With coalgebras usually being defined in terms of an endofunctor T on sets, this paper shows that modal logics for T-coalgebras can be naturally described as functors L on boolean algebras. Building on this idea, we study soundness, completeness and expressiveness of coalgebraic logics from the perspective of duality theory. That is, given a logic L for coalgebras of an endofunctor T, we construct an endofunctor L such that L-algebras provide a sound and complete (algebraic) semantics of the logic. We show that if L is dual to T, then soundness and completeness of the algebraic semantics immediately yield the corresponding property of the coalgebraic semantics. We conclude by characterising duality between L and T in terms of the axioms of L. This provides a criterion for proving concretely given logics to be sound, complete and expressive.
C. Kupke, A. Kurz, and D. Pattinson, “Algebraic Semantics for Coalgebraic Logics,” Electronic Notes in Theoretical Computer Science, vol. 106, pp. 219–241, Dec. 2004. DOI: 10.1016/j.entcs.2004.02.037
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