In [2], we discuss the incremental construction of the Lindenbaum algebra for a modal logic, which is the algebraic incarnation of its universal frame. Our approach there is based on the discrete duality between finite Boolean algebras with operators and finite Kripke frames, and builds on recent work of Gehrke and Bezhanishvili [1] and Ghilardi [3], who used this construction for intuitionistic propostional logic and the modal logic S4. We will discuss the possibility of extending their methods to other modal logics.
[1] Nick Bezhanishvili and Mai Gehrke. Finitely generated free Heyting algebras via Birkhoff duality and coalgebra. To appear in
Logical Methods in Computer Science, 2011. Available online at http://www.math.ru.nl/~mgehrke/
[2] Dion Coumans and Sam van Gool. Constructing the Lindenbaum algebra for a logic step-by-step using duality. To appear in the
proceedings of PhDs in Logic III, 2011. Available online at http://www.math.ru.nl/~vangool/
[3] Silvio Ghilardi. Continuity, Freeness, and Filtrations. Rapporto Interno 331-10, Dipartimento di Scienze dell’Informazione,
Universita` degli Studi di Milano, 2010. Available online at http://homes.dsi.unimi.it/~ghilardi/allegati/tr-mod-10.pdf