Summary: The operation of sum of a family \((\mathsf{F}_i \mid i \text{ in } \mathsf{I})\) of Kripke frames indexed by elements of another frame I provides a natural way to construct expressive polymodal logics with good semantic and algorithmic properties. This operation has had several important applications over the last decade: it was used by L. Beklemishev in the context of polymodal provability logic; two ways of combining modal logics, the refinement of modal logics introduced by S. Babenyshev and V. Rybakov [Log. J. IGPL 18, No. 6, 823–836 (2010; Zbl 1216.03035)], and the lexicographic product of modal logics proposed by P. Balbiani [Lect. Notes Comput. Sci. 5749, 165–180 (2009; Zbl 1193.03035)], can be defined in terms of sums of frames. This paper provides some general truth-preserving tools for operating with sums of Kripke frames, and then applies them to study properties of resulting modal logics, in particular, to investigate the finite model property.
For the entire collection see [Zbl 1398.03005].