relational system

relational system

A

relational system

, loosely speaking, is a pair

(

A

,

R

)

where

A

is a set and

R

is a set of finitary relations defined on

A

(a finitary relation is just an

n

-ary

relation

where

n

∈

ℕ

; when

n

=

1

, it is called a

property

). Since an

n

-ary

operator

on a set is an

(

n

+

1

)

-ary the set, a relational system can be thought of as a

generalization

of an

algebraic system

. We can formalize the notion of a relation system as follows:

Call a set

R

a relation set, if there is a

function

f

:

R

→

ℕ

, the set of natural numbers. For each

r

∈

R

, call

f

⁢

(

r

)

the arity of

r

.

Let

A

be a set and

R

a

relation set

. The pair

(

A

,

R

)

is called an

R

-relational system if there is a set

R

A

such that

•

R

A

is a set of finitary relations on

A

, called the

relation set

of

A

, and

•

there is a

one-to-one correspondence

between

R

and

R

A

, given by

r

↦

r

A

, such that the

f

⁢

(

r

)

=

the arity of

r

A

.

Since operators and

partial operators

are special types of relations. algebraic systems and partial algebraic systems can be treated as relational systems.

Below are some exmamples of relational systems:

•

any

algebraic

or partial algebraic system.

•

a poset

(

P

,

{

≤

P

}

)

, where

≤

P

is a

binary relation

, called the

partial ordering

, on

P

. A

lattice

, generally considered an algebraic system, can also be considered as a relational system, because it is a poset, and that

≤

alone defines the algebraic operations (

∨

and

∧

).

•

a

pointed set

(

A

,

{

a

}

)

is also a relational system, where a unary relation, or property, is the singled-out

element

a

∈

A

. A pointed set is also an algebraic system, if we treat

a

as the lone nullary operator (

constant

).

•

a

bounded poset

(

P

,

≤

P

,

0

,

1

)

is a relational system. It is a poset, with two unary relations

{

0

}

and

{

1

}

.

•

a

Buekenhout-Tits geometry

can be thought of as a relational system. It consists of a set

Γ

with two binary relations on it: one is an

equivalence relation

T

called type, and the other is a

symmetric

reflexive relation

#

called incidence, such that if

a

⁢

#

⁢

b

and

a

⁢

T

⁢

b

, then

a

=

b

(

incident

objects of the same type are identical).

•

ordered

algebraic structures

, such as

ordered groups

(

G

,

{

⋅

,

-

1

,

e

,

≤

G

}

)

and

ordered rings

(

R

,

{

+

,

-

,

⋅

,

-

1

,

0

,

≤

R

}

)

are also relational systems. They are not algebraic systems because of the additional

ordering relations

(

≤

G

and

≤

R

) defined on these objects. Note that these

orderings

are generally considered

total orders

.

•

ordered partial algebras such as ordered fields

(

D

,

{

+

,

-

,

⋅

,

-

1

,

0

,

1

,

≤

F

}

)

, etc…

•

structures

that are not relational are

complete

lattices (

http://planetmath.org/CompleteLattice

) and topological spaces, because the

operations

involved are infinitary.

Remark

. Relational systems and algebraic systems are both examples of structures in model theory. Although an algebraic system is a relational system in the sense discussed above, they are treated as distinct entities. A structure involves three objects, a set

A

, a set of

function symbols

F

, and a set of relation symbols

R

, so a relational system is a structure where

F

=

∅

and an algebraic system is a structure where

R

=

∅

.

References

1

G. Grätzer:

Universal Algebra

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

Title

relational system

Canonical

name

RelationalSystem

Date of creation

2013-03-22 16:35:33

Last modified on

2013-03-22 16:35:33

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

16

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A55

Classification

msc 03C07

Classification

msc 08A02

Synonym

relational structure

Related topic

AlgebraicSystem

Related topic

PartialAlgebraicSystem

Related topic

Structure

Related topic

StructuresAndSatisfaction

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