Graded monads and rings of polynomials

Abstract

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, the natural homomorphism $\mathbb{F}_{1^2 } [t] \to \mathbb{Z}[A,B], t \mapsto (A,B)$, where t is a binary variable, is not an inclusion even if t is subject to the alternating condition. Bibliography: 2 titles.