# 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.

## 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  we are thinking of a mathematical object whose articulation requires three pieces of data, specifying the set  the set  and a particular subset of their cartesian product  So far so good.

Let us write  to express what has been said so far.

When it comes to parsing the notation  everyone takes the part  to specify the type of the function, that is, the pair  but  is used equivocally to denote both the triple and the subset  that forms one part of it. One way to resolve the ambiguity is to formalize a distinction between a function and its graph, letting 

Another tactic treats the whole notation  as sufficient denotation for the triple, letting  denote 

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  bring to mind a mathematical object that is specified by three pieces of data, the set  the set  and a particular subset of their cartesian product  As before we have two choices, either let  or let  denote  and choose another name for the triple.

## Definition

It is convenient to begin with the definition of a -place relation, where  is a positive integer.

Definition. A -place relation  over the nonempty sets  is a -tuple  where  is a subset of the cartesian product 

## Remarks

Though usage varies as usage will, there are several bits of optional language that are frequently useful in discussing relations. The sets  are called the domains of the relation  with  being the  domain. If all of the  are the same set  then  is more simply described as a -place relation over  The set  is called the graph of the relation  on analogy with the graph of a function. If the sequence of sets  is constant throughout a given discussion or is otherwise determinate in context, then the relation  is determined by its graph  making it acceptable to denote the relation by referring to its graph. Other synonyms for the adjective -place are -adic and -ary, all of which leads to the integer  being called the dimension, adicity, or arity of the relation 

## Local incidence properties

A local incidence property (LIP) of a relation  is a property that depends in turn on the properties of special subsets of  that are known as its local flags. The local flags of a relation are defined in the following way:

Let  be a -place relation 

Select a relational domain  and one of its elements  Then  is a subset of  that is referred to as the flag of  with  at  or the -flag of  an object that has the following definition:

 

Any property  of the local flag  is said to be a local incidence property of  with respect to the locus 

A -adic relation  is said to be -regular at  if and only if every flag of  with  at  has the property  where  is taken to vary over the theme of the fixed domain 

Expressed in symbols,  is -regular at  if and only if  is true for all  in 

## Regional incidence properties

The definition of a local flag can be broadened from a point  in  to a subset  of  arriving at the definition of a regional flag in the following way:

Suppose that  and choose a subset  Then  is a subset of  that is said to be the flag of  with  at  or the -flag of  an object which has the following definition:

 

## 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,  is said to be -regular at  if and only if the cardinality of the local flag  is  for all  in  or, to write it in symbols, if and only if  for all 

In a similar fashion, one can define the NIPs, -regular at  -regular at  and so on. For ease of reference, a few of these definitions are recorded here:

 

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

 

If  is tubular at  then  is called a partial function or a prefunction from  to  This is sometimes indicated by giving  an alternate name, say,  and writing 

Just by way of formalizing the definition:

 

If  is a prefunction  that happens to be total at  then  is called a function from  to  indicated by writing  To say that a relation  is totally tubular at  is to say that it is -regular at  Thus, we may formalize the following definition:

 

In the case of a function  one has the following additional definitions:

 

## 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 -place relations, for  with some writers using the Greek forms, medadic, monadic, dyadic, triadic, -adic, and other writers using the Latin forms, nullary, unary, binary, ternary, -ary.

The number of relational domains may be referred to as the adicity, arity, or 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  then the relation may be described as a -adic relation, a -ary relation, or a -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 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.

## Examples

See the articles on relations, relation composition, relation reduction, sign relations, and triadic relations for concrete examples of relations.

Many relations of the greatest interest in mathematics are triadic relations, but this fact is somewhat disguised by the circumstance that many of them are referred to as binary operations, and because the most familiar of these have very specific properties that are dictated by their axioms. This makes it practical to study these operations for quite some time by focusing on their dyadic aspects before being forced to consider their proper characters as triadic relations.

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## Syllabus

### Relational concepts

 Relation theory Relative term Sign relation Triadic relation

## Document history

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