Failure of Interpolation in Combined Modal Logics

We investigate transfer of interpolation in such combinations of modal logic which lead to interaction of the modalities. Combining logics by taking products often blocks transfer of interpolation. The same holds for combinations by taking unions, a generalization of Humberstone's inaccessibility logic. Viewing first order logic as a product of modal logics, we derive a strong counterexample for failure of interpolation in the finite variable fragments of first order logic. We provide a simple condition stated only in terms of frames and bisimulations which implies failure of interpolation. Its us