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A generalization of The Dress construction for a Tambara functor, and polynomial Tambara functors

For a finite group $G$, (semi-)Mackey functors and (semi-)Tambara functors are regarded as $G$-bivariant analogs of (semi-)groups and (semi-)rings respectively. In fact if $G$ is trivial, they agree with the ordinary (semi-)groups and (semi-)rings, and many naive algebraic properties concerning rings and groups have been extended to these $G$-bivariant analogous notions. In this article, we investigate a $G$-bivariant analog of the semi-group rings with coefficients. Just as a coefficient ring $R$ and a monoid $Q$ yield the semi-group ring $R[Q]$, our constrcution enables us to make a Tambara functor $T[M]$ out of a semi-Mackey functor $M$, and a coefficient Tambara functor $T$. This construction is a composant of the Tambarization and the Dress construction. As expected, this construction is the one uniquely determined by the righteous adjoint property. Besides in analogy with the trivial group case, if $M$ is a Mackey functor, then $T[M]$ is equipped with a natural Hopf structure. Moreover, as an application of the above construction, we also obtain some $G$-bivariant analogs of the polynomial rings.