Let G be a finite group, and C be an abelian group. We introduce the notions of C-monomial G-sets and C-monomial G-posets, and state some of their categorical properties. This gives in particular a new description of the C-monomial Burnside ring BC (G). We also introduce Lefschetz invariants of C-monomial G-posets, which are elements of BC (G). These invariants allow for a definition of a generalized tensor induction multiplicative map TU,λ : BC (G) → BC (H) associated to any C-monomial (G, H)-biset (U, λ), which in turn gives a group homomorphism BC (G)× → BC (H)× between the unit groups of C-monomial Burnside rings.