There is a group in which is embedded the set of rational functions considering the formal composition binary operation? More generally how (and when) can we extend a non-commutative monoid (or semi-group) into a group ?
I was looking at the interesting case that the natural numbers can be extended to de integers by the matters of considering it as a monoid over the sum and constructing a group of the fractions and then extended again into the rational by applying the almost same method considering the multiplication operation this time.
Thinking in this context i try to apply the same ideia to the set of rational function (with rational coefficients - quotients of two rational polynomials) but considering the composition binary operation as it is associative, has a neutral element, etc...
But obviously isn't that simple and after searching for a answer i discover the concept of group completion of but not much more and specially not much for the non-commutative case.
I'm thinking in a abstract way, thinking in a rational function as a pair of finite sequence of rational numbers, as was already noted in comments that not all rational polynomial are invertible, but. Considering that the case is analogous to the fact that not all integers are multiplicatively invertible
Commutative semi group extends to group if and only if it has cancellation property. This condition is necessary in non-commutative case, but unfortunately not sufficient. The non-commutative case is quite complicated, read this: https://en.m.wikipedia.org/wiki/Cancellative_semigroup#Embeddability_in_groups
That being said I'm afraid your semi group does not have cancellation property, $P\circ Q=P\circ W$ does not imply that $Q=W$, e.g. when $P$ is constant.
And so it cannot be extended to group.