Two-dimensional modal logic

Propositional logics with many modalites, characterized by “two-dimensional” Kripke models, are investigated. The general problem can be formulated as follows: from two modal logics describing certain classes of Kripke modal lattices construct a logic describing all products of Kripke lattices from these classes. For a large number of cases such a logic is obtained by joining to the original logics an axiom of the form $\square_i\square_jp\equiv\square_j\square_ip$ and $\lozenge_i\square_jp\supset\square_j\lozenge_ip$. A special case of this problem, leading to the logic of a torus $S5\times S5$ was solved by Segerberg [1].