Changes

==Preliminaries==

==Preliminaries==

The first order of business is to define the [[operator|operation]] on [[relation (mathematics)|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 [[binary relation|2-adic]] and 3-adic relations.

The first order of business is to define the operation on [[relation (mathematics)|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 [[function (mathematics)|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.

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 [[function (mathematics)|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.

<p>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.</p>

* 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:

** LALA = left application, left association.

| LALA = left application, left association.

** LARA = left application, right association.

** RALA = right application, left association.

| LARA = left application, right association.

** RARA = right application, right association.

| RALA = right application, left association.

| RARA = right application, right association.

==Definition==

==Definition==

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

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

:* Composing ''on the right'', ''f''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;''Y'' followed by ''g''&nbsp;:&nbsp;''Y''&nbsp;&rarr;&nbsp;''Z'' results in a ''composite function'' formulated as ''fg''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;''Z''.

<p>Composing ''on the right'', <math>f : X \to Y</math> followed by <math>g : Y \to Z</math></p>

:* Composing ''on the left'', ''f''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;''Y'' followed by ''g''&nbsp;:&nbsp;''Y''&nbsp;&rarr;&nbsp;''Z'' results in a ''composite function'' formulated as ''gf''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;''Z''.

<p>results in a ''composite function'' formulated as <math>fg : X \to Z.</math></p>

<p>Composing ''on the left'', <math>f : X \to Y</math> followed by <math>g : Y \to Z</math></p>

 

<p>results in a ''composite function'' formulated as <math>gf : X \to Z.</math></p>

 

Note on notation.  The ordinary symbol for functional composition is the ''[[composition sign]]'', a small circle "<math>\circ</math>" written between the names of the functions being composed, as <math>f \circ g,</math> 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 ''center dot'' "<math>\cdot</math>", as <math>f \cdot g.</math>

Generalizing the paradigm along parallel lines, the ''composition'' of a pair of 2-adic relations is formulated in the following two ways:

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''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y'' followed by ''Q''&nbsp;&sube;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'' results in a ''composite relation'' formulated as ''PQ''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Z''.

<p>Composing ''on the right'', <math>P \subseteq X \times Y</math> followed by <math>Q \subseteq Y \times Z</math></p>

 

<p>results in a ''composite relation'' formulated as <math>PQ \subseteq X \times Z.</math></p>

<p>Composing ''on the left'', <math>P \subseteq X \times Y</math> followed by <math>Q \subseteq Y \times Z</math></p>

:* Composing ''on the left'', ''P''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Y'' followed by ''Q''&nbsp;&sube;&nbsp;''Y''&nbsp;&times;&nbsp;''Z'' results in a ''composite relation'' formulated as ''QP''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''Z''.

<p>results in a ''composite relation'' formulated as <math>QP \subseteq X \times Z.</math></p>

==Geometric construction==

==Geometric construction==