Semiproducts, products, and modal predicate logics: some examples

We study two kinds of combined modal logics, semiproducts and products with S5, and their correlation with modal predicate logics. We obtain examples of propositional modal logics when these semiproducts or products are axiomatized in the minimal way (semiproduct- or product-matching with S5), as well as counterexamples for these properties. The fmp for (semi)products together with (semi)product-matching allow us to show decidability of corresponding 1-variable modal predicate logics.