Bases of complete systems of rational functions with rational coefficients

A functional system is a set of functions endowed with a set of operations on these functions. The operations allow one to obtain new functions from the existing ones.
Functional systems are mathematical models of real and abstract control systems and thus are one of the main objects of discrete mathematics and mathematical cybernetic.
The problems in the area of functional systems are extensive. One of the main problems is deciding completeness that consists in the description of all subsets of functions that are complete, i.e. generate the whole set.
In our paper we consider the functional system of rational functions with rational coefficients endowed with the superposition operation. We investigate the problem of bases of complete systems, namely:

• Does every complete system have a (finite) basis?;
• For any positive integer $n$, is there a basis of a complete system consisting of $n$ functions?
• a number of examples of basis consisting of $n$ functions are presented explicitly $(n=1, 2, 3,\dots)$.

The answers to all these questions are positive, which is the main result of this article.