The Möbius algebra as a Burnside ring (Q1064398)

Given the category of subgroups of a finite group G, under inclusions, various Grothendieck type rings have been functorially associated to this category. Different properties and theorems have been proved: Mackey property, Frobenius reciprocity, Dress induction theorem, Green functors and transfer theorems. Dress pointed out that subgroups of G can be replaced by ''based categories''. Here the author states that broader generalization can be obtained by using an \({\mathcal S}_ f\)-topos (\({\mathcal S}_ f\) is a topos with finite hom-sets) as a basis; G-sets can be replaced by a regular category. In this paper, Burnside ring theory is developed for a poset P, viewed as a category. In Section 2, three definitions of Burnside rings are given for restricted regular categories. For the finite poset P category, these rings turn out to coincide with the Möbius ring A(P) and if P is the poset of subgroups of G, it yields the usual Burnside ring \(\Omega\) (G). In Section 3, various (adjoint) functors are defined making the Möbius ring a Green functor, and the properties mentioned at the end of paragraph one above are recovered.