Graded monads and rings of polynomials

Models for free graded monads over the category of sets are constructed. Certain rings of generalized noncommutative polynomials, generated by an operation of arbitrary arity, are implemented as subrings of classical rings of noncommutative polynomials. It is shown, that natural homomorphisms from rings of generalized polynomials to rings of the usual commutative polynomials are not inclusions as a rule. For instance, a natural homomorphism $\mathbb{F}_{1^2}[t]\to\mathbb{Z}[A,B]$, $t\mapsto(A,B)$, where $t$ is a binary variable, isn't an inclusion, even if $t$ is subjected to the alternating condition.