partial algebraic system

partial algebraic system

Let

λ

be a

cardinal

. A

partial function

f

:

A

λ

→

A

is called a

partial operation

on

A

.

λ

is called the arity of

f

. When

λ

is finite,

f

is said to be

finitary

. Otherwise, it is

infinitary

. A nullary partial operation is an element of

A

and is called a

constant

.

Definition

. A

partial algebraic system

(or

partial algebra

for short) is defined as a pair

(

A

,

O

)

, where

A

is a set, usually non-empty, and called the underlying set of the

algebra

, and

O

is a set of finitary partial operations on

A

. The partial algebra

(

A

,

O

)

is sometimes denoted by

𝑨

.

Partial algebraic systems sit between

algebraic systems

and

relational systems

; they are

generalizations

of algebraic systems, but special cases of relational systems.

The

type

of a partial algebra is defined exactly the same way as that of an algebra. When we speak of a partial algebra

𝑨

of type

τ

, we typically mean that

𝑨

is

proper

, meaning that the partial operation

f

𝑨

is non-empty for every

function symbol

f

∈

τ

, and if

f

is a constant symbol,

f

𝑨

∈

A

.

Below is a short list of partial algebras.

1.

Every algebraic system is automatically a partial algebraic system.

2.

A

division ring

(

D

,

{

+

,

⋅

,

-

,

-

1

⁢

,

⁢

0

⁢

,

⁢

1

}

)

is a prototypical example of a partial algebra that is not an algebra. It has type

⟨

2

,

2

,

1

,

1

,

0

,

0

⟩

. It is not an algebra because the unary operation

-

1

(

multiplicative inverse

) is only partial, not defined for

0

.

3.

Let

A

be the set of all non-negative integers. Let “

-

” be the ordinary

subtraction

. Then

(

A

,

{

-

}

)

is a partial algebra.

4.

A

partial groupoid

is a partial algebra of type

⟨

2

⟩

. In other words, it is a set with a partial

binary operation

(called the

product

) on it. For example, a

small category

may be viewed as a partial algebra. The product

a

⁢

b

is only defined when the source of

a

matches with the target of

b

. Special types of small categories are

groupoids

(

category

theoretic) (

http://planetmath.org/GroupoidCategoryTheoretic

), and

Brandt groupoids

, all of which are partial.

5.

A small category can also be thought of as a partial algebra of type

⟨

2

,

1

,

1

⟩

, where the two (total) unary

operators

are the source and target

operations

.

Remark

. Like algebraic systems, one can define

subalgebras

,

direct products

,

homomorphisms

, as well as

congruences

in partial algebras.

References

1

G. Grätzer:

Universal Algebra

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

Title

partial algebraic system

Canonical

name

PartialAlgebraicSystem

Date of creation

2013-03-22 18:42:10

Last modified on

2013-03-22 18:42:10

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

28

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03E99

Classification

msc 08A55

Classification

msc 08A62

Synonym

partial operator

Synonym

partial algebra

Related topic

RelationalSystem

Defines

partial operation

Defines

partial groupoid

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