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