examples of algebraic systems

examples of algebraic systems

Selected

examples of algebraic systems

are specified below.

1.

A set is an

algebra

where

τ

=

∅

.

2.

A

pointed set

is an algebra of type

⟨

0

⟩

, where

0

corresponds to the designated

element

in the set.

3.

An algebra of type

⟨

2

⟩

is called a

groupoid

. Another algebra of this type is a

semigroup

.

4.

A monoid is an algebra of type

⟨

2

,

0

⟩

. However, not every algebra of type

⟨

2

,

0

⟩

is a monoid.

5.

A group is an

algebraic system

of type

⟨

2

,

1

,

0

⟩

, where

2

corresponds to the arity of the

multiplication

,

1

the

multiplicative inverse

, and

0

the multiplicative identity.

6.

A ring is an algebraic system of type

⟨

2

,

2

,

1

,

0

,

0

⟩

, where the two

2

’s

represent

the arities of addition and multiplication,

1

the

additive inverse

, and

0

’s the

additive

and multiplicative identities.

7.

A

lattice

is an algebraic system of type

⟨

2

,

2

⟩

. The two

binary operations

are meet and join.

8.

A

bounded lattice

is an algebraic system of type

⟨

2

,

2

,

0

,

0

⟩

. Besides the meet and join

operations

, it has two

constants

, its top

1

and bottom

0

.

9.

A

uniquely complemented lattice

is an algebraic system of the type

⟨

2

,

2

,

1

,

0

,

0

⟩

. In addition to having the operations of a bounded lattice, there is a unary

operator

taking each element to its unique

complement

. Note that it has the same type as the type of a group.

10.

A

quandle

is an algebraic system of type

⟨

2

,

2

,

⟩

. It has the same type as a lattice.

11.

A

quasigroup

may be thought of as a algebraic system of type

⟨

2

⟩

, that of a groupoid, or

⟨

2

,

2

,

2

⟩

, depending on the definition used. A loop, as a quasigroup with an

identity

, is an algebraic system of type

⟨

0

,

q

⟩

, where

q

is the type of a quasigroup.

12.

An

n

-group (

http://planetmath.org/PolyadicSemigroup

) is an algebraic system of type

⟨

n

⟩

.

13.

A

left module

over a ring

R

is an algebraic system. Its type is

⟨

2

,

1

,

(

1

)

r

∈

R

,

0

⟩

, where

2

is the arity of addition, the first

1

the additive inverse, and the rest of the

1

’s represent the arity of left scalar multiplication by

r

, for each

r

∈

R

, and finally

0

the (arity) of

additive identity

.

14.

The set

V

¯

of all

well-formed formulas

over a set

V

of

propositional variables

can be thought of as an algebraic system, as each of the

logical connectives

as an operation on

V

¯

may be associated with a finitary operation on

V

¯

. In classical

propositional logic

, the algebraic system may be of type

⟨

1

,

2

⟩

, if we consider

¬

and

∨

as the only logical connectives; or it may be of type

⟨

1

,

2

,

2

,

2

,

2

⟩

, if the full set

{

¬

,

∨

,

∧

,

→

,

↔

}

is used.

Below are some non-examples of algebraic systems:

1.

A

complete lattice

is not, in general, an algebraic system because the arbitrary meet and join operations are not finitary.

2.

A field is not an algebraic system, since, in addition to the five operations of a ring, there is the multiplicative inverse operation, which is not defined for

0

.

3.

A

small category

may be defined as a set with one partial binary operation on it. Unless the

category

has only one object (so that the operation is everywhere defined), it is in general not an algebraic system.

References

1

G. Grätzer:

Universal Algebra

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

2

P. Jipsen:

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

structures

?HomePage

Mathematical Structures: Homepage

Title

examples of algebraic systems

Canonical name

ExamplesOfAlgebraicSystems

Date of creation

2013-03-22 18:40:11

Last modified on

2013-03-22 18:40:11

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

16

Author

CWoo (3771)

Entry type

Example

Classification

msc 08A05

Classification

msc 03E99

Classification

msc 08A62

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