A Coalgebraic Semantics for Fischer Servi Logic
Sarah M. Dukic

Abstract:
We present a new coalgebraic semantics for the intuitionistic modal logic known as IK or Fischer Servi logic, providing representations both for its modal spaces and for its image-finite Kripke frames. Our work is based on a recent construction by Almeida [Alm24], which has made coalgebraic analysis of intuitionistic modal logics possible. In particular, it provides a functorial method of turning coalgebras for a positive modal logic into coalgebras for its least intuitionistic extension, as shown by Almeida and Bezhanishvili [AB24]. This does not suffice on its own to treat Fischer Servi logic, as it is not the least intuitionistic extension of a positive modal logic. Thus, we fill this gap in the research by providing a modified approach, which yields coalgebraic completeness for IK. As an application of these results, we study bisimulations for Fischer Servi logic, describe the dual spaces of free IK algebras, and show how our approach can be used to capture extensions of IK with rank-1 axioms.