homomorphism between partial algebras

homomorphism between partial algebras

Definition

Like

subalgebras of partial algebras

, there are also three ways to define

homomorphisms between partial algebras

.

Similar

to the

definition

of

homomorphisms

between

algebras

, a homomorphism

ϕ

:

𝑨

→

𝑩

between two

partial algebras

of type

τ

is a

function

from

A

to

B

that

satisfies

the

equation

ϕ

⁢

(

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

)

=

f

𝑩

⁢

(

ϕ

⁢

(

a

1

)

,

…

,

ϕ

⁢

(

a

n

)

)

(1)

for every

n

-ary

function symbol

f

∈

τ

. However, because

f

𝑨

and

f

𝑩

are not everywhere defined in their respective domains, care must be taken as to what the equation means.

1.

ϕ

is a

homomorphism

if, given that

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

is defined, so is

f

𝑩

⁢

(

ϕ

⁢

(

a

1

)

,

…

,

ϕ

⁢

(

a

n

)

)

, and equation (1) is satisifed.

2.

ϕ

is a

full homomorphism

if it is a homomorphism and, given that

f

𝑩

⁢

(

b

1

,

…

,

b

n

)

is defined and in

ϕ

⁢

(

A

)

, for

b

i

∈

ϕ

⁢

(

A

)

, there exist

a

i

∈

A

with

b

i

=

ϕ

⁢

(

a

i

)

, such that

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

is defined.

3.

ϕ

is a

strong homomorphism

if it is a homomorphism and, given that

f

𝑩

⁢

(

ϕ

⁢

(

a

1

)

,

…

,

ϕ

⁢

(

a

n

)

)

is defined, so is

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

.

We have the following

implications

:

strong homomorphism

→

full homomorphism

→

homomorphism.

For example, field homomorphisms are strong homomorphisms.

Homomorphisms

preserve

constants

: for each constant symbol

f

in

τ

,

ϕ

⁢

(

f

𝑨

)

=

f

𝑩

. In fact, when restricted to constants,

ϕ

is a

bijection

between constants of

𝑨

and constants of

𝑩

.

When

𝑨

is an algebra (all partial operations are total), a homomorphism from

𝑨

is always strong, so that all three notions of homomorphisms coincide.

An

isomorphism

is a

bijective

homomorphism

ϕ

:

𝑨

→

𝑩

such that its

inverse

ϕ

-

1

:

𝑩

→

𝑨

is also a homomorphism. An

embedding

is an injective homomorphism. Isomorphisms and full embeddings are strong.

Homomorphic Images

The various

types of homomorphisms

and the various types of

subalgebras

are related. Suppose

𝑨

and

𝑩

are partial algebras of type

τ

. Let

ϕ

:

A

→

B

be a function, and

C

=

ϕ

⁢

(

A

)

. For each

n

-ary function symbol

f

∈

τ

, define

n

-ary partial operation

f

𝑪

on

C

as follows:

for

b

1

,

…

,

b

n

∈

C

,

f

𝑪

⁢

(

b

1

,

…

,

b

n

)

is defined iff the set

D

:=

{

(

a

1

,

…

,

a

n

)

∈

A

n

∣

ϕ

⁢

(

a

i

)

=

b

i

}

∩

dom

⁡

(

f

𝑨

)

is non-empty, where

dom

⁡

(

f

𝑨

)

is the

domain of definition

of

f

𝑨

, and when this is the case,

f

𝑪

⁢

(

b

1

,

…

,

b

n

)

:=

ϕ

⁢

(

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

)

, for some

(

a

1

,

…

,

a

n

)

∈

D

.

If

ϕ

preserves constants (if any), and

f

C

is non-empty for each

f

∈

τ

then

𝑪

is a partial algebra of type

τ

.

Fix an arbitrary

n

-ary symbol

f

∈

τ

. The following are the basic

properties

of

𝑪

:

Proposition 1

.

ϕ

is a homomorphism iff

𝐂

is a weak subalgebra of

𝐁

.

Proof.

Suppose first that

ϕ

is a homomorphism. If

n

=

0

, then

f

𝑨

∈

A

, and

f

𝑩

=

ϕ

⁢

(

f

𝑨

)

∈

C

. If

n

>

0

, then for some

a

1

,

…

,

a

n

∈

A

,

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

is defined, and consequently

f

𝑩

⁢

(

ϕ

⁢

(

a

1

)

,

…

,

ϕ

⁢

(

a

n

)

)

is defined, and is equal to

ϕ

⁢

(

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

)

∈

C

. By the definition for

f

𝑪

above,

f

𝑪

⁢

(

ϕ

⁢

(

a

1

)

,

…

,

ϕ

⁢

(

a

n

)

)

:=

ϕ

⁢

(

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

)

. This shows that

𝑪

is a

τ

-algebra.

To furthermore show that

𝑪

is a weak subalgebra of

𝑩

, assume

f

𝑪

⁢

(

b

1

,

…

,

b

n

)

is defined. Then there are

a

1

,

…

,

a

n

∈

A

with

b

i

=

ϕ

⁢

(

a

i

)

such that

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

is defined. Since

ϕ

is a homomorphism,

f

𝑩

⁢

(

ϕ

⁢

(

a

1

)

,

…

,

ϕ

⁢

(

a

n

)

)

, and hence

f

𝑩

⁢

(

b

1

,

…

,

b

n

)

, is defined. Furthermore,

f

C

⁢

(

b

1

,

…

,

b

n

)

=

ϕ

⁢

(

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

)

=

f

𝑩

⁢

(

ϕ

⁢

(

a

1

)

,

…

,

ϕ

⁢

(

a

n

)

)

=

f

𝑩

⁢

(

b

1

,

…

,

b

n

)

. This shows that

𝑪

is weak.

On the other hand, suppose now that

𝑪

is a weak subalgebra of

𝑩

. Suppose

a

1

,

…

,

a

n

∈

A

and

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

is defined. Let

b

i

=

ϕ

⁢

(

a

i

)

∈

C

. Then, by the definition of

f

𝑪

,

f

𝑪

⁢

(

b

1

,

…

,

b

n

)

is defined and is equal to

ϕ

⁢

(

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

)

. Since

𝑪

is weak,

f

𝑩

⁢

(

b

1

,

…

,

b

n

)

is defined and is equal to

f

𝑪

⁢

(

b

1

,

…

,

b

n

)

. As a result,

ϕ

⁢

(

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

)

=

f

𝑪

⁢

(

b

1

,

…

,

b

n

)

=

f

𝑩

⁢

(

b

1

,

…

,

b

n

)

=

f

𝑩

⁢

(

ϕ

⁢

(

a

1

)

,

…

,

ϕ

⁢

(

a

n

)

)

. Hence

ϕ

is a homomorphism.
∎

Proposition 2

.

ϕ

is a full homomorphism iff

𝐂

is a relative subalgebra of

𝐁

.

Proof.

Suppose first that

ϕ

is full. Since

ϕ

is a homomorphism,

𝑪

is weak. Suppose

b

1

,

…

,

b

n

∈

C

such that

f

𝑨

⁢

(

b

1

,

…

,

b

n

)

is defined and is in

C

. Since

ϕ

is full, there are

a

i

∈

A

such that

b

i

=

ϕ

⁢

(

a

i

)

and

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

is defined, and

ϕ

⁢

(

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

)

=

f

𝑩

⁢

(

ϕ

⁢

(

a

1

)

,

…

,

ϕ

⁢

(

a

n

)

)

=

f

𝑩

⁢

(

b

1

,

…

,

b

n

)

, so that

f

𝑩

⁢

(

b

1

,

…

,

b

n

)

is defined and thus

𝑪

is a relative subalgebra of

𝑩

.

Conversely

, suppose that

𝑪

is a relative subalgebra of

𝑩

. Then

𝑪

is a weak subalgebra of

𝑩

and

ϕ

is a homomorphism. To show that

ϕ

is full, suppose that

b

i

∈

C

such that

f

𝑩

⁢

(

b

1

,

…

,

b

n

)

is defined in

C

. Then

f

𝑪

⁢

(

b

1

,

…

,

b

n

)

is defined in

C

and is equal to

f

𝑩

⁢

(

b

1

,

…

,

b

n

)

. This means that there are

a

i

∈

A

such that

b

i

=

ϕ

⁢

(

a

i

)

, and

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

is defined, showing that

f

𝑨

is full.
∎

Proposition 3

.

ϕ

is a strong homomorphism iff

𝐂

is a subalgebra of

𝐁

.

Proof.

Suppose first that

ϕ

is strong. Since

ϕ

is full,

𝑪

is a relative subalgebra of

𝑩

. Suppose now that for

b

i

∈

C

,

f

𝑩

⁢

(

b

1

,

…

,

b

n

)

is defined. Since

b

i

=

ϕ

⁢

(

a

i

)

for some

a

i

∈

A

, and since

ϕ

is strong,

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

is defined. This means that

f

𝑩

⁢

(

b

1

,

…

,

b

n

)

=

f

𝑩

⁢

(

ϕ

⁢

(

a

1

)

,

…

,

ϕ

⁢

(

a

n

)

)

=

ϕ

⁢

(

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

)

, which is in

C

. So

𝑪

is a subalgebra of

𝑩

.

Going the other direction, suppose now that

𝑪

is a subalgebra of

𝑩

. Since

𝑪

is a relative subalgebra of

𝑩

,

ϕ

is full. To show that

ϕ

is strong, suppose

f

𝑩

⁢

(

ϕ

⁢

(

a

1

)

,

…

,

ϕ

⁢

(

a

n

)

)

is defined. Then

f

𝑪

⁢

(

ϕ

⁢

(

a

1

)

,

…

,

ϕ

⁢

(

a

n

)

)

is defined and is equal to

f

𝑩

⁢

(

ϕ

⁢

(

a

1

)

,

…

,

ϕ

⁢

(

a

n

)

)

. By definition of

f

𝑪

,

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

is therefore defined. So

ϕ

is strong.
∎

Definition

. Let

𝑨

and

𝑩

be partial algebras of type

τ

. If

ϕ

:

𝑨

→

𝑩

is a homomorphism, then

𝑪

, as defined above, is a partial algebra of type

τ

, and is called the

homomorphic image

of

A

via

ϕ

, and is sometimes written

ϕ

⁢

(

𝑨

)

.

References

1

G. Grätzer:

Universal Algebra

, 2nd Edition, Springer, New York (1978).

Title

homomorphism between partial algebras

Canonical

name

HomomorphismBetweenPartialAlgebras

Date of creation

2013-03-22 18:42:57

Last modified on

2013-03-22 18:42:57

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A55

Classification

msc 03E99

Classification

msc 08A62

Defines

homomorphism

Defines

full homomorphism

Defines

strong homomorphism

Defines

isomorphism

Defines

strong

Defines

homomorphic image

Defines

embedding

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