subalgebra of an algebraic system

subalgebra of an algebraic system

Let

(

A

,

O

)

be an

algebraic system

(

A

≠

∅

is the underlying set and

O

is the set of

operators

on

A

).

Subalgebras

of an

Algebra

Let

B

be a non-empty subset of

A

.

B

is

closed

under operators of

A

if for each

n

-ary operator

ω

A

on

A

, and any

b

1

,

…

,

b

n

∈

B

, we have

ω

A

⁢

(

b

1

,

…

,

b

n

)

∈

B

.

Suppose

B

is

closed under

operators of

A

. For each

n

-ary operator

ω

A

on

A

, we define

ω

B

:

B

n

→

B

by

ω

B

⁢

(

b

1

,

…

,

b

n

)

:=

ω

A

⁢

(

b

1

,

…

,

b

n

)

. Each of these operators is well-defined and is called a

restriction

(of the corresponding

ω

A

). Furthermore,

(

B

,

O

)

is a well-defined algebraic system, and is called the

subalgebra

of

(

A

,

O

)

. When

(

B

,

O

)

is a subalgebra of

(

A

,

O

)

, we also say that

(

A

,

O

)

is an

extension

of

(

B

,

O

)

.

(

A

,

O

)

is clearly a subalgebra of itself. Any other subalgebra of

(

A

,

O

)

is called a

proper subalgebra

.

Remark

. If

(

A

,

O

)

contains

constants

, then any subalgebra of

(

A

,

O

)

must contain the exact same constants. For example, the ring

ℤ

of integers is an algebraic system with no proper subalgebras. Indeed, if

R

is a

subring

of

ℤ

,

1

∈

R

, so

R

=

ℤ

.

Since we are operating under the same operator set, we can, for convenience, drop

O

and simply call

A

an algebra,

B

a subalgebra of

A

, etc… If

B

1

,

B

2

are subalgebras of

A

, then

B

1

∩

B

2

is also a subalgebra. In fact, given any set of subalgebras

B

i

of

A

, their

intersection

⋂

B

i

is also a subalgebra.

Generating Set of an Algebra

Let

C

be any subset of an algebra

A

. Consider the

collection

[

C

]

of all subalgebras of

A

containing

C

. This collection is non-empty because

A

∈

[

C

]

. The intersection of all these subalgebras is again a subalgebra containing the set

C

. Denote this subalgebra by

⟨

C

⟩

.

⟨

C

⟩

is called the subalgebra

spanned

by

C

, and

C

is called the

spanning set

of

⟨

C

⟩

.

Conversely

, any subalgebra

B

of

A

has a spanning set, namely itself:

B

=

⟨

B

⟩

.

Given a subalgebra

B

of

A

, a

minimal

spanning set

X

of

B

is called a

generating set

of

B

. By minimal we mean that the set obtained by deleting any

element

from

X

no longer spans

B

. When

B

has a generating set

X

, we also say that

X

generates

B

. If

B

can be generated by a

finite set

, we say that

B

is

finitely generated

. If

B

can be generated by a single element, we say that

B

is

cyclic

.

Remark

.

⟨

∅

⟩

=

the subalgebra generated by the constants of

A

. If no such constants exist,

⟨

∅

⟩

:=

∅

.

From the discussion above, the set of subalgebras of an algebraic system forms a

complete lattice

. Given subalgebras

A

i

,

⋁

A

i

is the intersection of all

A

i

, and

⋁

A

i

is the subalgebra

⟨

⋃

A

i

⟩

. The

lattice

of all subalgebras of

A

is called the

subalgebra latttice

of

A

, and is denoted by

Sub

⁡

(

A

)

.

Title

subalgebra of an algebraic system

Canonical name

SubalgebraOfAnAlgebraicSystem

Date of creation

2013-03-22 16:44:19

Last modified on

2013-03-22 16:44:19

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A30

Classification

msc 08A05

Classification

msc 08A62

Synonym

subalgebra lattice

Defines

subalgebra

Defines

generating set

Defines

subalgebra generated by

Defines

extension of an algebraic system

Defines

restriction

Defines

proper subalgebra

Defines

lattice of subalgebras

Defines

spanning set

Defines

finitely generated

Defines

cyclic

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