homomorphism between algebraic systems

homomorphism between algebraic systems

Let

(

A

,

O

)

,

(

B

,

O

)

be two

algebraic systems

with operator set

O

. Given

operators

ω

A

on

A

and

ω

B

on

B

, with

ω

∈

O

and

n

=

arity of

ω

, a

function

f

:

A

→

B

is said to be

compatible

with

ω

if

f

⁢

(

ω

A

⁢

(

a

1

,

…

,

a

n

)

)

=

ω

B

⁢

(

f

⁢

(

a

1

)

,

…

,

f

⁢

(

a

n

)

)

.

Dropping the

subscript

, we now simply identify

ω

∈

O

as an operator for both

algebras

A

and

B

. If a function

f

:

A

→

B

is compatible with every operator

ω

∈

O

, then we say that

f

is a

homomorphism

from

A

to

B

. If

O

contains a constant operator

ω

such that

a

∈

A

and

b

∈

B

are two

constants

assigned by

ω

, then any homomorphism

f

from

A

to

B

maps

a

to

b

.

Examples

.

1.

When

O

is the

empty set

, any function from

A

to

B

is a homomorphism.

2.

When

O

is a singleton consisting of a constant operator, a homomorphism is then a function

f

from one

pointed set

(

A

,

p

)

to another

(

B

,

q

)

, such that

f

⁢

(

p

)

=

q

.

3.

A homomorphism defined in any one of the well known algebraic systems, such as groups, modules, rings, and lattices (

http://planetmath.org/

Lattice

) is

consistent

with the more general definition given here. The essential thing to remember is that a homomorphism preserves constants, so that between two rings with 1, both the

additive identity

0 and the

multiplicative identity

1 are preserved by this homomorphism. Similarly, a homomorphism between two bounded lattices (

http://planetmath.org/BoundedLattice

) is called a

{

0

,

1

}

-

lattice homomorphism

(

http://planetmath.org/LatticeHomomorphism

) because it preserves both 0 and 1, the bottom and top elements of the lattices.

Remarks

.

•

Like the familiar algebras, once a homomorphism is defined, special

types of homomorphisms

can now be named:

–

a homomorphism that is

one-to-one

is a

monomorphism

;

–

an onto homomorphism is an

epimorphism

;

–

an

isomorphism

is both a monomorphism and an epimorphism;

–

a homomorphism such that its codomain is its domain is called an

endomorphism

;

–

finally, an

automorphism

is an endomorphism that is also an isomorphism.

•

All trivial algebraic systems (of the same type) are isomorphic.

•

If

f

:

A

→

B

is a homomorphism, then the image

f

⁢

(

A

)

is a

subalgebra

of

B

. If

ω

B

is an

n

-ary operator on

B

, and

c

1

,

…

,

c

n

∈

f

⁢

(

A

)

, then

ω

B

⁢

(

c

1

,

…

,

c

n

)

=

ω

B

⁢

(

f

⁢

(

a

1

)

,

…

,

f

⁢

(

a

n

)

)

=

f

⁢

(

ω

A

⁢

(

a

1

,

…

,

a

n

)

)

∈

f

⁢

(

A

)

.

f

⁢

(

A

)

is sometimes called the

homomorphic image

of

f

in

B

to emphasize the fact that

f

is a homomorphism.

Title

homomorphism between algebraic systems

Canonical name

HomomorphismBetweenAlgebraicSystems

Date of creation

2013-03-22 15:55:36

Last modified on

2013-03-22 15:55:36

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A05

Defines

compatible function

Defines

homomorphism

Defines

monomorphism

Defines

epimorphism

Defines

endomorphism

Defines

isomorphism

Defines

automorphism

Defines

homomorphic image

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