relation theory

relation theory

This article treats

relations

from the perspective of

combinatorics

, in other words, as a subject matter in

discrete mathematics

, with special attention to finite

structures

and concrete set-theoretic constructions, many of which arise quite naturally in applications. This approach to

relation theory

, or the

theory of relations

, is distinguished from, though closely related to, its study from the perspectives of

abstract algebra

on the one hand and

formal logic

on the other.

Contents:

1

Preliminaries

2

Definition

3

Remarks

4

Local incidence properties

5

Regional incidence properties

6

Numerical incidence properties

7

Species of 2-adic relations

8

Variations

9

Bibliography

1

Preliminaries

Two

definitions

of the relation

concept

are common in the literature. Although it is usually clear in context which definition is being used at a given time, it tends to become less clear as contexts collide, or as discussion moves from one context to another.

The same sort of ambiguity arose in the

development

of the

function

concept and it may save some effort to follow the pattern of

resolution

that worked itself out there.

When we speak of a function

f

:

X

→

Y

we are thinking of a mathematical object whose articulation requires three pieces of data, specifying the set

X

,

the set

Y

,

and a particular subset of their

cartesian product

X

×

Y

.

So far so good.

Let us write

f

=

(

obj

1

⁡

f

,

obj

2

⁡

f

,

obj

12

⁡

f

)

to express what has been said so far.

When it comes to parsing the notation

`

⁢

`

⁢

f

:

X

→

Y

⁢

"

,

everyone takes the part

`

⁢

`

⁢

X

→

Y

⁢

"

to specify the

type

of the function, that is, the pair

(

obj

1

⁡

f

,

obj

2

⁡

f

)

,

but

`

⁢

`

⁢

f

⁢

"

is used equivocally to denote both the triple and the subset

obj

12

⁡

f

that forms one part of it. One way to resolve the ambiguity is to formalize a distinction between a function and its

graph

, letting

graph

⁡

(

f

)

:=

obj

12

⁡

f

.

Another tactic treats the whole notation

`

⁢

`

⁢

f

:

X

→

Y

⁢

"

as

sufficient

denotation

for the triple, letting

`

⁢

`

⁢

f

⁢

"

denote

graph

⁡

(

f

)

.

In categorical and computational contexts, at least initially, the type is regarded as an essential

attribute

or an integral part of the function itself. In other contexts it may be desirable to use a more abstract concept of function, treating a function as a mathematical object that appears in

connection

with many different types.

Following the pattern of the

functional

case, let the notation

`

⁢

`

⁢

L

⊆

X

×

Y

⁢

"

bring to mind a mathematical object that is specified by three pieces of data, the set

X

,

the set

Y

,

and a particular subset of their cartesian product

X

×

Y

.

As before we have two choices, either let

L

=

(

X

,

Y

,

graph

⁡

(

L

)

)

or let

`

⁢

`

⁢

L

⁢

"

denote

graph

⁡

(

L

)

and choose another name for the triple.

2

Definition

It is convenient to begin with the definition of a

k

-place relation

, where

k

is a positive integer.

Definition.

A

k

-place relation

L

⊆

X

1

×

…

×

X

k

over the

nonempty sets

X

1

,

…

,

X

k

is a

(

k

+

1

)

-tuple

(

X

1

,

…

,

X

k

,

L

)

where

L

is a subset of the cartesian product

X

1

×

…

×

X

k

.

3

Remarks

Though usage varies as usage will, there are several bits of optional

language

that are frequently useful in discussing relations. The sets

X

1

,

…

,

X

k

are called the

domains

of the relation

L

⊆

X

1

×

…

×

X

k

,

with

X

j

being the

j

th

domain. If all of the

X

j

are the same set

X

,

then

L

⊆

X

1

×

…

×

X

k

is more simply described as a

k

-place relation over

X

.

The set

L

is called the

graph

of the relation

L

⊆

X

1

×

…

×

X

k

,

on

analogy

with the graph of a function. If the

sequence

of sets

X

1

,

…

,

X

k

is

constant

throughout a given discussion or is otherwise determinate in context, then the relation

L

⊆

X

1

×

…

×

X

k

is determined by its graph

L

,

making it acceptable to denote the relation by referring to its graph. Other synonyms for the adjective

k

-

place

are

k

-

adic

and

k

-

ary

, all of which leads to the integer

k

being called the

dimension

, the

adicity

, or the

arity

of the relation

L

.

4

Local incidence properties

A

local incidence property

(LIP) of a relation

L

is a property that depends in turn on the properties of special subsets of

L

that are known as its

local flags

. The local flags of a relation are defined in the following way:

Let

L

be a

k

-place relation

L

⊆

X

1

×

…

×

X

k

.

Select a relational domain

X

j

and one of its

elements

x

.

Then

L

x

⁢

@

⁢

j

is a subset of

L

that is referred to as the

flag

of

L

with

x

at

j

,

or the

x

⁢

@

⁢

j

-flag of

L

,

an object that has the following definition:

L

x

⁢

@

⁢

j

=

{

(

x

1

,

…

,

x

j

,

…

,

x

k

)

∈

L

:

x

j

=

x

}

.

Any property

C

of the local flag

L

x

⁢

@

⁢

j

⊆

L

is said to be a

local incidence property

of

L

with respect to the

locus

x

⁢

@

⁢

j

.

A

k

-adic relation

L

⊆

X

1

×

…

×

X

k

is said to be

C

-

regular

at

j

if and only if every flag of

L

with

x

at

j

has the property

C

,

where

x

is taken to vary over the

theme

of the fixed domain

X

j

.

Expressed in symbols,

L

is

C

-regular at

j

if and only if

C

⁢

(

L

x

⁢

@

⁢

j

)

is true for all

x

in

X

j

.

5

Regional incidence properties

The definition of a local flag can be broadened from a point

x

in

X

j

to a subset

M

of

X

j

,

arriving at the definition of a

regional flag

in the following way:

Suppose that

L

⊆

X

1

×

…

×

X

k

,

and choose a subset

M

⊆

X

j

.

Then

L

M

⁢

@

⁢

j

is a subset of

L

that is said to be the

flag

of

L

with

M

at

j

,

or the

M

⁢

@

⁢

j

-flag of

L

,

an object which has the following definition:

L

M

⁢

@

⁢

j

=

{

(

x

1

,

…

,

x

j

,

…

,

x

k

)

∈

L

:

x

j

∈

M

}

.

6

Numerical incidence properties

A

numerical incidence property

(NIP) of a relation is a local incidence property that depends on the

cardinalities

of its local flags.

For example,

L

is said to be

c

-regular at

j

if and only if the cardinality of the local flag

L

x

⁢

@

⁢

j

is

c

for all

x

in

X

j

,

or, to write it in symbols, if and only if

|

L

x

⁢

@

⁢

j

|

=

c

for all

x

∈

X

j

.

In a

similar

fashion, one can define the NIPs,

(

<

c

)

-regular at

j

,

(

>

c

)

-regular at

j

,

and so on. For ease of reference, a few of these definitions are recorded here:

L

is

c

⁢

-regular

at

⁢

j

if and only if

|

L

x

⁢

@

⁢

j

|

=

c

for all

⁢

x

∈

X

j

.

L

is

(

<

c

)

-regular

at

⁢

j

if and only if

|

L

x

⁢

@

⁢

j

|

<

c

for all

⁢

x

∈

X

j

.

L

is

(

>

c

)

-regular

at

⁢

j

if and only if

|

L

x

⁢

@

⁢

j

|

>

c

for all

⁢

x

∈

X

j

.

7

Species of 2-adic relations

Returning to 2-adic relations, it is useful to describe some familiar classes of objects in terms of their local and numerical incidence properties. Let

L

⊆

S

×

T

be an arbitrary 2-adic relation. The following properties of

L

can be defined:

L

is

total

at

⁢

S

if and only if

L

is

(

≥

1

)

-regular

at

⁢

S

.

L

is

total

at

⁢

T

if and only if

L

is

(

≥

1

)

-regular

at

⁢

T

.

L

is

tubular

at

⁢

S

if and only if

L

is

(

≤

1

)

-regular

at

⁢

S

.

L

is

tubular

at

⁢

T

if and only if

L

is

(

≤

1

)

-regular

at

⁢

T

.

If

L

⊆

S

×

T

is tubular at

S

,

then

L

is called a

partial function

or a

prefunction

from

S

to

T

.

This is sometimes indicated by giving

L

an alternate name, say, “

p

”, and writing

L

=

p

:

S

⇀

T

.

Just by way of formalizing the definition:

L

=

p

:

S

⇀

T

if and only if

L

is

tubular

at

⁢

S

.

If

L

is a prefunction

p

:

S

⇀

T

that happens to be total at

S

,

then

L

is called a

function

from

S

to

T

,

indicated by writing

L

=

f

:

S

→

T

.

To say that a relation

L

⊆

S

×

T

is

totally tubular

at

S

is to say that it is

1

-regular at

S

.

Thus, we may formalize the following definition:

L

=

f

:

S

→

T

if and only if

L

is

1

⁢

-regular

at

⁢

S

.

In the case of a function

f

:

S

→

T

,

one has the following additional definitions:

f

is

surjective

if and only if

f

is

total

at

⁢

T

.

f

is

injective

if and only if

f

is

tubular

at

⁢

T

.

f

is

bijective

if and only if

f

is

1

⁢

-regular

at

⁢

T

.

8

Variations

Because the concept of a relation has been developed quite literally from the beginnings of logic and mathematics, and because it has incorporated contributions from a diversity of thinkers from many different times and intellectual climes, there is a wide

variety

of terminology that the reader may run across in connection with the subject.

One dimension of

variation

is reflected in the names that are given to

k

-place relations, for

k

=

1

,

2

,

3

,

…

,

with some writers using the Greek forms,

medadic

,

monadic

,

dyadic

,

triadic

,

k

-

adic

, and other writers using the Latin forms,

nullary

,

unary

,

binary

,

ternary

,

k

-

ary

.

The cardinality of the relational ground, the set of relational domains, may be referred to as the

adicity

, the

arity

, or the

dimension

of the relation. Accordingly, one finds a relation on a finite number of domains described as a

polyadic

relation or a

finitary

relation, but others count infinitary relations among the polyadic. If the number of domains is finite, say equal to

k

,

then the relation may be described as a

k

-

adic

relation, a

k

-

ary

relation, or a

k

-

dimensional

relation, respectively.

A more conceptual than nominal variation depends on whether one uses terms like

predicate

,

relation

, and even

term

to refer to the

formal object

proper or else to the allied syntactic items that are used to denote them. Compounded with this variation is still another, frequently associated with philosophical

differences

over the status in reality accorded formal objects. Among those who speak of numbers, functions, properties, relations, and sets as being real, that is to say, as having objective properties, there are divergences as to whether some things are more real than others, especially whether particulars or properties are equally real or else which one is

derivative

in relationship to the other. Historically speaking, just about every

combination

of modalities has been used by one school of thought or another, but it suffices here merely to indicate how the options are generated.

9

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Category Theory

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•

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Elements of the History of Mathematics

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•

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Introduction

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•

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•

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Naive Set Theory

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•

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•

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•

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Title

relation theory

Canonical

name

RelationTheory

Date of creation

2013-10-08 21:04:11

Last modified on

2013-10-08 21:04:11

Owner

Jon Awbrey (15246)

Last modified by

Jon Awbrey (15246)

Numerical id

55

Author

Jon Awbrey (15246)

Entry type

Topic

Classification

msc 68R01

Classification

msc 68P15

Classification

msc 08A02

Classification

msc 05C65

Classification

msc 05B30

Classification

msc 05B20

Classification

msc 03E20

Classification

msc 03B10

Synonym

theory of relations

Related topic

RelationComposition2

Related topic

RelationConstruction

Related topic

RelationReduction

Related topic

LogicalMatrix

Related topic

TriadicRelation

Related topic

SignRelation

Related topic

SignRelationalComplex

Related topic

SemioticEquivalenceRelation

Defines

relation

Defines

adicity

Defines

arity

Defines

domain

Defines

graph

Defines

type

Defines

function

Defines

prefunction

Defines

partial function

Defines

bijective

Defines

injective

Defines

surjective

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