Decomposition of the Witt-Burnside ring and Burnside ring of an abelian profinite group

Let G, H be abelian profinite groups whose orders are coprime and assume that q ranges over the set of integers. The aim of this paper is to establish an isomorphism of functors , where denotes the q-deformed Witt-Burnside ring functor of G introduced in Y.-T. Oh, q-Deformation of Witt-Burnside rings, Math. Z. 207 (1) (2007) 151-191]. To do this, we first establish an isomorphism of functors , where denotes the q-deformed Burnside ring functor of G which was also introduced in Y.-T. Oh, q-Deformation of Witt-Burnside rings, Math. Z. 207 (1) (2007) 151-191]. As an application, we derive a pseudo-multiplicative property of the q-Möbius function associated to the lattice of open subgroups of the direct sum of G and H.