As binary relation (at least with Kuratowski definition) is set of ordered pairs, can relations be set of ordered n's? And if they can be, what is domain, and counter domain in such relation, for example with ordered 5's? And more, can be restriction and composition defined for such relations, and how?
More generally, i know the definition and properties and operations for binary relations, and i wonder if there is a way to expand this definition to n-argument relations.
On
As @Don Thousand and @Henno Brandsma have pointed out, $n$-ary relations are just subsets of $X_1, X_2, \ldots, X_n$. Henno metioned that they don't get studied much in mathematics, but that doesn't mean they don't get studied at all.
Relational databases (like SQL) use a relational model. Essentially, if you have a huge table of data, the rows of the table represent $n$-tuples in your relation. If you had a table where the first column was name, the second column was age, and the third column was weight, the tuple (John Doe, 37, 180) as a row of your data could represent the statement "John Doe is 37 years old and weighs 180 pounds". In general, $(x, y, z)$ would mean "Person $x$ is $y$ years old and weighs $z$ pounds".
Just like a binary relation is a subset of $X \times Y$, an $n$-ary relation is a set of $n$-tuples from a product $X_1 \times \ldots X_n$. See relational algebra for notions of composition etc. (joins). It has actually been studied in connection with relational databases..
Most maths concerns itself with binary relations ((partial) orders, equivalence relations, functions, incidence relations) and not so much with the higher order ones, as can be seen from examples here.