Geodesic
Two-dimensional modal logic
Matematičeskie zametki, Tome 23 (1978) no. 5, pp. 759-772 Cet article a éte moissonné depuis la source Math-Net.Ru

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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]. 
@article{MZM_1978_23_5_a12,
     author = {V. B. Shekhtman},
     title = {Two-dimensional modal logic},
     journal = {Matemati\v{c}eskie zametki},
     pages = {759--772},
     year = {1978},
     volume = {23},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1978_23_5_a12/}
}
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V. B. Shekhtman. Two-dimensional modal logic. Matematičeskie zametki, Tome 23 (1978) no. 5, pp. 759-772. http://geodesic.mathdoc.fr/item/MZM_1978_23_5_a12/