| Abstract: | The simplest combination of unimodal logics ({mathrm{L}_1 rm and mathrm{L}_2}) into a bimodal logic is their fusion, ({mathrm{L}_1 otimes mathrm{L}_2}), axiomatized by the theorems of ({mathrm{L}_1 rm for square_1 rm and of mathrm{L}_2 rm for square_{2}}). Shehtman introduced combinations that are not only bimodal, but two-dimensional: he defined 2-d Cartesian products of 1-d Kripke frames, using these Cartesian products to define the frame product({mathrm{L}_1 times mathrm{L}_2 rm of mathrm{L}_1 rm and mathrm{L}_2}). Van Benthem, Bezhanishvili, ten Cate and Sarenac generalized Shehtman’s idea and introduced the topological product({mathrm{L}_1 times_{t}mathrm{L}_2}), using Cartesian products of topological spaces rather than of Kripke frames. Frame products have been extensively studied, but much less is known about topological products. The goal of the current paper is to give necessary and sufficient conditions for the topological product to match the frame product, for Kripke complete extensions of ({mathrm{S}4: mathrm{L}_1 times_t mathrm{L}_2 = mathrm{L}_1 times mathrm{L}_2 rm iff mathrm{L}_1 supsetneq mathrm{S}5 rm or mathrm{L}_2 supsetneq mathrm{S}5 rm or mathrm{L}_1, mathrm{L}_2 = mathrm{S}5}). |
