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.
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 a measure of 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 as indicating the type of the function, in effect defining as the pair but is used equivocally to denote both the triple and the subset forming one part of it.
One way to resolve the ambiguity is to formalize a distinction between the function and its graph, defining
Another tactic treats the whole notation as a name for the triple, letting denote
In categorical and computational contexts, at least initially, the type is regarded as an essential attribute or integral part of the function itself. In other contexts we may wish to use a more abstract concept of function, treating a function as a mathematical object capable of being viewed under many different types.
Following the pattern of the functional case, let the notation bring to mind a mathematical object 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 be the triple or let denote and choose another name for the triple.
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
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 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 of a relation is a local incidence property predicated 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 numerical incidence properties, -regular at -regular at and so on. For ease of reference, a few definitions are recorded below.
Species of dyadic relations
Returning to 2-adic relations, it is useful to describe several 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 which happens to be total at then is called a function from to indicated by writing To say a relation is totally tubular at is to say it is -regular at Thus, we may formalize the following definition.
In the case of a function we have the following additional definitions.
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.
- 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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