Crossed Burnside Rings I. The Fundamental Theorem

Let G be a finite group and S a finite G-monoid. A crossed G-set over S is a finite G-set equipped with a G-map into S called a weight function. A crossed Burnside ring XΩ(G, S) is the Grothendieck ring of the category of crossed G-sets with respect to disjoint unions and tensor products. In this paper, we prove the fundamental theorem of crossed Burnside rings and an idempotent formula at characteristic 0.