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The theory of relations treats the subject matter of Relation s in its combinatorial aspect, as distinguished from, though related to, its more properly logical study on one side and its more generally mathematical study on another.

A relation, as conceived in the combinatorial theory of relations, is a mathematical object that in general can have a very complex type, the complexity of which is best approached in several stages, as indicated next.

In order to approach the combinatorial definition of a relation, it helps to introduce a few preliminary notions that can serve as stepping stones to the general idea.

A relation in mathematics is defined as an object that has its existence as such within a definite context or setting. It is literally the case that to change this setting is to change the relation that is being defined. The particular type of context that is needed here is formalized as a collection of elements from which are chosen the elements of the relation in question. This larger collection of '' Elementary Relation s'' or '' Tuple s'' is constructed by means of the set-theoretic product commonly known as the '' Cartesian Product ''.


PRELIMINARIES


A relation ''L'' is defined by specifying two mathematical objects as its constituent parts:

  • The first part is called the ''figure'' of ''L'', notated as ''figure''(''L'') or ''F''(''L'').


  • The second part is called the ''ground'' of ''L'', notated as ''ground''(''L'') or ''G''(''L'').


In the special case of a finitary relation, for concreteness a '''''k''-place relation''', the concepts of figure and ground are defined as follows:

  • The ground of ''L'' is a Sequence of ''k'' Nonempty Set s, ''X''1, …, ''X''''k'', called the ''domains'' of the relation ''L''.


  • The figure of ''L'' is a Subset of the Cartesian Product taken over the domains of ''L'', that is, ''F''(''L'') ⊆ ''G''(''L'') = ''X''1 × … × ''X''''k''.


Strictly speaking, then, the relation ''L'' consists of a couple of things, ''L'' = (''F''(''L''), ''G''(''L'')), but it is customary in loose speech to use the single name ''L'' in a systematically equivocal fashion, taking it to denote either the couple ''L'' = (''F''(''L''), ''G''(''L'')) or the figure ''F''(''L''). There is usually no confusion about this so long as the ground of the relation can be gathered from context.


DEFINITION


The formal definition of a finite arity relation, specifically, a '''k-ary relation''' can now be stated.

  • Definition. A '''k-ary relation''' ''L'' over the nonempty sets ''X''1, …, ''X''''k'' is a (1+''k'')-tuple ''L'' = (''F''(''L''), ''X''1, …, ''X''''k'') where ''F''(''L'') is a subset of the cartesian product ''X''1 × … × ''X''''k''. If all of the ''X''''j'' for ''j'' = 1 to ''k'' are the same set ''X'', then ''L'' is more simply called a '''''k''-ary relation over ''X'''''. The set ''F''(''L'') is called the ''figure'' of ''L'' and, providing that the sequence of sets ''X''1, …, ''X''''k'' is fixed throughout a given discussion or determinate in context, one may regard the relation ''L'' as being determined by its figure ''F''(''L'').


The formal definition simply repeats more concisely what was said above, merely unwrapping the conceptual packaging of the relation's ground to define the relation in 1 + ''k'' parts, as ''L'' = (''F''(''X''), ''X''1, …, ''X''''k''), rather than just the two, as ''L'' = (''F''(''L''), ''G''(''L'')).

A ''k''-ary predicate is a '' Boolean-valued Function '' on ''k'' variables.


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 which 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'' at ''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''.


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.

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