Topological-frame products of modal logics

The simplest bimodal combination of unimodal logics L1 and L2 is their fusion, L1 ⊗ L2, axiomatized by the theorems of L1 for 1 and of L2 for 2, and the rules of modus ponens, substitution and necessitation for 1 and for 2. Shehtman introduced the frame product L1 × L2, as the logic of the products of certain Kripke frames: these logics are two-dimensional as well as bimodal. Van Benthem, Bezhanishvili, ten Cate and Sarenac transposed Shehtman’s idea to the topological semantics and introduced the topological product L1 ×t L2, as the logic of the products of certain topological spaces. For almost all well-studies logics, we have L1⊗L2 ( L1×L2. Van Benthem et al show, by contrast, that S4×tS4 = S4⊗S4. It is straightforward to define the product of a topological space and a frame: the result is a topologized frame, i.e., a set together with a topology and a binary relation. In this paper, we introduce topological-frame products L1×tf L2 of modal logics, providing a complete axiomatization of S4 ×tf L, whenever L is a Kripke complete Horn axiomatizable extension of the modal logic D: these extensions include T,S4 and S5, but not K or K4. We leave open the problem of axiomatizing S4 ×tf K, S4×tf K4, and other related logics. When L = S4, our result is equivalent to a conjecture of van Benthem et al concerning the logic of products of Alexandrov spaces with arbitrary topological spaces.