Abstract
For any set-endofunctor \(T : {\mathcal S}et \rightarrow {\mathcal S}et\) there exists a largest sub-cartesian transformation μ to the filter functor \({\mathbb F}: {\mathcal S}et \rightarrow {\mathcal S}et\). Thus we can associate with every T-coalgebra A a certain filter-coalgebra \(A_{\mathbb F}\).
Precisely, when T (weakly) preserves preimages, μ is natural, and when T (weakly) preserves intersections, μ factors through the covariant powerset functor \({\mathbb P}\), thus providing for every T-coalgebra A a Kripke structure \(A_{\mathbb P}\).
We characterize preservation of preimages, preservation of intersections, and preservation of both preimages and intersections via the existence of natural, sub-cartesian or cartesian transformations from T to either \({\mathbb F}\) or \({\mathbb P}\).
Moreover, we define for arbitrary T-coalgebras \({\mathcal A}\) a next-time operator \(\bigcirc_{\mathcal A}\) with associated modal operators □ and \(\lozenge\) and relate their properties to weak limit preservation properties of T. In particular, for any T-coalgebra \({\mathcal A}\) there is a transition system \({\mathcal K}\) with \(\bigcirc_{A} = \bigcirc_{K}\) if and only if T preserves intersections.
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© 2005 Springer-Verlag Berlin Heidelberg
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Gumm, H.P. (2005). From T-Coalgebras to Filter Structures and Transition Systems. In: Fiadeiro, J.L., Harman, N., Roggenbach, M., Rutten, J. (eds) Algebra and Coalgebra in Computer Science. CALCO 2005. Lecture Notes in Computer Science, vol 3629. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11548133_13
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DOI: https://doi.org/10.1007/11548133_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28620-2
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