relation composition

relation composition

Relation composition

, or the

composition of relations

, is the

generalization

of function composition, or the

composition

of functions. The following treatment of relation composition takes the “strongly typed” approach to

relations

that is outlined in the entry on relation theory (

http://planetmath.org/RelationTheory

).

Contents:

1

Preliminaries

2

Definition

3

Geometric construction

4

Algebraic construction

5

Matrix representation

6

Graph-theoretic picture

7

Relation reduction

8

References

1

Preliminaries

The

first order

of business is to define the

operation on relations

that is variously known as the

composition of relations

,

relational composition

, or

relative multiplication

. In approaching the more general constructions, it pays to begin with the composition of 2-adic and 3-adic relations.

As an incidental observation on usage, there are many different conventions of syntax for denoting the application and composition of relations, with perhaps even more options in general use than are common for the application and composition of functions. In this case there is little chance of standardization, since the convenience of conventions is relative to the

context

of use, and the same writers use different styles of syntax in different settings, depending on the ease of

analysis

and

computation

.

•

The first

dimension

of

variation

in syntax has to do with the correspondence between the order of operation and the linear order of terms on the page.

•

The second dimension of variation in syntax has to do with the automatic

assumptions

in place about the associations of terms in the absence of associations marked by parentheses. This becomes a significant factor with relations in general because the usual property of

associativity

is lost as both the

complexities

of compositions and the dimensions of relations increase.

•

These two factors together generate the following four styles of syntax:

Left application, Left association (LALA).

Left application, Right association (LARA).

Right application, Left association (RALA).

Right application, Right association (RARA).

2

Definition

A notion of relational composition is to be defined that generalizes the usual notion of

functional

composition:

•

Composing

on the right

,

f

:

X

→

Y

followed by

g

:

Y

→

Z

results in a

composite function

formulated as

f

⁢

g

:

X

→

Z

.

•

Composing

on the left

,

f

:

X

→

Y

followed by

g

:

Y

→

Z

results in a

composite function

formulated as

g

⁢

f

:

X

→

Z

.

Note on notation. The ordinary symbol for functional composition is the

composition sign

, a small circle “

∘

” written between the names of the functions being composed, as

f

∘

g

, but the sign is often omitted if there is no risk of confusing the composition of functions with their algebraic

product

. In contexts where both compositions and products occur, either the composition is marked on each occasion or else the product is marked by means of a

raised dot sign

“

⋅

”, as

f

⋅

g

.

Generalizing the paradigm along parallel lines, the

composition

of a pair of 2-adic relations is formulated in the following two ways:

•

Composing

on the right

,

P

⊆

X

×

Y

followed by

Q

⊆

Y

×

Z

results in a

composite relation

formulated as

P

⁢

Q

⊆

X

×

Z

.

•

Composing

on the left

,

P

⊆

X

×

Y

followed by

Q

⊆

Y

×

Z

results in a

composite relation

formulated as

Q

⁢

P

⊆

X

×

Z

.

In the rest of this discussion 2-adic relations will be composed

on the right

, leading to the following definition of

P

⁢

Q

=

P

∘

Q

for the

composable pair

of relations,

P

⊆

X

×

Y

and

Q

⊆

Y

×

Z

.

Definition.

P

∘

Q

=

{

(

x

,

z

)

∈

X

×

Z

:

(

x

,

y

)

∈

P

⁢

and

⁢

(

y

,

z

)

∈

Q

}

.

3

Geometric construction

There is a neat way of defining relational compositions in geometric terms, not only showing their relationship to the

projection

operations

that come with any

cartesian product

, but also suggesting natural

directions

for generalizing relational compositions beyond the 2-adic case, and even beyond relations that have any fixed arity, in effect, to the general case of

formal languages

as generalized relations.

This way of looking at relational compositions is sometimes referred to as

Tarski’s trick

, on account of Alfred Tarski having put it to especially good use in his work (Ulam and Bednarek, 1977). It supplies the imagination with a geometric way of visualizing the relational composition of a pair of 2-adic relations, doing this by attaching concrete imagery to the basic set-theoretic operations, namely,

intersections

, projections, and a certain class of operations

inverse

to projections, here called

tacit extensions

(

http://planetmath.org/TacitExtension

)

.

See main entry (

http://planetmath.org/GeometricRepresentationOfRelationComposition

) for details.

4

Algebraic construction

The transition from a geometric picture of relation composition to an algebraic formulation is accomplished through the

introduction

of

coordinates

, in other words, identifiable names for the objects that are related through the various forms of relations, 2-adic and 3-adic in the present case.

See main entry (

http://planetmath.org/AlgebraicRepresentationOfRelationComposition

) for details.

5

Matrix representation

We have it within our reach to pick up another way of representing 2-adic relations, namely, the

representation

as

logical matrices

, and also to grasp the

analogy

between relational composition and ordinary

matrix multiplication

as it appears in

linear algebra

.

See main entry (

http://planetmath.org/MatrixRepresentationOfRelationComposition

) for details.

6

Graph-theoretic picture

There is another form of representation for 2-adic relations that is useful to keep in mind, especially for its ability to render the logic of many complex

formulas

almost instantly understandable to the mind’s eye. This is the representation in terms of

bipartite graphs

(

http://planetmath.org/BipartiteGraph

)

, or

bigraphs

for short.

See main entry (

http://planetmath.org/GraphTheoreticRepresentationOfRelationComposition

) for details.

7

Relation reduction

See main entry (

http://planetmath.org/RelationReduction

) for details.

8

References

•

Ulam, Stanislaw Marcin; and Bednarek, A.R. (1977), “On the Theory of

Relational Structures

and Schemata for

Parallel

Computation”. Reprinted, pp. 477–508 in Ulam (1990).

•

Ulam, Stanislaw Marcin (1990),

Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators

, A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA.

Title

relation composition

Canonical name

RelationComposition

Date of creation

2013-10-24 16:58:43

Last modified on

2013-10-24 16:58:43

Owner

Jon Awbrey (15246)

Last modified by

Jon Awbrey (15246)

Numerical id

40

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

composition of relations

Synonym

relational composition

Synonym

relative multiplication

Related topic

RelationTheory

Related topic

RelationConstruction

Related topic

RelationReduction

Related topic

TacitExtension

Related topic

LogicalMatrix

Defines

relation composition

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