A generalization of the Dress construction for a Tambara functor, and its relation to polynomial Tambara functors

For a finite group GG, (semi-)Mackey functors and (semi-)Tambara functors are regarded as GG-bivariant analogs of (semi-)groups and (semi-)rings respectively. In fact if GG 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 GG-bivariant analogous notions.In this article, we investigate a GG-bivariant analog of the semi-group rings with coefficients. Just as a coefficient ring RR and a monoid QQ yield the semi-group ring R[Q]R[Q], our construction enables us to make a Tambara functor T[M]T[M] out of a semi-Mackey functor MM, and a coefficient Tambara functor TT. 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 MM is a Mackey functor, then T[M]T[M] is equipped with a natural Hopf structure.Moreover, as an application of the above construction, we also obtain some GG-bivariant analogs of the polynomial rings.