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Algebraic semantics for coalgebraic logics. (English) Zbl 1271.03031
Adámek, J. (ed.) et al., Proceedings of the 7th workshop on coalgebraic methods in computer science (CMCS), Barcelona, Spain, March 27–29, 2004. Amsterdam: Elsevier. Electronic Notes in Theoretical Computer Science 106, 219-241 (2004).
Summary: 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 $$\mathcal{L}$$ 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 $$\mathcal{L}$$. This provides a criterion for proving concretely given logics to be sound, complete and expressive.
For the entire collection see [Zbl 1271.68014].

##### MSC:
 03B45 Modal logic (including the logic of norms) 03G30 Categorical logic, topoi 18C50 Categorical semantics of formal languages
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