algebraic system

algebraic system

An

algebraic system

, loosely speaking, is a set, together with some

operations

on the set. Before formally defining what an algebraic system is, let us recall that a

n

-ary operation (or

operator

) on a set

A

is a

function

whose domain is

A

n

and whose range is a subset of

A

. Here,

n

is a non-negative

integer

. When

n

=

0

, the operation is usually called a nullary operation, or a

constant

, since one element of

A

is singled out to be the (sole) value of this operation. A finitary operation on

A

is just an

n

-ary operation for some non-negative integer

n

.

Definition

. An

algebraic system

is an

ordered pair

(

A

,

O

)

, where

A

is a set, called the underlying set of the algebraic system, and

O

is a set, called the operator set, of finitary operations on

A

.

We usually write

𝑨

, instead of

(

A

,

O

)

, for brevity.

A prototypical example of an algebraic system is a group, which consists of the underlying set

G

, and a set

O

consisting of three operators: a constant

e

called the

multiplicative identity

, a unary operator called the multiplicative inverse, and a

binary operator

called the

multiplication

.

For a more comprehensive listing of examples, please see this entry (

http://planetmath.org/ExamplesOfAlgebraicSystems

).

Remarks

.

•

An algebraic system is also called an

algebra

for short. Some authors require that

A

be non-empty. Note that

A

is automatically non-empty if

O

contains constants. A

finite algebra

is an algebra whose underlying set is finite.

•

By definition, all operators in an algebraic system are finitary. If we allow

O

to contain infinitary operations, we have an

infinitary algebraic system

. Other

generalizations

are possible. For example, if the operations are allowed to be

multivalued

, the algebra is said to be a

multialgebra

. If the operations are not everywhere defined, we get a

partial algebra

. Finally, if more than one underlying set is involved, then the algebra is said to be

many-sorted

.

The study of algebraic systems is called the theory of

universal algebra

. The first important thing in studying algebraic system is to compare systems that are of the same ‘‘type’’. Two algebras are said to have the same

type

if there is a

one-to-one correspondence

between their operator sets such that an

n

-ary operator in one algebra is mapped to an

n

-ary operator in the other algebra. A more formal way of doing this is to define what a

type

is:

Definition

. A

type

is a set

τ

, whose

elements

are called operator symbols, such that there is a function

a

:

τ

→

ℕ

∪

{

0

}

. Given an operator symbol

f

, its image

a

⁢

(

f

)

is called the arity of

f

.

Remark

. It is often the practice to well-order

τ

, and write

τ

as a

sequence

of non-negative integers

⟨

a

(

f

1

)

,

a

(

f

2

)

,

…

,

⟩

. When

τ

is finite, the convention is to order the sequence in non-increasing order:

a

⁢

(

f

1

)

≥

a

⁢

(

f

2

)

≥

⋯

≥

a

⁢

(

f

n

)

.

Definition

. An algebraic system

𝑨

is said to be of type

τ

if there is a bijection between

O

and

τ

so that every operator symbol

f

in

τ

corresponds to an operator

f

𝑨

of arity

a

⁢

(

f

)

in

O

. When the algebra

𝑨

is said to be of type

τ

, we also say that

𝑨

is a

τ

-algebra.

For example, a group is an algebraic system of type

⟨

2

,

1

,

0

⟩

, where

2

is the arity of the group multiplication,

1

is the arity of the group inverse, and

0

is the arity of the group multiplicative identity.

References

1

А. И. Мальцев:

Алгебраические
системы

.  Издательство
‘‘Наука’’. Москва (1970).

2

P. M. Cohn:

Universal Algebra

, Harper & Row, (1965).

3

G. Grätzer:

Universal Algebra

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

4

P. Jipsen:

http://math.chapman.edu/cgi-bin/

structures

?HomePage

Mathematical Structures: Homepage

Title

algebraic system

Canonical name

AlgebraicSystem

Date of creation

2013-03-22 15:44:37

Last modified on

2013-03-22 15:44:37

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

50

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A05

Classification

msc 03E99

Classification

msc 08A62

Synonym

algebraic structure

Synonym

universal algebra

Synonym

signature

Synonym

trivial algebra

Related topic

RelationalSystem

Related topic

Model

Related topic

StructuresAndSatisfaction

Related topic

PartiallyOrderedAlgebraicSystem

Defines

n

-ary operator

Defines

finitary operator

Defines

infinitary operator

Defines

operator set

Defines

constant operator

Defines

operator symbol

Defines

nullary operator

Defines

type

Defines

trivial algebraic system

Defines

finite algebra

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