Geodesic
Axiomatization of Modal Logic Squares with Distinguished Diagonal
Matematičeskie zametki, Tome 88 (2010) no. 2, pp. 261-274

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Modal logics of squared Kripke frames with distinguished diagonal are considered. It is shown that many such logics, unlike ordinary two-dimensional products, cannot be axiomatized by formulas with finitely many variables. The method resembles that used to obtain a similar result for $\ge3$-dimensional products of modal logics. The proof uses, in particular, generalized Sahlquist formulas.
Keywords: $\delta$-square of a Kripke frame, $\delta$-square of a modal logic, $\delta$-logic of a class of frames, axiomatizability, Kripke frame, Kripke model, variety of a modal logic, modal logic. 
@article{MZM_2010_88_2_a8,
     author = {S. P. Kikot'},
     title = {Axiomatization of {Modal} {Logic} {Squares} with {Distinguished} {Diagonal}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {261--274},
     publisher = {mathdoc},
     volume = {88},
     number = {2},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_88_2_a8/}
}
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S. P. Kikot'. Axiomatization of Modal Logic Squares with Distinguished Diagonal. Matematičeskie zametki, Tome 88 (2010) no. 2, pp. 261-274. http://geodesic.mathdoc.fr/item/MZM_2010_88_2_a8/