Geodesic
On squares of modal logics with additional connectives
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algorithmic aspects of algebra and logic, Tome 274 (2011), pp. 343-351

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The paper gives an overview of new results on two-dimensional modal logics of special type, “Segerberg squares.” They are defined as usual squares of modal logics with additional connectives corresponding to the diagonal symmetry and two projections onto the diagonal. In many cases these logics are finitely axiomatizable, complete and have the finite model property. Segerberg squares are interpreted in the classical predicate logic. 
@article{TM_2011_274_a19,
     author = {V. B. Shehtman},
     title = {On squares of modal logics with additional connectives},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {343--351},
     publisher = {mathdoc},
     volume = {274},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2011_274_a19/}
}
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V. B. Shehtman. On squares of modal logics with additional connectives. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algorithmic aspects of algebra and logic, Tome 274 (2011), pp. 343-351. http://geodesic.mathdoc.fr/item/TM_2011_274_a19/