arXiv:2611.14667v1 Announce Type: new Abstract: We develop a general monoid-theoretic framework to construct and study graded and color Hopf algebras of formal power series of bounded degree invariant under group actions. Rather than working with external alphabets of commuting, noncommuting, or anti-commuting variables, we start from an abstract monoid $M$ equipped with a $2$-cocycle twisting $\tau:M\times M\to \mathbb{K}$, which is allowed to vanish, a bigrading over a commutative monoid $\mathcal{G}\times\mathbb{N}$ endowed with a bicharacter $\phi$, and a group $G$ acting on $M$ via a bigraded action $\rho$ that is quasi-multiplicative with respect to $\tau$, coupled with a $1$-cocycle weight function $\omega:G\times M\to\mathbb{K}^\times$. By formulating local finiteness, gradings, quasi-multiplicativity, and ideals relative to the non-vanishing of $\tau$, we prove that the invariant space $\mathbb{K}\langle\kern-0.2em\langle M\rangle\kern-0.2em\rangle^G$ forms a well-defined bigraded subalgebra of bounded-degree formal power series with a canonical monomial basis indexed by $\omega$-admissible orbits. Furthermore, we establish four functorial constructions: embeddings into larger invariant algebras via restricted actions, color tensor product decompositions for direct products and diagonal actions, connected color Hopf algebra structures induced by internal monoid duplication, and quotient color Hopf algebras governed by invariant-retraction pairs with an explicit description of their kernel bi-ideals. As applications, we uniformly recover and extend the classical and twisted Hopf algebras of symmetric and quasisymmetric functions in commuting and noncommuting variables, MacMahon symmetric functions, the full square of symmetric and quasisymmetric Hopf algebras in superspace, symmetric and quasisymmetric functions in partially commutative variables, and signed-symmetric functions associated with the hyperoctahedral group.
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