subalgebra of a partial algebra

subalgebra of a partial algebra

Unlike an

algebraic system

, where there is only one way to define a

subalgebra

, there are several ways to define a subalgebra of a

partial algebra

.

Suppose

𝑨

and

𝑩

are partial algebras of type

τ

:

1.

𝑩

is a

weak subalgebra

of

𝑨

if

B

⊆

A

, and

f

𝑩

is a

subfunction

of

f

𝑨

for every operator symbol

f

∈

τ

.

In words,

𝑩

is a weak subalgebra of

𝑨

iff

B

⊆

A

, and for each

n

-ary symbol

f

∈

τ

, if

b

1

,

…

,

b

n

∈

B

such that

f

B

⁢

(

b

1

,

…

,

b

n

)

is defined, then

f

A

⁢

(

b

1

,

…

,

b

n

)

is also defined, and is equal to

f

B

⁢

(

b

1

,

…

,

b

n

)

.

2.

𝑩

is a

relative subalgebra

of

𝑨

if

B

⊆

A

, and

f

𝑩

is a restriction of

f

𝑨

relative to

B

(

http://planetmath.org/Subfunction

) for every operator symbol

f

∈

τ

.

In words,

𝑩

is a relative subalgebra of

𝑨

iff

B

⊆

A

, and for each

n

-ary symbol

f

∈

τ

, given

b

1

,

…

,

b

n

∈

B

,

f

B

⁢

(

b

1

,

…

,

b

n

)

is defined iff

f

A

⁢

(

b

1

,

…

,

b

n

)

is and belongs to

B

, and they are equal.

3.

𝑩

is a

subalgebra

of

𝑨

if

B

⊆

A

, and

f

𝑩

is a restriction (

http://planetmath.org/Subfunction

) of

f

𝑨

for every operator symbol

f

∈

τ

.

In words,

𝑩

is a subalgebra of

𝑨

iff

B

⊆

A

, and for each

n

-ary symbol

f

∈

τ

, given

b

1

,

…

,

b

n

∈

B

,

f

B

⁢

(

b

1

,

…

,

b

n

)

is defined iff

f

A

⁢

(

b

1

,

…

,

b

n

)

is, and they are equal.

Notice that if

𝑩

is a weak subalgebra of

𝑨

, then every

constant

of

𝑩

is a constant of

𝑨

, and vice versa.

Every subalgebra is a relative subalgebra, and every relative subalgebra is a weak subalgebra. But the

converse

is false for both statements. Below are two examples.

1.

Let

F

be a field. Then every subalgebra of

F

is a subfield, and every relative subalgebra of

F

is a subring.

2.

Let

A

be the set of all non-negative integers, and

-

A

the ordinary

subtraction

on integers. Consider the partial algebra

(

A

,

-

A

)

.

–

Let

B

=

A

and

-

B

the usual subtraction on integers, but

x

-

B

y

is only defined when

x

,

y

∈

B

have the same parity. Then

(

B

,

-

B

)

is a weak subalgebra of

(

A

,

-

A

)

.

–

Let

C

be the

set of all positive integers

, and

-

C

the ordinary subtraction. Then

(

C

,

-

C

)

is a relative subalgebra of

(

A

,

-

A

)

.

–

Let

D

be the set

{

0

,

1

,

…

,

n

}

and

-

D

the ordinary subtraction. Then

(

D

,

-

D

)

is a subalgebra of

(

A

,

-

A

)

.

Notice that

(

B

,

-

B

)

is not a relative subalgebra of

(

A

,

-

A

)

, since

7

-

B

6

is not defined, even though

7

-

A

⁢

6

=

1

∈

B

, and and

(

C

,

-

C

)

is not a subalgebra of

(

A

,

-

A

)

, since

1

-

C

1

is not defined in

C

, even though

1

-

A

⁢

1

is defined in

A

.

Remarks

.

1.

A weak subalgebra

𝑩

of

𝑨

is a relative subalgebra iff given

b

1

,

…

,

b

n

∈

B

such that

f

A

⁢

(

b

1

,

…

,

b

n

)

is defined and is in

B

, then

f

B

⁢

(

b

1

,

…

,

b

n

)

is defined. A relative subalgebra

𝑩

of

𝑨

is a subalgebra iff whenever

f

A

⁢

(

b

1

,

…

,

b

n

)

is defined for

b

i

∈

B

, it is in

B

.

2.

Let

𝑨

be a partial algebra of type

τ

, and

B

⊆

A

. For each

n

-ary

function symbol

f

∈

τ

, define

f

𝑩

on

B

as follows:

f

𝑩

⁢

(

b

1

,

…

,

b

n

)

is defined in

B

iff

f

𝑨

⁢

(

b

1

,

…

,

b

n

)

is defined in

A

and

f

𝑨

⁢

(

b

1

,

…

,

b

n

)

∈

B

. This turns

𝑩

into a partial algebra. However,

𝑩

may not be of type

τ

, since

f

𝑩

may not be defined at all on

B

. When

𝑩

is a partial algebra of type

τ

, it is a relative subalgebra of

𝑨

.

3.

When

𝑨

is an

algebra

, all three notions of subalgebras are

equivalent

(assuming that the partial operations on a weak subalgebra are all total).

References

1

G. Grätzer:

Universal Algebra

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

Title

subalgebra of a partial algebra

Canonical

name

SubalgebraOfAPartialAlgebra

Date of creation

2013-03-22 18:42:54

Last modified on

2013-03-22 18:42:54

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A55

Classification

msc 03E99

Classification

msc 08A62

Defines

weak subalgebra

Defines

relative subalgebra

Defines

subalgebra

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