The purpose of this this paper is to generalize the functors arising from the theory of Witt vectors duto to Cartier. Given a polynomial $g(q)in mathbb Z[q]$, we construct a functor ${overline {W}}^{g(q)}$ from the category of $mathbb Z[q]$-algebras to that of commutative rings. When $q$ is specialized into an integer $m$, it produces a functor from the category of commutative rings with unity to that of commutative rings. In a similar way, we also construct several functors related to ${overline { W}}^{g(q)}$. Functorial and structural properties such as induction, restriction, classification and unitalness will be investigated intensively.
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