Some prospects for semiproducts and products of modal logics

We consider products and semiproducts of propositional modal logics L with S5 and present new examples of product and semiproduct logics axiomatized in the minimal way and enjoying the product (or semiproduct) FMP. An essential part of the proof is local tabularity of these (semi)products for L of finite depth; it is obtained by using bisimulation games. These results readily imply decidability for 1-variable fragments of predicate modal logics QL and QL+Barcan formula. We also present new counterexamples, i.e. (semi)products not axiomatizable in the simplest way.