kernel of a homomorphism between algebraic systems

kernel of a homomorphism between algebraic systems

Let

f

:

(

A

,

O

)

→

(

B

,

O

)

be a

homomorphism

between two

algebraic systems

A

and

B

(with

O

as the operator set). Each

element

b

∈

B

corresponds to a subset

K

⁢

(

b

)

:=

f

-

1

⁢

(

b

)

in

A

. Then

{

K

⁢

(

b

)

∣

b

∈

B

}

forms a

partition

of

A

. The

kernel

ker

⁡

(

f

)

of

f

is defined to be

ker

⁡

(

f

)

:=

⋃

b

∈

B

K

⁢

(

b

)

×

K

⁢

(

b

)

.

It is easy to see that

ker

⁡

(

f

)

=

{

(

x

,

y

)

∈

A

×

A

∣

f

⁢

(

x

)

=

f

⁢

(

y

)

}

. Since it is a subset of

A

×

A

, it is relation on

A

. Furthermore, it is an

equivalence relation

on

A

:

1

1

In general,

{

N

i

}

is a partition of a set

A

iff

⋃

N

i

2

is an equivalence relation on

A

.

1.

ker

⁡

(

f

)

is

reflexive

: for any

a

∈

A

,

a

∈

K

⁢

(

f

⁢

(

a

)

)

, so that

(

a

,

a

)

∈

K

⁢

(

f

⁢

(

a

)

)

2

⊆

ker

⁡

(

f

)

2.

ker

⁡

(

f

)

is

symmetric

: if

(

a

1

,

a

2

)

∈

ker

⁡

(

f

)

, then

f

⁢

(

a

1

)

=

f

⁢

(

a

2

)

, so that

(

a

2

,

a

1

)

∈

ker

⁡

(

f

)

3.

ker

⁡

(

f

)

is

transitive

: if

(

a

1

,

a

2

)

,

(

a

2

,

a

3

)

∈

ker

⁡

(

f

)

, then

f

⁢

(

a

1

)

=

f

⁢

(

a

2

)

=

f

⁢

(

a

3

)

, so

(

a

1

,

a

3

)

∈

ker

⁡

(

f

)

.

We write

a

1

≡

a

2

(

mod

ker

⁡

(

f

)

)

to denote

(

a

1

,

a

2

)

∈

ker

⁡

(

f

)

.

In fact,

ker

⁡

(

f

)

is a

congruence relation

: for any

n

-ary operator symbol

ω

∈

O

, suppose

c

1

,

…

,

c

n

and

d

1

,

…

,

d

n

are two sets of elements in

A

with

c

i

≡

d

i

mod

ker

⁡

(

f

)

. Then

f

(

ω

A

(

c

1

,

…

,

c

n

)

=

ω

B

(

f

(

c

1

)

,

…

,

f

(

c

n

)

)

=

ω

B

(

f

(

d

1

)

,

…

,

f

(

d

n

)

)

=

f

(

ω

A

(

d

1

,

…

,

d

n

)

)

,

so

ω

A

⁢

(

c

1

,

…

,

c

n

)

≡

ω

A

⁢

(

d

1

,

…

,

d

n

)

(

mod

ker

⁡

(

f

)

)

. For this reason,

ker

⁡

(

f

)

is also called the

congruence

induced by

f

.

Example

. If

A

,

B

are groups and

f

:

A

→

B

is a

group homomorphism

. Then the kernel of

f

, using the definition above is just the union of the square of the cosets of

N

=

{

x

∣

f

⁢

(

x

)

=

e

}

,

the traditional definition of the kernel of a group homomorphism (where

e

is the

identity

of

B

).

Remark

. The above can be generalized. See the analog (

http://planetmath.org/KernelOfAHomomorphismIsACongruence

) in

model theory

.

Title

kernel of a homomorphism between algebraic systems

Canonical name

KernelOfAHomomorphismBetweenAlgebraicSystems

Date of creation

2013-03-22 16:26:20

Last modified on

2013-03-22 16:26:20

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A05

Synonym

induced congruence

Related topic

KernelOfAHomomorphismIsACongruence

Related topic

KernelPair

Defines

congruence induced by a homomorphism

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