Consider distinct integer polynomials of distinct degree. Also they are all univariate polynomials of the variable $x$.
Let $*$ denote composition and $*$ is the operation for the monoid. Consider when such polynomials $a,b,c,d$ forms an abelian monoid.
Many questions arise naturally. For instance
1) can we classify them ?
2) Is every finitely generated abelian monoid a submonoid of a nonfinitely generated abelian monoid ?
3) if the distinct integer polynomials $x,y$ satisfy
$$ x * y = y * x $$
Does that imply they belong to an abelian monoid with polynomials of degree other than $1$ and No common divisors with $deg(y)$ or $deg(x)$ ?
4) If an Abelian monoid is not finitely generated and is not an abelian submonoid , does that imply that the degrees of the polynomials are dense in the positieve integers ?
5) If the Abelian monoid is not finitely generated does that imply that there exists an analytic function $F $ and the functional inverse ( of $F$) $G $ such that all elements are given by
$$ F( n \space G(x) ) $$
For some integer $n$ ,where $n$ is the degree ? ( I did not say Every integer $n $ is a Solution ! )
6) I only know solutions
$$exp( n \space ln(x)), sin( n \space arcsin(x)) , cos( n \space arccos(x) ) $$
as Abelian monoids. What are the others ?? [ MAIN QUESTION ! ]
