Tensor products of modal logics

We consider shifted products of modal algebras and logics first introduced by Y. Hasimoto in 2000. For logics this operation is similar to the well-known usual product but it is logically invariant. We prove the conjecture of D. Gabbay that shifted products act on Boolean algebras exactly as tensor products, so we call them tensor products of modal algebras. We also propose a filtration technique for models based on tensor products and obtain some decidability results.