Dirichlet product from an action-type point of view [00DO]

    Not yet sure if this is usefull in any way.

    Definition 1. cartesian decomposition [00DQ]

      Given a finite set , we define the type of cartesian decompositions by

      Where cartesian product in FinSet.

      Definition 2. Dirichlet Product Species [00DP]

        Given species define the dirichlet product by

        where is the type of cartesian decompositions of .

        3. [00DR]

          For and , the transported Dirichlet product is

          where denotes the product of the underlying finite sets.

          • Aut(A) xx Aut(B) acting on F(a,A) xx G(b,B).
          • In the case where A = [a] and B = [b], we get an -set
          • given an equivalence we can view S_a xx S_b as subgroup of S_n.
          • the S_n acts on the dirichlet product
          • the stabelizer of is