Topological-Frame Products of Modal Logics

The simplest bimodal combination of unimodal logics (text {L} _1) and (text {L} _2) is their fusion, (text {L} _1 otimes text {L} _2), axiomatized by the theorems of (text {L} _1) for (square _1) and of (text {L} _2) for (square _2), and the rules of modus ponens, necessitation for (square _1) and for (square _2), and substitution. Shehtman introduced the frame product (text {L} _1 times text {L} _2), 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 (text {L} _1 times _t text {L} _2), as the logic of the products of certain topological spaces. For almost all well-studies logics, we have (text {L} _1 otimes text {L} _2 subsetneq text {L} _1 times text {L} _2), for example, (text {S4} otimes text {S4} subsetneq text {S4} times text {S4} ). Van Benthem et al. show, by contrast, that (text {S4} times _t text {S4} = text {S4} otimes text {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 (text {L} _1 times _ tf text {L} _2) of modal logics, providing a complete axiomatization of (text {S4} times _ tf text {L} ), whenever (text {L} ) is a Kripke complete Horn axiomatizable extension of the modal logic D: these extensions include (text {T} , text {S4} ) and (text {S5} ), but not (text {K} ) or (text {K4} ). We leave open the problem of axiomatizing (text {S4} times _ tf text {K} ), (text {S4} times _ tf text {K4} ), and other related logics. When (text {L} = text {S4} ), our result confirms a conjecture of van Benthem et al. concerning the logic of products of Alexandrov spaces with arbitrary topological spaces.