Matching topological products and frame products of modal logics

The simplest combination of unimodal logics L1 and L2 into a bimodal logic is their fusion, L1 ⊗ L2, axiomatized by the theorems of L1 for ⬛1 and of L2 for ⬛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 L1 × L2 of L1 and L2. Van Benthem, Bezhanishvili, ten Cate and Sarenac generalized Shehtman's idea and introduced the topological product L1 ×t L2, 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 S4: L1 ×t L2 = L1 × L2 iff L1 ⊋ S5 or L2 ⊋ S5 or L1, L2 = S5.

Keywords: bimodal logic, multimodal logic, topological semantics, topological product, product space.