Tensor products of modal logics

Products were introduced in the 1970s as a natural type of combined modal logics. They arise in different areas of pure and applied logics — spatial reasoning, multi-agent systems, quantified modal and intuitionistic logics etc. The theory of products was systematized and essentially developed first in the paper [GS98] and then in the book [GKWZ03], but during the past 10 years new important results were proved and the research is going on, cf. [Kur07]. Recall that the product of modal logics is defined as the logic of the class of products of their Kripke frames. On the one hand, this definition is quite natural, and in some cases products can be simply axiomatized and have nice properties. On the other hand, products are always Kripke complete. However, Kripke semantics sometimes may be inadequate. So different logics L1,L ′ 1 can have the same frames; then L1 × L2 = L1 × L2 for any L2 — which looks strange. Another peculiarity is logical non-invariance: it may happen that for some frames Log(F) = Log(F′), while Log(F× G) 6= Log(F′ × G). Also, if logics L1 and L2 are consistent, but L1 has an empty class of frames, then L1 × L2 is inconsistent. To amend the situation, we can try to define products of Kripke of models (or, equivalently, general Kripke frames or modal algebras). The following problem was mentioned in [Kur07, p. 877]: There are several attempts for extending the product construction from Kripke complete logics to arbitrary modal logics, mainly by considering product-like constructions on Kripke models. All the suggested methods so far result in sets of formulas that are not closed under the rule of Substitution.