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From T-Coalgebras to Filter Structures and Transition Systems

  • H. Peter Gumm
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3629)

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.

Keywords

Transition System Natural Transformation Kripke Structure Filter Structure Preservation Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • H. Peter Gumm
    • 1
  1. 1.Philipps-Universität MarburgMarburgGermany

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