equational class

equational class

Let

K

be a class of

algebraic systems

of the same type. Consider the following “

operations

” on

K

:

1.

S

⁢

(

K

)

is the class of

subalgebras

of

algebras

in

K

,

2.

P

⁢

(

K

)

is the class of

direct products

of non-empty

collections

of algebras in

K

, and

3.

H

⁢

(

K

)

is the class of

homomorphic images

of algebras in

K

.

It is clear that

K

is a

subclass

of

S

⁢

(

K

)

,

P

⁢

(

K

)

, and

H

⁢

(

K

)

.

An

equational class

is a class

K

of algebraic systems such that

S

⁢

(

K

)

,

P

⁢

(

K

)

, and

H

⁢

(

K

)

are subclasses of

K

. An equational class is also called a

variety

.

A subclass

L

of a variety

K

is called a

subvariety

of

K

if

L

is a variety itself.

Examples

.

•

In the variety of groups, the classes of

abelian groups

is equational. However, the following classes are not:

simple groups

,

cyclic groups

,

finite groups

, and

divisible groups

.

•

In the variety of rings, the classes of

commutative rings

and

Boolean rings

are varieties. Most classes of rings, however, are not equational. For example, the class of

Noetherian rings

is not equational, as infinite products of Noetherian rings are not

Noetherian

.

•

In the variety of lattices, the classes of

modular

lattices and

distributive

lattices are equational, while

complete

lattices and

complemented

lattices are not.

•

The class of

Heyting algebras

is equational, and so is the subclass of

Boolean algebras

.

•

The class of

torsion free

abelian groups is

not

equational. For example, the homomorphic image of

ℤ

under the

canonical map

ℤ

↦

ℤ

n

is not torsion free.

Remarks

.

•

If

A

,

B

are any of

H

,

S

,

P

, we define

A

⁢

B

⁢

(

K

)

:=

A

⁢

(

B

⁢

(

K

)

)

for any class

K

of algebras, and write

A

⊆

B

iff

A

⁢

(

K

)

⊆

B

⁢

(

K

)

. Then

S

⁢

H

⊆

H

⁢

S

,

P

⁢

H

⊆

H

⁢

P

and

P

⁢

S

⊆

S

⁢

P

.

•

If

C

is any one of

H

,

S

,

P

, then

C

2

:=

C

⁢

C

=

C

.

•

If

K

is any

class of algebras

, then

H

⁢

S

⁢

P

⁢

(

K

)

is an equational class.

•

For any class of algebras, let

P

S

⁢

(

K

)

be the family of all

subdirect products

of all non-empty collections of algebras of

K

. Then

H

⁢

S

⁢

P

⁢

(

K

)

=

H

⁢

P

S

⁢

(

K

)

.

•

The reason for call such classes “equational” is due to the fact that algebras within the same class all

satisfy

a set of “

equations

”, or “

identities

(

http://planetmath.org/IdentityInAClass

)”. Indeed, a famous

theorem

of Birkhoff says:

a class

V

of algebras is equational iff there is a set

Σ

of identities (or equations) such that

K

is the smallest class of algebras where each algebra

A

∈

V

is satisfied by every identity

e

∈

Σ

. In other words,

V

is the set of all models of

Σ

:

V

=

Mod

(

Σ

)

=

{

A

is a structure

∣

(

∀

e

∈

Σ

)

→

(

A

⊧

e

)

}

.

References

1

G. Grätzer:

Universal Algebra

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

Title

equational class

Canonical name

EquationalClass

Date of creation

2013-03-22 16:48:02

Last modified on

2013-03-22 16:48:02

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

19

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08B99

Classification

msc 03C05

Synonym

variety of algebras

Synonym

primitive class

Related topic

VarietyOfGroups

Defines

variety

Defines

subvariety

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