Relation is a canonic class of any ontology. It is characterized by
substantial properties and formal attributes. Of the material properties,
there are their reality, nature and type and direction of dependency. Of
the second, there are transitivity, symmetry, reflexivity, and n-ary, or
cardinality, terms, or tuples, of domains, elements, components, or
arguments).

The typical mathematical reading of relation is an extensive set
of ordered elements (as ordered pairs, Kuratowski, Wiener, Skolem; well-ordering
axiom).

There are two formal definitions of relationship deserving attention, http://en.wikipedia.org/wiki/Relation_(mathematics):

i. [A **relation** *R* over the sets *X*_{1}, …, *X*_{k} is a subset of their
Cartesian product, written *R*
⊆ *X*_{1} × … × *X*_{k}.].

ii. [A relation *R*
over the sets *X*_{1}, …, *X*_{k} is a (*k*+1)-tuple *R* = (*X*_{1},
…, *X*_{k}, *G*(*L*)),
where *G*(*L*) is a subset of the Cartesian product *X*_{1} × … × *X*_{k}. *G*(*L*) is called
the *graph* of *L*.]

So, one can say "an n-ary relation is an ordered class of
n-tuples, or it is an ordered class of (n+1) tuple". Three things are of
importance here:

1. the components of relations are of the same kind and sorts, objects,
persons, qualities, quantities, times;

2. ordering of relations, their direction, a triadic 'giving',
tetradic 'paying' or triadic 'betweenness';

3. the key sense of relationship is represented by the graph,
indicating its nature and kind: if it's causal relation, temporal relation,
spatial relation, semantic relation, logical relation, etc.

Think of the complex case of social networks, where social
relationships described in terms of nodes (agents) and ties
(relationships), of different sorts and kinds, like as emotional,
friendly, economical, political, or commercial links and connections.

Any general ontology missing the class of relation as the fundamental
category of reality is internally defective. For example, there is a
comprehensive scheme of categories of reality proposed by Chisholm, who divided
Entia into Contingent things (States (Events) and Individuals (Boundaries
and Substances) and Necessary things (States and Nonstates (Attributes and
Substances). Since the class of relationship is deprecated, the
scheme is missing the adhesive of all things, Relationship, as well as
Time and Space.

Azamat Abdoullaev