

b

′

)

′

=

(

(

f

(

a

′

)

∨

f

(

b

′

)

)

′

=

f

(

a

′

)

′

∧

f

(

b

′

)

′

=

f

(

a

)

′′

∧

f

(

b

)

′′

=

f

(

a

)

∧

f

(

b

)

. The second assertion can also be easily proved. For example, to see that the LHS implies RHS, we have that

f

⁢

(

a

′

)

∨

f

⁢

(

a

)

=

f

⁢

(

a

′

∨

a

)

=

f

⁢

(

1

)

=

1

and

f

⁢

(

a

′

)

∧

f

⁢

(

a

)

=

f

⁢

(

a

′

∧

a

)

=

f

⁢

(

0

)

=

0

. Together, this implies that

f

⁢

(

a

′

)

is the

complement

of

f

⁢

(

a

)

, which is

f

⁢

(

a

)

′

.

If a function

satisfies

one, and hence all, of the above conditions also satisfies the property that

f

⁢

(

0

)

=

0

, for

f

⁢

(

0

)

=

f

⁢

(

a

∧

a

′

)

=

f

⁢

(

a

)

∧

f

⁢

(

a

′

)

=

f

⁢

(

a

)

∧

f

⁢

(

a

)

′

=

0

. Dually,

f

⁢

(

1

)

=

1

.

As a Boolean algebra is an

algebraic system

, the definition of a Boolean algebra homormphism is just a special case of an algebra homomorphism between two algebraic systems. Therefore, one may similarly define a Boolean algebra

monomorphism

,

epimorphism

, endormophism, automorphism, and

isomorphism

.

Let

f

:

A

→

B

be a Boolean algebra homomorphism. Then the

kernel

of

f

is the set

{

a

∈

A

∣

f

⁢

(

a

)

=

0

}

, and is written

ker

⁡

(

f

)

. Observe that

ker

⁡

(

f

)

is a

Boolean ideal

of

A

.

Let

κ

be a

cardinal

. A Boolean algebra homomorphism

f

:

A

→

B

is said to be

κ

-

complete

if for any subset

C

⊆

A

such that

1.

|

C

|

≤

κ

, and

2.

⋁

C

exists,

then

⋁

f

⁢

(

C

)

exists and is equal to

f

⁢

(

⋁

C

)

. Here,

f

⁢

(

C

)

is the set

{

f

⁢

(

c

)

∣

c

∈

C

}

. Note that again, by de Morgan’s laws, if

⋀

C

exists, then

⋀

f

⁢

(

C

)

exists and is equal to

f

⁢

(

⋀

C

)

. If we place no

restrictions

on the

cardinality

of

C

(i.e., drop condition 1), then

f

:

A

→

B

is said to be a

complete Boolean algebra homomorphism

. In the

categories

of

κ

-

complete Boolean algebras

and complete Boolean algebras, the

morphisms

are

κ

-complete homomorphisms and complete homomorphisms respectively.

Title

Boolean algebra homomorphism

Canonical name

BooleanAlgebraHomomorphism

Date of creation

2013-03-22 18:02:05

Last modified on

2013-03-22 18:02:05

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06E05

Classification

msc 03G05

Classification

msc 06B20

Classification

msc 03G10

Synonym

Boolean homomorphism

Defines

kernel

Defines

complete Boolean algebra homomorphism

Defines

κ

-complete

Boolean

algbra homomorphism

Generated on Fri Feb 9 10:02:28 2018 by

LaTeXML